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Study Guide
Algebra Math-a-thon
Interpret Box-and-Whisker Plots
Box-and-whisker plots are graphs that are very helpful in interpreting the distribution of data. A box-andwhisker plot is created using five specific numbers from a set of data: the largest number, the smallest number,
the median, the first quartile, and the third quartile. These five numbers are called the five number summary of
the data.
Definitions:
• The median of a data set is the number that is in the middle of the data set (it separates the data into
two equal-sized groups). The median can be found by arranging the numbers in numerical order and
finding the middle term.
• The quartiles break the data into four equal-sized groups.
• The first quartile is the median of the lower group of data.
• The second quartile is the median of the entire set of data.
• The third quartile is the median of the upper group of data.
• The interquartile range is the range between the third quartile (Q3) and the first quartile (Q1). It is
found using the formula Q3 - Q1.
The box-and-whisker plot below shows how each number is used in the plot.
The values of the five number summary for the box-and-whisker plot above are as follows.
smallest number = 34
first quartile = 45
median = 54
third quartile = 70
largest number = 78
interquartile range = 70 - 45 = 25
Box-and-whisker plots can also provide percentages. Since the quartiles break the data into four pieces
of equal size, each quartile represents 25% of the data. From the smallest number to the first quartile,
25% of the data is represented. From the smallest number to the median (second quartile), 50% of the
data is represented. From the smallest number to the third quartile, 75% of the data is represented. The
plot below shows this breakdown.
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Once the student understands the numbers and percentages that are represented by box-and-whisker
plots, he or she can start to answer questions about plots that illustrate real data.
Example 1: The attendance rate for Concerts in the Park this summer is represented in the box-andwhisker plot below. What is the first quartile for this data?
Answer: 240 people
Example 2: The attendance rate for Concerts in the Park this summer is represented in the box-andwhisker plot below. What is the interquartile range for this data?
(1) third quartile = 290
(2) first quartile = 240
(3) 290 - 240 = 50
Step 1: Determine the third quartile for the data set.
Step 2: Determine the first quartile for the data set.
Step 3: Subtract the first quartile from the third quartile.
Answer: 50
Example 3: The attendance rate for Concerts in the Park this summer is represented in the box-andwhisker plot below. What percentage of the concerts had attendance of 290 or higher?
Solution: The question is asking for an attendance rate of 290 or higher. Since 290 is the third quartile,
the percentage of the plot that is 290 or higher is 25%.
Answer: 25%
Interpret Stem-and-Leaf Plots
A stem-and-leaf plot summarizes the shape of a set of data (the distribution) and provides extra detail regarding
individual values. The data is arranged by place value. The digits with the largest place value are referred to as
the stem (stems) and the digits with the smallest place value are referred to as the leaf (leaves). The leaves are
always displayed to the right of the stem. A stem can contain one or more digits, but a leaf can only be a onedigit number. Stem-and-leaf plots are generally used for organizing large amounts of information. An example
of a stem-and-leaf plot is below.
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The stems of a stem-and-leaf plot are always listed in numerical order from smallest at the top to largest
at the bottom. The leaves of the stem-and-leaf plot are always listed in numerical order from left to
right. The stem is read with each leaf. In this case, the stems are the tens place and the leaves are the
ones place, so 3 | 1 4 9 represents 31, 34, and 39.
Example 1: What is the maximum value of the data set represented in the stem-and-leaf plot?
Solution: The largest stem is 10 and the largest leaf for that stem is 0, so the largest number represented
by the plot is 100.
Answer: 100
Example 2: The stem-and-leaf plot below shows the test scores from students in Mr. Nguyen's class. In
which of the following ranges did most students score?
A. between 50 and 59
B. between 60 and 69
C. between70 and 79
D. between 80 and 89
Solution:
A. is not the correct answer because only 2 students scored between 50 and 59.
B. is not the correct answer because only 2 students scored between 60 and 69.
C. is not the correct answer because only 4 students scored between70 and 79.
D. is the correct answer because 5 students scored between 80 and 89. The next highest range only had
4 students score in it.
Answer: D.
Example 3: The stem-and-leaf plot shows numbers of miles driven while on a vacation by a group of
families. How many families drove 235 miles while on vacation?
Solution: In this stem-and-leaf plot, the stems represent the hundreds and tens places and the leaves
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represent the ones place. To figure out how many families drove 235 miles while on vacation, look at
the stem of 23 and count the number of 5s listed in the corresponding leaves column. There are 3 leaves
that fit these two conditions, so 3 families drove 235 miles while on vacation.
Answer: 3 families
Example 4: The stem-and-leaf plot shows numbers of miles driven while on a vacation by a group of
families. How many families drove more than 282 miles while on vacation?
Solution: Look at the stem of 28. There are a total of four numbers represented, but only two of them
would be greater than 282. The stem of 29 has another four numbers that would be greater than 282 and
the stem of 30 has five numbers that would be greater than 282. So, 2 + 4 + 5 = 11 families drove more
than 282 miles while on vacation.
Answer: 11 families
An activity to reinforce this skill is to create a stem-and-leaf plot with the student. Gather data about the
ages of family members (from siblings to grandparents) and use the data to create a stem-and-leaf plot
together. Then ask the student questions about the stem-and-leaf plot such as the ones below.
• How many people in our family are older than 30?
• Which age occurs the most/least?
Direct Variation
Variation equations are formulas that show how one quantity changes in relation to one or more other
quantities. There are four types of variation: direct, indirect (or inverse), joint, and combined.
Direct variation equations show a relationship between two quantities such that when one quantity increases, the
other also increases, and when one quantity decreases, the other also decreases. We can say that y varies
directly as x, or y is proportional to x. Direct variation formulas are of the form y = kx, where the number
represented by k does not change and is called a constant of variation.
Indirect variation equations are of the form y = k/x and show a relationship between two quantities such that
when one quantity increases, the other decreases, and vice versa.
This skill focuses on direct variation. The following is an example of a direct variation problem.
The amount of money in a paycheck, P, varies directly as the number of hours, h, that are worked. In
this case, the constant k is the hourly wage, and the formula is written P = kh. If the equation is solved
for k, the resulting equation shows that P and h are proportional to each other.
Therefore, when two variables show a direct variation relationship, they are proportional to
each other. Direct variation problems can be solved by setting up a proportion in the form below.
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Example 1: The amount of fuel needed to run a textile machine varies directly as the number of hours
the machine is running. If the machine required 8 gallons of fuel to run for 24 hours, how many gallons
of fuel were needed to run the machine for 72 hours? Round your answer to the nearest tenth of a
gallon, if necessary.
Step 1: Set up the proportion. Since the machine used 8 gallons of fuel in 24 hours, the left side of the
proportion should be 8 gallons over 24 hours. The number of gallons that the machine used in 72 hours
needs to be found, so the right side of the proportion should be g gallons over 72 hours.
Step 2: Cross-multiply across the equal sign.
Step 3: Set up the cross-multiplication equation.
Step 4: Divide both sides of the equation by 24 hours to isolate g gallons.
Step 5: Reduce the fractions on both sides of the equal sign.
Step 6: Simplify by multiplying the numbers remaining on the right side of the equal sign (8 gallons
3).
Answer: 24 gallons
Example 2: The price of jellybeans, j, varies directly as the number of pounds, p, that are purchased.
Find the equation that relates the two variables if jellybeans are $1.95 per pound.
(1) y = kx, j = kp
(2) j = 1.95p
Step 1: Remember that the formula for direct variation is: y = kx and substitute the variables from the
question into the appropriate places.
Step 2: Since the jellybeans are always $1.95 per pound, the constant, k, equals 1.95. Substitute 1.95
into the equation for k.
Answer: j = 1.95p
Activities that can help reinforce the concept of direct variation are as follows.
1. Have students solve the equation y = kx for k, and then substitute two sets of (x, y) values into
the equation and compare the values for k. If they are the same, then x and y have a direct variation
relationship.
2. Have the student think of scenarios that show a direct variation relationship. Then, make up
numbers to go with the relationships and have the students practice solving them.
Domain/Range
A relation is a collection of (x, y) coordinate points. For example:
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The domain of the relation is the set of x-values in the collection of coordinate points that make up the
relation, and the range is the set of y-values. The domain and range of a relation are written as sets of
numbers in ascending order within brackets and each number is only written once. For example:
This skill focuses on giving students practice determining the domain and
range of relations.
Example 1: Find the domain of the following relation r.
Solution: The domain is the set of all x-values of the coordinate points that make up the relation.
Answer:
Example 2: Find the range of the following relation r.
Solution: The range is the set of all y-values of the coordinate points that make up the relation. NOTE:
Although the - 4 is used as a y-value twice, it only needs to be written once in the range.
Answer:
An activity that can help reinforce the concept of domain and range is to show students a collection of
coordinate points and ask them to name the member of the domain of each coordinate point (x-value)
and the member of the range of each coordinate point (y-value).
Exponential Functions
An exponential function is an equation that has a variable in the exponent. The functions below are all
exponential functions, where f(x) represents the letter y and is dependent on the value of x.
When x-values are substituted into the equation and the right side of the equal sign is simplified, a list of
f(x) (or y) values is obtained. This process gives you a set of (x, y) coordinate pairs that can be plotted
(or graphed) on a coordinate plane.
Step 1: Make a table of values with the following column headings.
Step 2: Choose values to enter into the x column (remember to
choose the smallest possible values to keep computations as simple as possible). It is generally
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necessary for the values to contain both negative and positive numbers and the number zero. As few as
three values could be used, but this example will use these five values: -2, -1, 0, 1, 2. Enter the values
into the table.
Step 3: Substitute each x-value into the equation in the center column for x and simplify. This will
determine the y-values for the coordinate points.
Step 4: Write the (x, y) pairs generated in Step 3 into the (x, y)
column, using the values from the x column and the y-values from the center column.
Step 5: Plot the points on a coordinate plane to graph the
exponential function.
Step 6: Compare this graph to the solution choices and choose the correct
answer, which is choice D.
Answer: D.
Step 1: Make a table of values and choose x-values to substitute into the equation.
Step 2: Substitute the values in the x column into the equation in
the center column to determine the corresponding y-values. Write the coordinate points in the (x, y)
column.
Step 3: Plot the points on a coordinate plane to graph the
exponential function.
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Step 4: Compare this graph to the curves in the question and choose the
correct answer.
Answer: W
Comparing Graphs of Exponential Functions:
The standard form for an exponential function is shown below.
For purposes of comparing graphs of exponential functions, two more
variables need to be added, such that the standard form becomes the form shown below.
The value of c determines whether the graph shifts upward or downward
and the value of d determines whether the graph shifts right or left. See the table below.
Example 3:
Step 1:
Step 2: Look at the table above to determine that the graph of the exponential
function should move upward 3 units and to the right 4 units.
Step 3: Compare the actual graphs below to see if these predictions were right.
An activity that will help reinforce the skill of graphing exponential functions is to explore exponential
functions with the student using a graphing calculator. Make up a list of exponential functions like the
one below.
Follow the steps to graph each function using a graphing calculator. Notice the way changing a sign or
adding a term alters the graph.
Step 1: Follow the instructions in your graphing calculator manual to set the viewing window to the
following values: x min = -10, x max = 10, xscl = 1; y min = -10, y max = 10, y scl = 1. Exit the
window.
Step 2: Press the y = key to open the equation editor. Delete any equations that are listed. Enter the
following keystrokes for the first exponential equation on the list: 5^x. Press Enter.
Step 3: Press the Graph key to display the graph of the function.
**NOTE: This activity could be completed using graph paper and tables of values to graph the
exponential functions.
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Add Matrices
A matrix is an array of numbers arranged in rows and columns. Rows are horizontal and columns are
vertical. An entry is a number in the matrix.
The plural of matrix is "matrices." Two matrices can be added together, if they have the same number
of rows and the same number of columns. Addition of matrices is done by adding corresponding
entries, or entries that are in the same position within their respective matrix.
A matrix with 2 rows and 3 columns is identified as a 2
3 matrix, and a matrix with 3 rows and 2
columns is a 3
2 matrix. The number of rows is indicated first, and the number of columns is
indicated second.
Example 1: Add the following 2
2 matrices.
Step 1: Rewrite the problem.
Step 2: Add corresponding entries.
Step 3: Simplify.
Answer:
Example 2: Add the following matrices.
Step 1: Rewrite the problem.
Step 2: Add corresponding entries.
Step 3: Simplify.
Answer:
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An activity that can help reinforce the concept of matrices is to ask the value of the entry in a given row
and column in a matrix. After the student is able to identify entries, ask him or her to add the
corresponding entries.
Exponential Notation - F
In the expression 32 , the number 2 is called an exponent and the number 3 is called a base. The exponent
determines the number of times the base is multiplied by itself. For example: 32 = (3)(3) = 9.
Rational exponents are exponents that are in fraction form. The following are examples of expressions
with rational exponents.
Expressions with rational exponents can be written in radical form. The numerator is the number of
times the base, which is placed under the radical sign, is multiplied by itself. The denominator is the root
of the radical expression. A root is the inverse of an exponent. The examples below show how
expressions with rational exponents are written in radical form.
A root of 2 is called a square root and is the inverse of the exponent 2. A root of 3 is called a cube root
and is the inverse of the exponent 3. The following examples show how to simplify expressions with
rational exponents in two ways.
Simplifying by converting to radical form:
Example 1: Simplify.
Step 1: Rewrite the rational expression in radical form.
Step 2: Simplify: 8 to the first power equals 8.
Step 3: Factor 8 into (2)(2)(2).
Step 4: Since there are three factors of 2 under the cube root, the radical expression will simplify to 2.
Answer: 2
Simplifying using exponent laws:
Step 1: Rewrite the expression and factor 8 into (2)(2)(2).
Step 2: Write (2)(2)(2) in exponential form.
Step 3: Multiply 3 and 1/3 (remember that when an exponent is taken to another power, the two powers
are multiplied) to get 1. Simplify 2 to the first power to 2.
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Answer: 2
Example 2: Simplify.
Step 1: Rewrite the expression as a radical expression.
Step 2: Simplify 322 : 32
32 = 1,024.
Step 3: Factor 1,024 into (4)(4)(4)(4)(4). Always try to factor the number under the radical into the
same number of factors as the root.
Step 4: Since this is a 5th root and there are five 4s under the radical, the radical expression will simplify
to 4.
Answer: 4
Activities that can help reinforce the concept of rational exponents are as follows.
1. Take 30 index cards, and make three piles of ten cards each. Write expressions with rational
exponents on the first set of ten cards. On the second set, write the radical form for each expression you
wrote on the first ten cards. On the third set, write the simplified answer to the ten expressions. Shuffle
all of the cards and lay them face down on a table. Each player takes turns flipping three cards. If all
three cards go together, the player has made a match and gets to try again. If not, it is the next player's
turn.
2. Name expressions with rational roots, and ask the student to factor the base into the same number of
identical factors as the root. For example, if the root is 1/6, the student should factor the base into 6
identical factors, so that when the factors are rewritten in exponential form, and the exponent is
multiplied by the root, the product is 1.
Equations: Order of Operations
An equation is a statement in which two numbers or two expressions are set equal to each other. For example,
5 + 3 = 8 and 16 = 3c + 4 are equations.
When solving equations, find the value of the variable by getting the variable alone on one side of the
equal sign. To do this, undo any operations on the variable by using the inverse operation. Any operation
done on one side of the equal sign must be done on the other side of the equal sign in order to keep the
statement true.
If a number has been added to the variable, subtract the number from both sides of the equation.
m + 3 = 5
- 3 - 3
m = 2
If a number has been subtracted from the variable, add the number to both sides of the equation.
b - 7 = 9
+ 7 + 7
b = 16
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If a variable has been multiplied by a nonzero number, divide both sides by the number.
If a variable has been divided by a number, multiply both sides by the number.
When solving 2-step equations, we must first undo the addition or subtraction using the inverse
operation, then undo the multiplication or division:
Example 1: Solve the equation for t.
5(t - 4) = t + 12
Step 1: Multiply 5 times the terms inside the parenthesis.
Step 2: Add 20 to both sides of the equation
Step 3: Subtract t from both sides of the equation
Step 4: Divide both sides of the equation by 4.
Answer: t = 8
Example 2: Evaluate the expression for c = 3:
2(c + 4) + 2(15)
(1) 2(3 + 4) + 2(15)
(2) 2(7) + 2(15)
(3) 14 + 30
(4) 44
Step 1: Substitute 3 in place of 'c' in the expression.
Step 2: Add the numbers in parentheses.
Step 3: Rewrite the equation after performing all multiplications in order from left to right.
Step 4: Add 14 and 30 to get 44.
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Answer: 44
Graph Absolute Value
The absolute value of a real number is the distance the real number, x, is from 0 on a number line. The absolute
value of a real number is denoted by placing the real number within two vertical lines: |x|. In other words, |- 5|
(read: the absolute value of - 5) is 5 because - 5 is 5 units from 0 on a number line.
There are two major principles of absolute value:
1. The absolute value of a negative or positive number is always a positive number.
2. The absolute value of 0 is 0.
Graphs of Absolute Value Functions:
The standard form for an absolute value function (y = a|bx + c| + d) is needed to compare graphs of
absolute value functions with the graph of y = |x|.
• a determines whether the graph opens up or opens down
• b determines how wide open the graph is
• c determines whether the graph shifts to the right or the left
• d determines whether the graph shifts upward or downward
When determining the shifts, it is helpful to use the vertex of the graph as the point of reference. In this
skill, the absolute value equations are given in the form where a is equal to 1 or - 1 and d changes. The
variables b and c do not affect the equations in this skill.
When comparing graphs of absolute value functions, start with the graph of y = |x|, which is shown
below.
Then, determine how the given equation alters this graph. When a is positive, the graph opens up, and
when a is negative, the graph opens down.
When d is positive, the graph has a vertical shift up d units, and when d is negative, the graph has a
vertical shift down d units.
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Example 1:
Determine the equation that represents the graph below.
Solution:
The graph opens down, so the equation must have a negative value of a. It is shifted up five units on the
y-axis, so d must equal 5. Therefore, the equation that represents the graph is y = - |x| + 5.
Answer: y = - |x| + 5
Example 2:
Choose the correct graph for the equation below.
y = |x| - 3
Solution:
The value of a is 1, so the graph opens up, which is true only in graphs C and D. The value of d is -3, so
the graph is shifted down three units on the y-axis. Therefore, the correct answer is D.
Answer: D
A possible activity to reinforce this skill is to have the student create their own graphs on graph paper.
Have the student make a T-chart (or x - y chart) of values for different equations. Then have the student
plot the points on the graph paper to graph the absolute value equation. As an enrichment activity,
provide him or her with equations that contain different values of b and c in the equation as discussed
earlier in this tutorial. For example, the student could graph the equations y = |2x| and y = |x - 3| and
determine the effects they have on the graph of y = |x|.
Classifying Triangles - B
Triangles can be classified using their side lengths and angle measures, as shown in the diagrams below.
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• An obtuse triangle contains one obtuse angle. An
obtuse angle is any angle greater than 90º . Because the sum of the angles in a triangle is equal to 180º ,
there is never more than one obtuse angle in a triangle.
• An acute triangle contains three acute angles. An acute angle is any angle less than 90º .
• An equiangular triangle contains three angles with the same measure. That is, all of the angles are 60º .
An equiangular triangle is also an acute triangle.
• A right triangle contains one right angle. Right angles measure 90º . Because the sum of the angles in
a triangle is equal to 180º , there is never more than one right angle in a triangle
• A scalene triangle contains three sides of different lengths.
• An isosceles triangle contains two sides of the same length. All isosceles triangles also have two equal
angles.
• An equilateral triangle contains three sides of the same length. All equilateral triangles are also
equiangular.
Example 1:
Classify the following triangle.
Solution:
The triangle has three sides of equal length, and therefore it is an equilateral triangle. All equilateral
triangles have angle measures of 60º , 60º , and 60º . Therefore, this triangle can also be classified as
equiangular or acute.
Answer: The triangle is equilateral, equiangular, and acute.
Example 2:
Classify the following triangle.
Solution:
The triangle contains a 90º angle, and therefore it is a right triangle. The remaining two angles are
equivalent so the triangle is an isosceles triangle as well.
Answer: The triangle is a right isosceles triangle.
A possible activity to help reinforce this skill could be to set up a memory game. Draw a variety of the
different types of triangles on index cards, making sure to draw pairs of identical triangles. Spread the
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cards out, face down, on a table. Each player takes a turn flipping over two cards, announcing the type
of triangle on each of the cards he or she has turned over. If the two triangles have the same description,
the player who chose the two cards keeps those cards. Once all of the pairs have been taken, the player
with the most cards wins.
Angles and Bisectors
Before working with angles and bisectors, it is important to understand several basic definitions and concepts.
Bisectors:
•An angle bisector is a line, segment, or ray that divides an angle into two congruent adjacent angles. In
other words, it cuts the angle into two equal angles.
•A perpendicular bisector is any line, segment, or ray that forms a 90º angle to a segment at its midpoint.
In other words, it cuts the segment in half and forms right angles at the point of intersection.
Angles:
• A straight line is an angle that contains 180º .
• A circle contains 360º .
• Interior angles are the corner angles or inside angles of a polygon. Examples of interior angles follow:
• When the sides of a polygon are extended, the Exterior angles
are adjacent to the interior angles. Examples of exterior angles follow:
• The sum of the angles in a triangle is 180º .
• The sum of the interior angles of any polygon can be found by taking the number of sides of the
polygon, subtracting two, and multiplying that result by 180º : 180(n - 2), where n = the number of sides
in the polygon.
• To calculate the measure of the exterior angles of a regular polygon, simply divide 360º by the number
of sides. The formula for finding the measure of each of the exterior angles in a regular polygon is:
Example 1:
What is the measure of any exterior angle of a regular dodecagon (twelve-sided figure)?
Step 1: Use the equation for finding the measure of an exterior angle of a regular polygon.
Step 2: Substitute the value of n into the equation and simplify. The value of n for the dodecagon is
twelve, because it has twelve sides.
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Answer: 30º
Example 2:
Step 1: Recall that the sum of the angles of a straight line is 180º .
Step 2: Substitute the known values of m MRN and m
subtracting the sum from both sides of the equation.
Step 3: By definition of an angle bisector, m
half of the final angle measure from Step 2.
Answer: m
PRQ. Simplify by adding 26 and 9 and
NRO and m
ORP are equal. Therefore, each must be
ORP = 72.5º
Example 3:
Step 1: Recall that the sum of the angles in a triangle is 180º . Therefore, m
subtracting the other two angles from 180º .
Step 2: Because ray BD bisects
the measure of
Answer: m
ABC. Divide m
ABC, m
DBA = m
ABC can be found by
DBC, and the measure of each angle is half
ABC by 2 or multiply by 1/2.
DBA = 70º
Several of the questions in this skill relate the concepts of bisectors and angles to real world problems.
An example is given below.
Example 4:
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The hour hand on a clock travels the same distance each hour. How many degrees does the hour hand
travel each hour?
Solution:
360º ÷ 12 = 30º
Recall that the total number of degrees in a circle is 360º . Since the hour hand of the clock travels the
entire 360º distance in 12 hours, divide 360º by 12 to determine the number of degrees the hour hand
travels in one hour.
Answer: The hour hand travels 30º each hour.
Angles - D
This study guide shows the student how to define and apply the characteristics of alternate interior angles,
alternate exterior angles, corresponding angles, and vertical angles. Learning to recognize these types of angles
is helpful when trying to prove that two angles are congruent or that two lines are parallel.
Important Definitions:
Parallel lines: Two lines in the same plane are parallel when, if extended in both directions forever, they
will never intersect. Some examples of parallel lines are railroad tracks, columns in a building, and rows
in a field. The symbol for parallel is ||.
Transversal: A line that cuts through a set of lines (that are usually parallel).
Notice the transversal intersects each line in exactly one point.
Congruent: Objects that have exactly the same size and shape. Synonymous with equal.
Alternate interior angles: When two lines are cut by a transversal, the inner angles on opposite sides of
the transversal are called alternate interior angles. When parallel lines are cut by a transversal, alternate
interior angles are congruent. In the figure below, line a is parallel to line b (a || b) and line c is parallel
to line d (c || d).
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Alternate exterior angles: When two lines are cut by a transversal, the angles touching the outer side of
each line on opposite sides of the transversal are called alternate exterior angles. When the parallel lines
are cut by a transversal, alternate exterior angles are congruent. In the figure below, a || b and c || d.
Corresponding angles: When two lines are cut by a transversal, the angles on the same side of the
transversal are corresponding angles. When parallel lines are cut by a transversal, corresponding angles
are congruent. In the figure below, a || b and c || d.
Vertical angles: When two lines intersect, the angles that are opposite one another (diagonal) are called
vertical angles. Vertical angles are always congruent.
Example 1:
Step 1: Determine if
2 and
6 fit any of the special types of angles. Since r || s,
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2 and
6 are
corresponding angles.
Step 2: Since corresponding angles are congruent when parallel lines are cut by a transversal, m
m
2. Therefore, m
Answer: m
6=
6 = 121º .
6 = 121 º
Example 2:
Step 1: Determine if the given angles fit any of the special types of angles and pick out the information
that is needed (example the m
8 is not needed).
Step 2: Since g and h are parallel,
5 and
11 are alternate exterior angles.
Step 3: Since alternate exterior angles are congruent when parallel lines are cut by a transversal, m
=m
5. Therefore, m
Answer: m
11
11 = 112 º .
11 = 112 º
Example 3: Madison plays soccer on a rectangular field. During practice she saved the ball from going
out of bounds by kicking it across the field. The ball traveled in a diagonal path from sideline to sideline
forming a 63º angle. If no one is able to retrieve the ball on the other side, what would be the angle
measure where it would exit the field (m
s)?
Step 1: Recognize that rectangles have parallel sides, and the diagonal created a transversal. This
makes the angles in question alternate interior angles.
Step 2: Since alternate interior angles are congruent when parallel lines are cut by a transversal, the
angles in question are equal. Therefore m
Answer: m
s = 63º .
s = 63º .
Example 4:
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Step 1: Recognize that
1 and
14 do not fit any of the special angles.
Step 2: Recall that supplementary angles have a sum of 180º .
1 and
2 are supplementary angles
because together they make a straight angle, which measures 180º . To find the measure of
subtract: 180º - m
1=m
2. m
Step 3: Since g and h are parallel,
2 = 180º - 59º = 121º
2 and
14 are corresponding angles. Corresponding angles are
congruent when parallel lines are cut by a transversal, so m
Answer: m
2,
2=m
14 = 121º .
14 = 121º
An active way to explore the angles created by parallel lines that are cut by a transversal is laid out as
follows. First, draw several sets of parallel lines on a piece of paper and include a transversal with each
set. Next, trace some of the angles formed by this transversal onto a piece of tracing or wax paper.
Have the student slide the tracing/wax paper around the diagram and compare the sets of angles by
placing a traced angle over another angle in the diagram. Are they congruent? Have the student identify
the types of angles he or she sees.
Interior Angles: Polygons - A
This study guide will show students how to identify the interior angles of polygons. It will also show students
how to find the measure of an interior angle of a regular polygon.
A polygon is a closed figure made up of at least three line segments. The number of sides of a polygon
is equal to its number of interior angles.
These are polygons:
These are not polygons:
Distinguishing Between Interior Angles and Other Angles:
Interior angles are the angles inside or the corners of the polygon.
These are interior angles:
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These are not interior angles:
Example 1:Name all possible interior angles in the figure below.
Step 1: Recall that interior angles are inside the polygon. These angles include 3, 8, 9, and 15.
Answer:
3,
8,
9, and
15
Example 2:Which figure shows an interior angle?
Step 1: Look for the figure that shows an angle inside the polygon (this eliminates figures b and c).
Step 2: Interior angles are in the corners of the polygon so that eliminates figure d.
Step 3: By process of elimination, we are left with figure a. Verify that figure a fits the definition of an
interior angle.
Answer: figure a
Determining the Measure of an Interior Angle of a Regular Polygon:
A formula is needed to find the measure of an interior angle.
Simply substitute the number of sides of the polygon into the formula and evaluate.
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A regular polygon has sides of the same length and has the same interior angle measures. Students
should be familiar with the following regular polygons:
Triangle: a polygon with 3 sides
Square: a polygon with 4 sides
Pentagon: a polygon with 5 sides
Hexagon: a polygon with 6 sides
Heptagon: a polygon with 7 sides
Octagon: a polygon with 8 sides
Nonagon: a polygon with 9 sides
Decagon: a polygon with 10 sides
Hendecagon (also an undecagon): a polygon with 11 sides
Dodecagon: a polygon with 12 sides
Example 3: Find the interior angle measure of a regular hendecagon. Round your answer to the nearest
degree.
Step 1: Recognize that a hendecagon has 11 sides and replace n with 11 in the formula.
Step 2: Simplify using the order of operations, so parentheses are first, (11 - 2 = 9).
Step 3: Multiply 180º by 9. Divide 1,620 by 11 to get 147.2727273º . Finally, round 147.2727273 to
the nearest degree to get 147º
Answer: 147º
A fun activity to reinforce these concepts is laid out as follows. Have the student draw or trace various
regular polygons. Have the student measure the interior angles of these polygons with a protractor and
record their measures. Have the student verify his or her data by using the formula given above. If data
and the measurements match exactly, the student has calculated correctly.
Congruence (AAS/ASA/SAS) - B
Engineers and people in the construction field use triangle congruency on a daily basis to make sure that things
like rafters are congruent so the roof of a house does not sink in. This study guide will introduce students to
postulates, which are used to determine triangle congruency. A postulate is a statement which is taken to be
true without proof.
Review of Triangle Congruency Symbols
Tick marks are used to show that sides are congruent. All sides in a diagram that are marked with the
same number of tick marks are congruent.
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Arcs are used to show that angles are congruent. All angles in a diagram that are marked with the same
number of arcs are congruent.
Angles in a triangle are named by their vertex, the point at which two line segments meet. They are also
named with the vertex in the middle.
Postulates for Congruency:
ASA (Angle, Side, Angle) Postulate: If two angles and the included side (the side between the two
angles) of one triangle are congruent to two angles and the included side of another triangle, then the
triangles are congruent.
SAS (Side, Angle, Side) Postulate: If two sides and an included angle of one triangle are congruent to
two sides and an included angle of another triangle, then the triangles are congruent.
AAS (Angle, Angle, Side) Postulate: If two angles and a non-included side of one triangle are
congruent to two angles and a non-included sides of another triangle, then the triangles are congruent.
Determining that Triangles are Congruent:
Example 1: What is the measure of
YXZ?
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Step 1: Write out all the information given in the diagram.
Step 2: Determine if the triangles are congruent and, if congruent, by what postulate? These triangles
are congruent by the SAS postulate.
Step 3: Since the triangles are congruent, m
TUV equals m
XYZ. m
XYZ = 34º .
Step 4: Remember that the sum of the 3 angles of a triangle is 180º . To determine m
180 - 57 - 34 = 89.
YXZ, subtract
Answer: 89º
Example 2: Using only the information presented in the diagram, determine if the following triangles
are congruent and state which congruence theorem was used.
Step 1: State the given information as shown in the diagram.
Step 2: The triangles are congruent by the AAS postulate.
Answer: The triangles are congruent by the AAS postulate.
Example 3: Using only the information below, and your knowledge of triangle congruence postulates,
which pair of triangles are congruent?
A is not correct. The angles of one triangle are congruent to the angles of another. This proves
similarity, not congruence.
B is correct. Two angles and a non-included side of one triangle are congruent to two angles and a nonincluded side of another triangle. The triangles are congruent by the AAS postulate.
C is not correct. SSA is not true for every triangle, that is why it is not a postulate.
D is not correct. There is not enough information to prove congruency.
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Example 4:
A.
B.
C.
D.
2 and 4
1 only
1, 3, and 4
4 only
A is not correct. Although SAS does prove that the triangles are congruent, SSS cannot be proven using
only what is presented in the drawing.
B is not correct. While AAS does prove that the triangles are congruent, SAS does too, and it is not part
of option B.
C is correct. AAS, ASA, and SAS will all prove that the two triangles are congruent using only what is
presented in the drawing.
D is not correct. SAS is not the only postulate that will prove these triangles congruent using only what
is presented in the drawing.
Answer: C
Example 5: Jacob is constructing rafters for a house. The middle piece of wood stands 4 feet tall and is
perpendicular to the base. He attaches two pieces of wood at the top of the middle piece which create a
68º angle with the base on either side. Using only the information presented, which postulate would be
used to prove that the sides of the rafters are equal in length?
A) ASA B) SSS C) AAS D) SAS
A) is not correct. The side that is congruent is not included between the congruent angles.
B) is not correct. Only one side is marked congruent.
C) is correct. The base angles are congruent, the right angles are congruent, and the middle piece of
wood is congruent to itself. The triangles are congruent by the AAS postulate.
D) is not correct. Only one side is marked congruent.
The following is a fun activity to help reinforce the concept of triangle congruency. Take a walk with
the student and observe where triangles occur in the neighborhood. Look at houses, buildings, parks,
and landscapes and pick out triangles that look congruent. If possible, take a ruler or tape measure and a
protractor along to a public place or a place where it will not matter if measurements are taken. Take
measurements of the sides and angles of triangular objects in pairs. Verify that they are congruent by
comparing their measures. After the walk, have the student build a roof rafter out of toothpicks. Use a
ruler and a protractor to measure the sides and angles. Verify that all the triangles are the same using the
congruency postulates. Ask the student questions like, "What would happen if the triangles were not
congruent? What would happen if engineers built bridges with triangles that were not congruent?"
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Have the student research on the Internet or at the library why congruent triangles are important in real
life situations.
Symmetry - C
Symmetry occurs all around us, in nature, at home, and at school. It occurs in crystals, plants, animals, and
even in some letters of the alphabet. This study guide will help students determine whether a figure is
symmetric about a point or a line.
For this study guide, students need to be familiar with the coordinate plane.
Determining Line Symmetry
Line Symmetry: A figure is symmetric about a line when one half of the object is the mirror image of
the other.
In order to determine if an object is symmetric about a line, try to find the line of symmetry, the line that
divides the figure into two mirror images. Line symmetry is sometimes referred to as reflection
symmetry.
Once symmetry has been determined, it is necessary to name the line of symmetry. If the line is vertical,
it will have the form x = a constant (a number). If it is horizontal, it will have the form y = a constant.
Recognizing Horizontal and Vertical Lines
•If the line crosses the x-axis, the line will be vertical. The intersection point names the line. For
example, if the line crosses the x-axis at 7, then the line is named x = 7.
•If the line crosses the y-axis, the line will be horizontal. The intersection point names the line. If the
line crosses the y-axis at 5, then the line is named y = 5.
Example 1: About which line is the figure symmetric?
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Step 1:
Step 2:
Step 3:
Step 4:
Draw a line where the figure is symmetrical and imagine folding the paper along the line.
The opposing sides are the same, so they are symmetrical.
Name the line. Since this is a vertical line it has the form x = a constant.
The line crosses the x-axis at (2, 0) so the line is x = 2.
Answer: x = 2.
Example 2: About which line are the triangles symmetric?
A. x = - 4
B. y = - 4
C. x = - 2
D. y = - 2
Step 1: In order for the triangles to be symmetric, the line of symmetry must pass through the midpoint
of the segment between the two triangles. Determine the endpoints of the segment between the
triangles: (- 6, 0) and (- 6, - 4).
Step 2: Use the midpoint formula (below) to determine the
coordinates of the midpoint.
Substitute the correct values into the formula and simplify to
determine the coordinates of the midpoint: (- 6, - 2).
Step 3: In this case, there are two lines that would pass through this midpoint and separate the figure into
two equal parts. Those lines are: x = - 6 and y = - 2.
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Step 4: Since the question provides four answer choices, x = - 6 can
be eliminated because it is NOT one of the four choices. The line y = - 2 is an answer choice, so must be
the line of symmetry. Test the answer by folding the graph along the line y = - 2. The triangles should
match up with each other. NOTE: If the problem did not give answer choices, there would be two lines
of symmetry.
Answer: D. y = - 2
Determining Point Symmetry:
Point Symmetry: A figure is symmetric about a point if it is built around a single point, called the
center. That is, for each point in the figure, there is another one directly opposite on the other side of the
center.
A simple test to determine point symmetry is to turn the figure upside down. If it does not change, point
symmetry is achieved.
The following figure is symmetric about a center point.
Example 3: Which of the following figures is not symmetric about a point?
a) is not correct. Each way the figure is turned, there is a point exactly opposite any given point. It is
symmetric about a point.
b) is correct. Turned upside down, the flower does not look the same. It is not symmetric about a point.
c) is not correct. Each way the figure is turned, there is a point exactly opposite any given point. It is
symmetric about a point.
d) is not correct. Each way the figure is turned, there is a point exactly opposite any given point. It is
symmetric about a point.
There are many activities that a student could do to reinforce symmetry. Such as...
Have the student stand in front of a full length mirror and study the symmetry in his or her face and
body. The image on the mirror is an exact reflection so there is a line of symmetry between the student
and the mirror.
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Take a piece of dot or graph paper and draw a line splitting it in half (horizontally, vertically, or
diagonally). Create a symmetrical figure by drawing geometrical shapes on one side and reflecting them
onto the other side. Color may be added to enhance the lines of symmetry.
Cut out pictures in a magazine that look symmetrical. Fold them horizontally, vertically, or diagonally
to determine if they are symmetrical.
Mutually Exclusive/Inclusive
Two events are said to be mutually exclusive if they cannot occur at the same time. Examples of mutually
exclusive events are: rolling a 3 and a 4 at the same time when rolling one die OR tossing one coin and having it
land on heads and tails at the same time. These examples show situations that cannot happen at the same time
because if one happens, the other has no chance of occurring. A special formula is needed to determine the
probability of mutually exclusive events.
P(A or B) = P(A) + P(B), where A and B are mutually exclusive events
If two events are not mutually exclusive, they are said to be inclusive and could possibly occur at the
same time. For these cases, a more general formula must be used.
P(A or B) = P(A) + P(B) - P(A and B), where A and B are any events
It is important to note that if it cannot be determined that A and B are mutually exclusive, use the
formula for inclusive events because it will work even if A and B are mutually exclusive.
Example 1: Javier is on a television game show. In order to win the $25,000.00 showcase, he must
draw a king or a queen from a standard deck of cards. What is the probability that Javier will win the
showcase?
Step 1: These events are mutually exclusive because it is not possible to draw a king and a queen out of
a standard deck of cards if you only draw one card. Determine the probability of drawing a king from
the deck of cards. There are 4 kings in a deck of cards and a total of 52 cards in the deck, so the
probability of drawing a king is 4/52 or 1/13.
Step 2: Determine the probability of drawing a queen from the deck of cards. There are 4 queens in a
deck of cards and a total of 52 cards in the deck, so the probability of drawing a queen is 4/52 or 1/13.
Step 3: Use the formula for mutually exclusive events. Fill in the correct probabilities and add.
Example 2: Michele is competing in a swim meet. The probability that she will win her first race is
0.82 and the probability that she will win her second race is 0.23. If the probability that she will win
both of her races is 0.19, what is the probability that Michele will win either the first or second race?
(1)
P(first) = 0.82
P(second) = 0.23
P(first and second) = 0.19
(2) P(first or second) = P(first) + P(second) - P(first and second)
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P(first or second) = 0.82 + 0.23 - 0.19 = 0.86
Step 1: Use the formula for inclusive events, since it is possible that she could win both races.
Determine the probabilities needed for the formula. P(first), P(second), and P(first and second).
Step 2: Substitute the correct probabilities into the formula for inclusive events and simplify.
Answer: 0.86
As an extension activity, have the student make up probabilities for events from his or her life such as
tests, chores, games, etc. Use these probabilities to make up scenarios using the formulas for mutually
exclusive and inclusive events. (To determine P(A and B), multiply the probability of A by the
probability of B. In other words, P(A and B) = P(A)
P(B).)
Estimate Line of Best Fit: Scatter Plot
A scatter plot is a graph that shows how much one variable is affected by another. One variable is plotted along
the horizontal axis and the other variable is plotted along the vertical axis. The relationship between the two
variables is called their correlation. Scatter plots usually consist of a large amount of data. The closer the data
points come to making a straight line, the higher the correlation between the two variables (or the stronger the
relationship). If the data points fall along a straight line from a low value to a high value, then the values are
said to have a positive correlation. If the data points fall along a straight line going from a high value to a low
value, then the variables are said to have a negative correlation. When looking for positive or negative
correlation, look at the graph from left to right.
A line of best fit shows the relationship
between two variables in a scatter plot. The line that is drawn represents an average of all of the data
points, most of which will probably not lie on the line itself. Remember, the line is an estimation and
not an exact measure. The points on each side of the line of best fit should be as close to the line as
possible. Once the line of best fit is drawn, it is possible to estimate the value of one variable if the
value of the other variable is known. This study guide will focus on drawing an estimated line of best fit
for a scatter plot.
Example 1: Draw an estimated line of best fit for the Barometric Pressure vs. Precipitation scatter plot
below.
Step 1: Determine if there is a positive or a negative correlation between the two variables. Since the
precipitation increases as the barometric pressure increases, there is a positive correlation. This means
that the line of best fit will go up from left to right.
Step 2: Determine where to place the line of best fit. Recall that the line of best fit represents the
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average of all of the data points. This means that there should be close to the same number of data
points above the line of best fit as there are below the line of best fit, and the points on each side of the
line should be as close to the line as possible.
The placement of this line of best fit includes 7 points above the line and 6 points below the line. This is
the estimated line of best fit.
Example 2: The graphs below show the relationship between the number of years a couple has been
married and the amount of debt the couple is in. Which graph illustrates the line of best fit for this data?
Step 1: Determine whether there is a positive or a negative correlation between the two variables. Since
the data points in this case get lower as the graph is read from left to right, there is a negative
correlation. This means that the line of best fit will go down from left to right.
Step 2: Eliminate answer choices A. and C. because their lines of best fit do not go down from left to
right.
Step 3: Of the two graphs left, determine which one represents the average of the two variables.
Remember that this means that there should be approximately the same number of data points above the
line of best fit as there are below the line of best fit and the points on each side of the line should be as
close to the line as possible.
Answer: B. This graph shows the data points with a line of best fit that is drawn in the correct direction
and has approximately the same number of data points above the line and below the line.
An activity that could help the student with this skill is to find scatter plots in newspapers, magazines,
and on the Internet and have the student draw estimated lines of best fit for the data. Have the student
justify the reasoning behind drawing the line by asking questions such as:
• Why did he or she draw a positive (or negative) correlation?
• Why did he or she place the line where it is?
• What does the line of best fit tell about the data?
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Circle Graphs - C
A graph is a visual aid used to show and compare information. A circle graph, or pie chart, is often used to
show data given in a percentage format.
An interesting method for increasing the student's understanding of graphs is to help him or her develop
a graph for a school project or event.
Have him or her create a circle graph for the events that typically occur in the student's day. If he or she
spends half of the day at school, then half (50%) of the circle would be filled with the title "School."
Similarly, if one-quarter (25%) of his or her day is spent at the soccer field, then a quarter of the pie
would be titled "Soccer." The following is an example:
Example 1: Six hundred people were asked what color their eyes are. The circle graph below represents
the results of the survey.
How many people said that they have green eyes?
(1) 600(17%) = the number of people with green eyes
(2) 600(0.17) = 102
Step 1: A total of 600 people were surveyed. To determine the number who said they have green eyes,
multiply 600 by 17%.
Step 2: Before we can multiply, 17% must be converted into a decimal number. This involves moving
the decimal point 2 places to the left.
Now multiply 600 by 0.17 to determine the number of people who have green eyes.
Answer: 102 people
Equations: Systems
A system of equations contains at least two equations that may be linear, non-linear, or a combination of the
two types. A graphical interpretation of the solution of a system of equations is that point (or points) where the
graphs of the equations intersect. One method of finding the solution(s) of a system of equations involves
adding the two equations together.
Example 1:
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Step 1: Write the equations in a vertical format, aligning the x-terms, y-terms, equal signs, and constant
terms.
Step 2: The objective is to add the corresponding parts of the two equations together and eliminate either
the x- or y-term. Multiplying each term in the top equation by 4 will create the necessary conditions for
eliminating the y-term.
Step 3: Add like terms in the two equations, and notice the resulting y-term will have a coefficient of
zero and be eliminated.
Step 4: Solve the resulting equation for x. In this case, that means divide both sides of the equation by
14. This results with x = 5.
Step 5: To solve for y, substitute the value of x (5) in either of the original equations. The second
equation was chosen for this example.
Step 6: Subtract 10 from both sides of the equation.
Step 7: Solve the resulting equation for y. In this case, that means divide both sides of the equation by 4. This results with y = -2.
As with all other equations, substitute the values of x and y into the original equations to ensure they are
correct solutions.
Both values check out. Therefore, the solution to the system of equations are x = 5 and y = -2. This can
be interpreted as the ordered pair (5, -2) of the point of intersection of the graphs of these two equations.
Equations of a Line
Every line on any coordinate graph has a corresponding equation which describes every point on the line.
Every linear equation (equation of the line) contains a slope. The slope of a line is the same between any two
points on the line.
Before you can find the equation of a line, you must first be able to find the slope of a line when given
two coordinate points on the line. These two points are named:
The formula for the slope of
a line follows.
Example 1: Find the slope of the line between Point R (2, 4) and Point S (1, 3).
Step 1: Substitute the given coordinate points into the formula.
Step 2: Simplify the fraction.
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Answer: The slope of the line is 1.
The Point-Slope form for the equation of a line:
Example 2: Use the following points to find the equation of the line.
Point T (7, -3)
Point U (-4, 6)
Step 1: Solve for the slope of the line between Point T and Point U.
Step 2: Use one of the coordinate points and the slope and substitute them into the Point-Slope form for
the equation of a line.
Step 3: Simplify both sides of the equation.
Step 4: Subtract 3 from both sides of the equation.
The equation of the line that passes through (7, -3) and (-4, 6) is y = -9/11x + 30/11.
Quadratic Formula
A quadratic equation is a polynomial equation in which the highest power of the unknown variable is two.
An example of a quadratic equation is below.
The format of a quadratic equation is
Quadratic equations can be solved by factoring,
graphing, or by using the quadratic formula. The quadratic formula is as follows:
It can be found in any algebra textbook. This formula should be memorized.
To apply the formula to a quadratic equation, use the quadratic equation format given above as a
guideline.
Example 1: Solve the quadratic equation.
Step 1: Determine the values of a, b, and c and substitute them into the quadratic formula. a = 1, b = 6,
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and c = -91
Step 2: Determine the value under the radical symbol. 6 squared is 36 and -91 times -4 equals 364.
36 + 364 = 400
Step 3: The square root of 400 is 20 (20 x 20 = 400).
Step 4: Split the remaining problem into two problems: (-6 + 20) ÷ 2 and (-6 - 20) ÷ 2 and solve the two
problems.
The answers are x = 7 and x = -13.
Example 2: Solve the quadratic equation.
Step 1: Write the equation.
Step 2:
This will put the equation in standard form.
Step 3: Determine the values of a, b, and c.
Step 4: Substitute the values of a, b, and c into the quadratic formula.
Step 5: Determine the value under the radical sign. The square root of 0 is 0.
Step 6: Solve for x.
Answer: x = -2
The discriminant is the portion of the quadratic equation under the radical sign
properties below will give you vital information about quadratic equations.
The discriminant
1. If the discriminant is a perfect square, then the quadratic equation can be factored.
2. If the discriminant is greater than 0, then the equation has two real solutions.
3. If the discriminant is less than 0, then the equation has no real solutions.
4. If the discriminant is equal to 0, then the equation has one real solution.
Example 3: How many solutions does the following quadratic equation have?
Step 1: Determine the values of a, b, and c.
Step 2: Substitute the values for a, b, and c into
Step 3: Simplify the discriminant.
Since the discriminant is greater than zero, there are two real solutions.
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If the discriminant is a perfect square, then the solutions are rational.
If the discriminant is not a perfect square, then the solutions are irrational.
Polynomials: Addition
A monomial is the product of a number and an unknown variable or unknown variables. 6xy is a
monomial. The sum of two or more monomials is called a polynomial. Here is an example of a
polynomial:
Adding and subtracting polynomials includes simplifying and combining "like" terms. Like terms are
monomials that have the same variable or variables for which the variable or variables have the same
exponent.
To add polynomials, combine similar terms.
Example 1:
Step 1: Set the two polynomials up as one long polynomial.
Step 2: Combine the like terms.
Step 3: Add the results of combining the like terms to determine the answer.
The sum is
Example 2:
Step 1: The area of the large rectangle would be the area of the shaded region added to the area of the
small rectangle. Since we know both areas, we simply add them together.
Step 2: Collect like terms so they can be added.
Step 3: Add together any like terms that were collected to determine the final answer.
The area of the large rectangle is
Sometimes it is necessary to use the distributive property before we can combine like terms.
Example 3:
Step 1: Multiply each term of the first polynomial by 3. Then multiply each term of the second
polynomial by -4.
Step 2: Group like terms together.
Step 3: Combine like terms.
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Answer:
Example 4: Solve for a, b, and c.
Step 1: Group the similar terms on the left side of the equation together.
Step 2: Now, group like terms from both sides of the equal sign together.
Step 3: Solve for a, b, and c.
Answer: a = -4, b = -2, and c = 1
Exponential Notation - E
Exponents communicate the number of times a base number is used as a factor. The base number 5 to the 3rd
power (with an exponent of 3) translates to 5 x 5 x 5. (5 to the 3rd power is not 5 x 3.) The result of 5 to the 3rd
power is 125. To perform operations with exponents, all exponential properties must be understood.
To simplify an expression or to find the missing term in a simplified expression, apply the following
exponential properties which are listed below.
Product of Powers:
When multiplying two (or more) numbers with the same base that have exponents, the base remains the
same and the exponents are added.
Power to a Power:
When taking a number with an exponent to another power, the base remains the same and the exponents
are multiplied.
The Power of Zero:
Any number taken to the power of zero (except zero) equals 1.
Negative Exponents:
There is a rule for evaluating negative exponents.
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Example 1: Simplify.
Step 1: Separate the expression into products of whole numbers times products of variables with like
bases. A variable or number that is written without an exponent automatically has an exponent of 1.
Step 2: First, multiply the whole numbers to get 6. Use the "Product of Powers" rule to evaluate the
terms in the second set of parentheses and the terms in the third set of parentheses.
Step 3: Add the exponents to complete the expression.
Answer:
Example 2: Simplify.
Step 1: Separate the expression into products of each of its terms. The exponent (2) is given to each term
inside the parentheses.
Step 2: -2 to the second power is (-2)(-2). Use the "Power to a Power" rule to evaluate the rest of the
expression.
Step 3: Multiply the powers to determine the exponents on the variables. The final expression can be
written with the parentheses still in it, or without the parentheses. It is more acceptable to remove the
parentheses.
Distance Formula
The distance formula can be used to find the distance between two points on a coordinate plane.
If you need to find the distance, d, between Point A and Point B on a coordinate plane, you can use the
distance formula:
Some distance problems can be solved by using the Pythagorean theorem. Pythagoras was a Greek
mathematician, philosopher, and theologian who lived around 580 to 500 B.C. He proved the universal
validity of a theorem later called the Pythagorean theorem.
The Distance Formula can be applied to find the distance between any two points.
Example: On a coordinate plane, point D is (3, 7), and point E is (6, 2). How far is point D from point
E?
Step 1: Substitute the values of the variables into the distance formula.
Step 2: Simplify (6 - 3) and (2 - 7).
Step 3: Square the two results of step 2.
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Step 4: Simplify the radicand (the expression under the radical sign).
Answer:
Absolute Value: Solve
The absolute value of a real number is the distance the real number x is from 0 on a number line. The absolute
value of a real number is denoted by placing the real number within two vertical lines: |x|. In other words, |-5|
("the absolute value of -5") is 5 because -5 is 5 units from 0 on a number line.
There are two major principles of absolute value:
1. The absolute value of a negative number is always a positive number.
2. The absolute value of 0 is 0.
One way to find the absolute value of single digit numbers such as |-5|, is to draw a number line and plot
-5 and 0 on the number line.
Notice that -5 is 5 points from 0, so the absolute value of |-5| is 5.
The student must also be able to find the absolute value of single digit numbers, solve absolute value
equations, evaluate absolute value expressions, and recognize absolute value expressions on number
lines.
The following example demonstrates how to solve equations with absolute value.
Example 1: Solve: |2x - 3| = 17
Step 1: Set the equation up to equal both 17 and -17. This is because 2x - 3 can equal either 17 or -17 to
make the original equation true.
Step 2: Solve the equation 2x - 3 = 17. Add 3 to both sides of the equation to get 2x = 20. Then divide
each side of the equation by 2 to get x = 10.
Step 3: Solve the equation 2x - 3 = -17. Add 3 to both sides of the equation to get 2x = -14. Then divide
each side of the equation by 2 to get x = -7.
The solutions for |2x - 3| = 17 are x = 10 or x = -7. However, to be sure, always check solutions by
substituting both solutions (10 and -7) for x (in |2x - 3| = 17). For x = 10 (shown in Step 4) and for
x = -7 (shown in Step 5):
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Example 2: Evaluate expression |x - 3| when x = -2.
(1) |-2 - 3|
(2) |-5| = 5
Step 1: Substitute -2 in the absolute value expression for x.
Step 2: The expression |-2 - 3| becomes |-5|. The absolute value of |-5| is 5 because -5 is 5 points from 0
on a number line.
It may be useful to review the inequality symbols.
Inequalities can be represented as a value on a number line. The following number line represents the
inequality
Example 3: Solve and graph the inequality. |3x + 5| < 10
Step 1: Set up the inequality as being less than 10 and greater than -10. (The second inequality should
have the "is greater than" symbol because: when we make the 10 negative, we turn the inequality
symbol the opposite direction. In this case, we change it from "is less than" to "is greater than.")
Step 2: Solve the inequality 3x + 5 < 10. Subtract 5 from both sides of the inequality, then divide both
sides of the inequality by 3.
Step 3: Solve the inequality 3x + 5 > -10. Subtract 5 from both sides of the inequality, then divide both
sides of the inequality by 3.
The answer is: x < 5/3 and x > -5, which can also be written as -5 < x < 5/3.
Now, graph the inequality. Place an open dot on the -5 and an open dot on 5/3 because the inequality is
strictly less than 5/3 and strictly greater than -5 (x cannot equal 5/3 or -5). Shade the portion of the graph
that is greater than -5 and less than 5/3.
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