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CONTENTS
1.1.1 Definition of Absolute Value ........................................................ 3
1.1.3 Lines of Symmetry ........................................................................ 3
1.2.5 Functions ........................................................................................ 4
A.1.1 Non-Commensurate .................................................................... 5
A.1.2 Expressions and Terms ................................................................ 5
A.1.3 Order of Operations..................................................................... 5
A.1.4 Evaluating Expressions and Equations ...................................... 6
A.1.5 Combining Like Terms ................................................................ 6
A.1.6 Simplifying an Expression ........................................................... 7
A.1.8 Using an Equation Mat ................................................................ 7
A.1.9 Checking a Solution ..................................................................... 7
2.1.4 The Slope of a Line ..................................................................... 10
2.2.2 Writing the Equation of a Line from a Graph ......................... 11
2.2.3 x- and y-Intercepts ..................................................................... 11
3.1.2 Laws of Exponents ...................................................................... 13
3.2.1 Using Algebra Tiles to Solve Equations .................................... 14
3.2.2 Multiplying Algebraic Expressions with Tiles ......................... 14
3.2.3 Vocabulary for Expressions ....................................................... 14
3.2.4 Properties of Real Numbers ....................................................... 15
3.3.1 The Distributive Property .......................................................... 16
3.3.2 Linear Equations from Slope and/or Points ............................. 16
3.3.3 Using Generic Rectangles to Multiply....................................... 17
4.1.1 Line of Best Fit ............................................................................ 18
4.1.2 The Equal Values Method .......................................................... 19
4.2.1 Describing Association................................................................ 19
4.2.2 The Substitution Method ............................................................ 20
4.2.3 Systems of Linear Equations ..................................................... 20
4.2.4 Forms of a Linear Function ....................................................... 21
4.2.5 Intersection, Parallel, and Coincide .......................................... 21
5.1.1 The Elimination Method ............................................................ 22
5.1.2 Continuous and Discrete Graphs .............................................. 23
5.3.2 Types of Sequences ..................................................................... 23
6.1.2 Interpreting Slope and y-Intercept on Scatterplots ................. 24
6.1.4 Residuals ...................................................................................... 25
6.2.1 Least Squares Regression Line .................................................. 25
6.2.3 Residual Plots .............................................................................. 25
6.2.4 Correlation Coefficient ............................................................... 25
6.2.5 Non-Linear Models ..................................................................... 26
7.1.1 Graphs with Asymptotes ............................................................ 28
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7.1.3 Compound Interest ..................................................................... 28
7.1.5 Step Functions ............................................................................. 29
7.2.2 Negative and Fractional Exponents .......................................... 29
7.2.3 Equations for Sequences............................................................. 29
8.1.1 More Vocabulary for Expressions ............................................. 32
8.1.2 Diagonals of Generic Rectangles ............................................... 32
8.1.3 Standard Form of a Quadratic Expression .............................. 32
8.1.4 Factoring Quadratic Expressions .............................................. 32
8.2.2 Zero Product Property ............................................................... 33
8.2.4 Forms of a Quadratic Function ................................................. 33
9.1.1 Zeros and Roots of Quadratics .................................................. 35
9.1.2 The Quadratic Formula ............................................................. 36
9.1.3 Solving a Quadratic Equation ................................................... 36
9.1.4 Simplifying Square Roots ........................................................... 37
9.2.1 Inequality Symbols...................................................................... 38
9.3.1 Curve Fitting an Exponential Function .................................... 38
9.3.2 Solving One-Variable Inequalities............................................. 39
9.4.1 Graphing Inequalities with Two Variables .............................. 39
10.2.1 Equivalent Equations ............................................................... 41
10.2.2 Solving Equations with Fractions (Fraction Busters) ........... 41
10.2.3 Methods to Solve Single-Variable Equations ......................... 42
10.2.4 Forms of a Quadratic Equation ............................................... 42
10.2.5 The Number System ................................................................. 43
10.2.6 Solving Absolute-Value Equations .......................................... 43
11.1.2 Number of Solutions to a Quadratic Equation ...................... 45
11.2.1 Interquartile Range and Boxplots ........................................... 45
11.2.2 Describing Shape (of a Data Distribution).............................. 46
11.2.3 Describing Spread (of a Data Distribution) ............................ 46
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Name: ________________________________
1.1.1 Definition of Absolute Value
Absolute value, represents the numerical value of a number without regard to its
sign. The symbol for absolute value is two vertical bars, | | . Absolute value can
represent the distance on a number line between a number and zero. Since a distance is
always positive, the absolute value is always either a positive value or zero. The
absolute value of a number is never negative.
For example, the number −3 is 3 units away
from 0, as shown on the number line at
right. Therefore, the absolute value of −3 is
3. This is written 3  3 .
Likewise, the number 5 is 5 units away
from 0. The absolute value of 5 is 5,
written
5 5
1.1.2 Families of Relations
There are several “families” of
special functions that you will study
in this course. One of these is
called direct variation (also
called direct proportion) which is
a linear function. The data you
gathered in the “Sign on the Dotted
Line” lab (in problem 1-9) is an
example of a linear relation.
Another function
is inverse
variation (also
called inverse
proportion). The
data collected in the
“Hot Tub Design” lab
(in problem 1-9) is an
example of
inverse variation.
You also observed
an exponential fu
nction. The
growth of infected
people in the
“Local Crisis” (in
problem 1-9)
was exponential.
1.1.3 Lines of Symmetry
When a graph or picture can be folded so that both sides of the fold will perfectly
match, it is said to have reflective symmetry. The line where the fold would be is
called the line of symmetry. Some shapes have more than one line of symmetry. See
the examples below.
This shape
has one line
of symmetry.
This shape has two
lines of symmetry.
This shape has
eight lines of
symmetry.
3
This graph has
two lines of
symmetry.
This shape has
no lines of
symmetry
1.2.5 Functions
A relationship between inputs and outputs is a function if there is no more than
one output for each input. We often write a function as y = some expression
involving x, where x is the input and y is the output. The following is an example of a
function.
In the example above the value of y depends on x, so y is also called the dependent
variable and x is called the independent variable.
Another way to write a function is with the notation “f(x) =” instead of “y =”. The
function named “f” has output f(x). The input is x.
In the example at right, f(5) = 9. The input is 5 and the output
is 9. You read this as, “f of 5 equals 9.”
The set of all inputs for which there is an output is called
the domain. The set of all possible outputs is called
the range. In the example above, notice that you can input
any x-value into the equation and get an output. The domain of
this function is “all real numbers” because any number can be an input. But the
outputs are all greater than or equal to zero. The range is y ≥ 0.
x2 + y2 = 1 is not a function because there are two y-values (outputs) for some x-values,
as shown below.
Prob #
1-32
Date
Learning Log Entry
Graph Investigation Questions
4
Prob #
1-65
Date
Learning Log Entry
Functions
A.1.1 Non-Commensurate
Two measurements are called non-commensurate if no combination of one
measurement can equal a combination of the other. For example, your algebra tiles are
called non-commensurate because no combination of unit squares will ever be exactly
equal to a combination of x-tiles (although at times they may appear close in
comparison). In the same way, in the example below, no combination of x-tiles will
ever be exactly equal to a combination of y-tiles.
A.1.2 Expressions and Terms
A mathematical expression is a combination of numbers, variables, and operation
symbols. Addition and subtraction separate expressions into parts called terms. For
example, 4x2 − 3x+ 6 is an expression. It has three terms: 4x2, 3x, and 6.
A more complex expression is 2x + 3(5 − 2x) + 8, which has three terms: 2x, 3(5 − 2x),
and 8. However, the term 3(5 − 2x) also has an expression inside the parentheses: 5 –
2x. 5 and 2x are terms of this inside expression.
A.1.3 Order of Operations
Mathematicians have agreed on an order of operations for simplifying expressions.
5
(10  3 2) 22 
Original expression:
(10  3 2) 22 
Circle expressions that are grouped within
parentheses or by a fraction bar:
13  3
2
2
13  3
2
2
6
6
Simplify within circled terms using the order of
operations:
Evaluate exponents.
(10  3 2) 22 
Multiply and divide from left to right.
(10  6) 22 
Combine terms by adding and
subtracting from left to right.
(4) 22   6
13  3  3
2
13  9
2
6
6
4
2
4
(4) 22   6
Circle the remaining terms:
2
4
Simplify within circled terms using the order of
operations as above.
4 2 2 6
2
16  2  620
A.1.4 Evaluating Expressions and Equations
The word evaluate indicates that the value of an expression should be calculated when
a variable is replaced by a numerical value.
For example, when you evaluate the expression x2 + 4x – 3 for x = 5, the result is:
(5)2 + 4(5) –3
25 + 20 – 3
42
When you evaluate the equation y = 3x2 – 5x + 2 for x = 4, the result is:
y = 3(4)2 – 5(4) + 2
y = 3(16) – 5(4) + 2
y = 48 – 20 + 2
y = 30
In each case remember to follow the Order of Operations.
A.1.5 Combining Like Terms
Combining tiles that have the same area to write a simpler
expression is called combining like terms. See the example
shown at right.
When you are not working with actual tiles, it can help to
picture the tiles in your mind. You can use these images to combine the terms that are
the same. Here are two examples:
6
Example 1: 2x2 + xy + y2 + x + 3 + x2 + 3xy + 2
⇒
3x2 + 4xy + y2 + x + 5
Example 2: 3x2 − 2x + 7 − 5x2 + 3x − 2
⇒
−2x2 + x + 5
Remember that addition and subtraction separate expressions into terms.
A.1.6 Simplifying an Expression
In addition to combing like terms, the following two ways to simplify an expression are
common.

Flip tiles and move them from the negative region to the
positive region (that is, find the opposite). For example,
the two unit tiles in the “–” region can be flipped and
placed in the “+” region

Remove an equal number of opposite tiles that are in the
same region. For example, the positive and negative tile
to the right can be removed.
A.1.8 Using an Equation Mat
An equation mat can help you
visually represent an equation
with algebra tiles.
For example, the equation
2x − 1 − (−x + 3) = 6 − 2x
can be represented by the equation mat at right.
(Note that there are other possible ways to represent
this equation correctly on the equation mat.)
A.1.9 Checking a Solution
To check a solution to an equation, substitute the solution into the equation and verify
that it makes the two sides of the equation equal.
For example, to verify that x = 10 is a solution to the equation 3(x −5) = 15, substitute
10 into the equation for x and then verify that the two sides of the equation are equal.
As shown at right, x = 10 is a solution to the
equation 3(x −5) = 15.
7
What happens if your answer is incorrect? To
investigate this, test any solution that is not
correct. For example, try substituting x = 2 into
the same equation. The result shows that x =
2 is not a solution to this equation.
Prob #
Date
Learning Log Entry
A-5
Variables (And Algebra Tiles)
A-16
Finding Perimeter and Combining Like Terms
8
Prob #
Date
Learning Log Entry
A-23
Meaning of Minue
A-29
Representing Expressions on an Expression Mat
A-41
Using Zeroes to Simplify
9
Prob #
Date
Learning Log Entry
A-52
“Legal” Moves for Simplifying and Comparing Expressions
A-91
Solutions of an Equation
2.1.4 The Slope of a Line
The slope of a line is the ratio of the vertical distance to
the horizontal distance of a slope triangle formed by two
points on a line. The vertical part of the triangle is
called y (Δy ) (read “change in y”), while the horizontal
part of the triangle is called x (Δx ) (read “change in x”).
It indicates both how steep the line is and its direction,
upward or downward, left to right.
Note that “Δ ” is the Greek letter “delta” that is often
used to represent a difference or a change.
Note that lines pointing upward from left to right have
positive slope, while lines pointing downward from left to right have negative slope. A
horizontal line has zero slope, while a vertical line has undefined slope.
To calculate the slope of a line, pick two points on the line, draw a slope triangle (as
shown in the example above), determine Δy and Δx, and then write the slope ratio.
You can verify that your slope correctly resulted in a negative or positive value based
on its direction. In the example above, Δy = 2 and Δx = 5, so the slope is 2 .
5
10
2.2.2 Writing the Equation of a Line from a Graph
One of the ways to write the equation of a line
directly from a graph is to find the slope of the
line (m) and the y-intercept (b). These values can
then be substituted into the general slope-intercept
form of a line: y = mx + b.
For example, the slope of the line at right
is m = 1 , while the y-intercept is (0, 2). By
3
substituting m = 1 and b = 2 into y = mx + b, the
3
equation of the line is:
2.2.3 x- and y-Intercepts
Recall that the x-intercept of a line is the point where
the graph crosses the x-axis (where y = 0). To find
the x-intercept, substitute 0 for y and solve for x. The
coordinates of the x-intercept are (x, 0).
Similarly, the y-intercept of a line is the point where
the graph crosses the y-axis, which happens when x =
0. To find the y-intercept, substitute 0 forx and solve
for y. The coordinates of the y-intercept are (0, y).
Example: The graph of 2x + 3y = 6 is a line, as shown above right.
To calculate the x-intercept,
Let y  0;2 x  3(0)  6
2x  6
x3
x-intercept: (3, 0)
Prob #
2-30
Date
To calculate the y-intercept,
Let x  0;2(0)  3x  6
3x  6
x2
y-intercept: (0, 2)
Learning Log Entry
Multiple Representations Web for Linear Relationships
TABLE
EQUATION
GRAPH
SITUATION
11
Prob #
Date
Learning Log Entry
2-40
y = mx + b
2-58
Rates of Change and Slope
What does it mean if the line is steeper? Less steep?
What does a positive slope mean? What about a negative slope?
What does a line with slope zero look like?
What does a zero slope mean in your situation?
Why does a vertical line have undefined slope?
2-81
Multiple Representations Web for Linear Relationships
TABLE
EQUATION
GRAPH
SITUATION
12
Prob #
Date
2-89
Learning Log Entry
Finding the Equation of a Line Through Two Points
5
5
-5
-5
3.1.2 Laws of Exponents
In the expression x3, x is the base and 3 is the exponent.
x3 = x · x · x
The patterns that you have been using during this section of the book are called
the laws of exponents. Here are the basic rules with examples:
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3.2.1 Using Algebra Tiles to Solve Equations
Algebra tiles are a physical and visual representation of an equation. For example, the
equation 2x + (−1) − (−x) −3 = 6 − 2x can be represented by the Equation Mat below.
An Equation Mat can be used to represent the process of solving an equation. The
“legal” moves on an Equation Mat correspond with the mathematical properties used to
algebraically solve an equation.
“Legal” Tile Move
Group tiles which are alike together.
Flip all tiles from subtraction region to
addition region
Flip everything on both sides
Corresponding Algebra
Combine like terms
Change subtraction to “adding the
opposite”
Multiply (or divide) both sides by -1
Remove zero pairs (pairs of tiles that
are opposites) within a region of the
mat
Place or remove the same tiles from
both sides
Arrange the tiles into equal size groups
A number plus its opposite equals 0
Add or subtract the same value from
both sides
Divide both sides by the same value
3.2.2 Multiplying Algebraic Expressions with Tiles
The area of a rectangle can be written two different ways. It can
be written as a product of its width and length or as a sum of its
parts. For example, the area of the shaded rectangle at right can
be written two ways:
3.2.3 Vocabulary for Expressions
A mathematical expression is a combination of numbers, variables, and operation
symbols. Addition and subtraction separate expressions into parts called terms. For
example, 4x2 − 3x + 6 is an expression. It has three terms: 4x2, 3x, and 6.
The coefficients are 4 and 3. 6 is called a constant term.
A one-variable polynomial is an expression which only has terms of the form:
(any real number) x(whole number)
For example, 4x2 − 3x1 + 6x0 is a polynomials, so the simplified form, 4x2 − 3x + 6 is a
polynomial.
14
1
The function f ( x)  7 x5  2.5 x3  x  7 is a polynomial function.
2
The following are not polynomials: 2x − 3,
1
x 2
2
and
x2
A binomial is a polynomial with only two terms, for example, x3 − 0.5x and 2x + 5.
3.2.4 Properties of Real Numbers
The legal tiles moves have formal mathematical names, called the properties of real
numbers.
The Commutative Property states that when adding or multiplying two or more
numbers or terms, order is not important. That is:
a + b = b + a For example, 2 + 7 = 7 + 2
a·b=b·a
For example, 3 · 5 = 5 · 3
However, subtraction and division are not commutative, as shown below.
7 − 2 ≠ 2 − 7 since 5 ≠ −5
50 ÷ 10 ≠ 10 ÷ 50 since 5 ≠ 0.2
The Associative Property states that when adding or multiplying three or more
numbers or terms together, grouping is not important. That is:
(a + b) + c = a + (b + c) For example, (5 + 2) + 6 = 5 + (2 + 6)
(a · b) · c = a ·(b · c)
For example, (5 · 2) · 6 = 5 · (2 · 6)
However, subtraction and division are not associative, as shown below.
(5 − 2) − 3 ≠ 5 −(2 − 3) since 0 ≠ 6
(20 ÷ 4) ÷ 2 ≠ 20 ÷(4 ÷ 2) since 2.5 ≠ 10
The Identity Property of Addition states that adding zero to any expression gives the
same expression. That is:
a+0=a
For example, 6 + 0 = 6
The Identity Property of Multiplication states that multiplying any expression by one
gives the same expression. That is:
1·a=a
For example,1 · 6 = 6
The Additive Inverse Property states that for every number a there is a number – a
such that a + – a = 0. A common name used for the additive inverse is the opposite.
For example, 3  (3)  0 and 5  5  0
The Multiplicative Inverse Property states that for every nonzero number a there is a
number 1 such that 1 1  1 . A common name used for the multiplicative inverse is
a
a
the reciprocal. That is,
1
a
is the reciprocal of a. For example, 6  1 
6
15
3.3.1 The Distributive Property
The Distributive Property states that for any three terms a, b, and c:
a(b + c) = ab + ac
That is, when a multiplies a group of terms, such as (b + c), then it multiplies each term
of the group. For example, when multiplying 2x(3x + 4y), the 2x multiplies both the
3x and the 4y.This can be shown with algebra tiles or in a generic rectangle (see
below).
3.3.2 Linear Equations from Slope and/or Points
If you know the slope, m, and y-intercept, (0, b), of a line, you can write the equation of
the line as y = mx + b.
You can also find the equation of a line when you know the slope and one point on the
line. To do so, rewrite y = mx + b with the known slope and substitute the coordinates
of the known point for x and y. Then solve for b and write the new equation.
For example, find the equation of the line with a slope of –4 that passes through the
point (5, 30). Rewrite y = mx + b as y = –4x + b. Substituting (5, 30) into the equation
results in 30 = –4(5) + b. Solve the equation to find b = 50. Since you now know the
slope and y-intercept of the line, you can write the equation of the line as y = –4x + 50.
Similarly you can write the equation of the line when you
know two points. First use the two points to find the slope.
Then substitute the known slope and either of the known
points into y = mx + b. Solve for b and write the new
equation.
For example, find the equation of the line through (3, 7) and
(9, 9). The slope is
y
x

2
6

1
. Substituting m =
3
1
and
3
(x, y) = (3, 7) into y = mx + b results in 7 
1
3
(3)  b . Then solve the equation to
find b = 6. Since you now know the slope and y-intercept, you can write the equation
of the line as y 
1
3
xb.
16
3.3.3 Using Generic Rectangles to Multiply
A generic rectangle can be used to find products because it helps to organize the
different areas that make up the total rectangle. For example, to multiply (2x + 5)(x +
3), a generic rectangle can be set up and completed as shown below. Notice that each
product in the generic rectangle represents the area of that part of the rectangle.
Note that while a generic rectangle helps organize the problem, its size and scale are
not important. Some students find it helpful to write the dimensions on the rectangle
twice, that is, on both pairs of opposite sides.
Prob #
Date
Learning Log Entry
3-18
Zero and Negative Exponents
3-47
Area as a Product and as a Sum
17
Prob #
3-106
Date
Learning Log Entry
Summary of Solving Equations
4.1.1 Line of Best Fit
A line of best fit is a straight line that represents, in general, the
data on a scatterplot, as shown in the diagram. This line does
not need to touch any of the actual data points, nor does it need
to go through the origin. The line of best fit is a model of
numerical two-variable data that helps describe the data in order
to make predictions for other data.
To write the equation of a line of best fit, find the coordinates of any two convenient
points on the line (they could be lattice points where the gridlines intersect, or they
could be data points, or the origin, or a combination). Then write the equation of the
line that goes through these two points.
18
4.1.2 The Equal Values Method
The equal values method is a method to find the solution to a system of equations.
For example, solve the system of equations below:
2x + y = 5
y=x–1
Put both equations into y = mx + b form. The two
equations are now y = −2x + 5 and y = x − 1.
2x  y  5
y  x 1
y  2 x  5
Take the two expressions that equal y and set them equal
to each other. Then solve this new equation to find x.
See the example at right
Once you know x, substitute your solution
for x into either original equation to find y. In this
example, the second equation is used.
Check your solution by evaluating for x and y in both of
the original equations.
2x  y  5
y  x 1
2(2)  1  5
1  2 1
55
11
4.2.1 Describing Association
An association (relationship) between two numerical variables can be described by its
form, direction, strength, and outliers. The shape of the pattern is called the form of
the association: linear or non-linear. The form can be made of clusters of data.
If one variable increases as the other variable increases, the direction is said to be
a positive association. If one variable increases as the other variable decreases, there
is said to be a negative association. If there is no apparent pattern in the scatterplot,
then the variables have no association.
Strength is a description of how much scatter there is in the data away from the line of
best fit. See some examples below.
An outlier is a piece of data that does not seem to fit
into the overall pattern. There is one obvious outlier in
the association graphed at right.
19
4.2.2 The Substitution Method
The Substitution Method is a way to change two equations with two variables into one
equation with one variable. It is convenient to use when only one equation is solved for
a variable.
For example, to solve the system:
Use substitution to rewrite the two equations as one. In
other words, replace x in the second equation with (−3y +
1) from the first equation to get 4(−3y + 1) − 3y = −11.
This equation can then be solved to find y. In this case, y =
1.
To find the point of intersection, substitute the value you
found into either original equation to find the other value.
In the example, substitute y = 1 into x = −3y + 1 and write
the answer for x and y as an ordered pair.
To check the solution, substitute x = −2 and y = 1
into both of the original equations.
4.2.3 Systems of Linear Equations
A system of linear equations is a set of two or more
linear equations that are given together, such as the
example at right:
y  2x
y  3x  5
If the equations come from a real-world context, then
each variable will represent some type of quantity in
both equations. For example, in the system of equations
above, y could stand for a number of dollars
in both equations and x could stand for the number of
weeks.
(1,2)
To represent a system of equations graphically, you can
simply graph each equation on the same set of axes. The
graph may or may not have a point of intersection, as
shown circled at right.
Also notice that the point of intersection lies
on both graphs in the system of equations. This means
that the point of intersection is a solution to both
equations in the system. For example, the point of
intersection of the two lines graphed above is (1, 2). This
point of intersection makes both equations true, as shown
at right.
y  2x
(2)  2(1)
22
y  3x  5
(2)  3(1)  5
2  3  5
22
The point of intersection makes both equations true; therefore the point of intersection
is a solution to both equations. For this reason, the point of intersection is sometimes
called asolution to the system of equations.
20
4.2.4 Forms of a Linear Function
There are three main forms of a linear function: slope-intercept form, standard form,
and point-slope form. Study the examples below.
Slope-Intercept form: y = mx + b. The slope is m, and the y-intercept is (0, b).
Standard form: ax + by = c
Point-Slope form: y − k = m(x − h). The slope is m, and (h, k) is a point on the line.
For example, if the slope is –7 and the point (20, –10) is on the line, the equation of the
line can be written y − (−10) = −7(x − 20) or y + 10 = −7(x − 20).
4.2.5 Intersection, Parallel, and Coincide
When two lines lie on the same flat surface (called a plane), they may intersect (cross
each other) once, an infinite number of times, or never.
For example, if the two lines are parallel, then they never intersect.
The system of these two lines does not have a solution. Examine the
graph of two parallel lines at right. Notice that the distance between
the two lines is constant and that they have the same slope but
different y-intercepts..
However, what if the two lines lie exactly on top of each other? When this happens, we
say that the two lines coincide. When you look at two lines that coincide, they appear
to be one line. Since these two lines intersect each other at all points along the line,
coinciding lines have an infinite number of intersections. The system has an infinite
number of solutions. Both lines have the same slope
and y-intercept.
While some systems contain lines that are parallel and others
coincide, the most common case for a system of equations is when
the two lines intersect once, as shown at right. The system has one
solution, namely, the point where the lines intersect, (x, y).
Prob #
4-48
Date
Learning Log Entry
Solutions to Two-Variable Equations
21
Prob #
4-80
Date
Learning Log Entry
Solving Systems of Equations
5.1.1 The Elimination Method
When solving a system of equations, it may be easier to eliminate one of the
variables by adding multiples of the two equations. This process is
called elimination.
The first step is to rewrite the equations so that the x and y variables are lined up
vertically. Next, decide what number to multiply each equation by, if necessary, in
order to make the coefficients of either the x-terms or the y-terms add up to zero. Be
sure that you can justify each step in the solution.
10y − 3x = 14
4y + 2x = −4
For example, consider the system at right.
You can eliminate the x-terms by
multiplying the top equation by 2 and the
bottom equation by 3 and then adding the
equations, as shown at right.
(10y − 3x = 14) · 2 → 20y − 6x = 28
(4y + 2x = −4) · 3 → 12y + 6x = −12
Add the resulting equations: 32y = 16
Divide: y = 0.5
Finally, substitute 0.5 for y in either original equation:
10(0.5) − 3x = 14
5 − 3x = 14
−3x = 9
x = −3
Thus, the solution to the original system is (−3, 0.5).
Check your solution by evaluating for x and y in both of the original equations.
22
5.1.2 Continuous and Discrete Graphs
When the points on a graph are connected, and it makes sense to connect them, the
graph is said to be continuous. If the graph is not continuous, and is just a sequence of
separate points, the graph is
called discrete. For example, the graph
below left represents the cost of
buying x shirts, and it is discrete because
you can only buy whole numbers of shirts.
The graph below farthest right represents
the cost of buying x gallons of gasoline,
and it is continuous because you can buy
any (non-negative) amount of gasoline.
5.3.2 Types of Sequences
An arithmetic sequence is a sequence with an addition (or subtraction) generator. The
number added to each term to get the next term is called the common difference.
A geometric sequence is a sequence with a multiplication (or division) generator. The
number multiplied by each term to get the next term is called the common ratio or
the multiplier.
A multiplier can also be used to increase or decrease by a given percentage. For
example, the multiplier for an increase of 7% is 1.07. The multiplier for a decrease of
7% is 0.93.
A recursive sequence is a sequence in which each term depends on the term(s) before
it. The equation of a recursive sequence requires at least one term to be specified. A
recursive sequence can be arithmetic, geometric, or neither.
For example, the sequence –1, 2, 5, 26, 677, … can be defined by the recursive
equation:
t(1) = –1, t(n + 1) = (t(n))2 + 1
An alternative notation for the equation of the sequence above is:
a1 = –1, an + 1 = (an)2 + 1
Prob #
5-5
Date
Learning Log Entry
Exponential Functions
23
Prob #
Date
Learning Log Entry
5-94
Multipliers
5-113
Sequences vs. Functions
6.1.2 Interpreting Slope and y-Intercept on Scatterplots
The slope of a linear association plays the same role as the slope of a line in
algebra. Slope is the amount of change we expect in the dependent variable
(Δy) when we change the independent variable (Δx) by one unit. When describing the
slope of a line of best fit, always acknowledge that you are making a prediction, as
opposed to knowing the truth, by using words like “predict,” “expect,” or “estimate.”
The y-intercept of an association is the same as in algebra. It is the predicted value of
the dependent variable when the independent variable is zero. Be careful. In statistical
scatterplots, the vertical axis is often not drawn at the origin, so the y-intercept can be
someplace other than where the line of best fit crosses the vertical axis in a scatterplot.
Also be careful about extrapolating the data too far—making predictions that are far
to the right or left of the data. The models we create can be valid within the range of
24
the data, but the farther you go outside this range, the less reliable the predictions
become.
When describing a linear association, you can use the slope, whether it is positive or
negative, and its interpretation in context, to describe the direction of the association.
6.1.4 Residuals
We measure how far a prediction made by our model is from the actual observed value
with a residual:
 residual = actual – predicted
A residual has the same units as the y-axis. A residual can be graphed with a vertical
segment that extends from the point to the line or curve made by the best-fit model.
The length of this segment (in the units of the y-axis) is the residual. A positive residual
means that the actual value is greater than the predicted value; a negative residual
means that the actual value is less than the predicted value.
6.2.1 Least Squares Regression Line
There are two reasons for modeling scattered data with a best-fit line. One is so that the
trend in the data can easily be described to others without giving them a list of all the
data points. The other is so that predictions can be made about points for which we do
not have actual data.
A unique best-fit line for data can be found by determining the line that makes the
residuals, and hence the square of the residuals, as small as possible. We call this line
the least squares regression line and abbreviate it LSRL. A calculator can find the
LSRL quickly. Statisticians prefer the LSRL to some other best-fit lines because there
is one unique LSRL for any set of data. All statisticians, therefore, come up with
exactly the same best-fit line and can use it to make similar descriptions of, and
predictions from, the scattered data.
6.2.3 Residual Plots
A residual plot is created in order to analyze the appropriateness of a best-fit model.
A residual plot has an x-axis that is the same as the independent variable for the data.
The y-axis of a residual plot is the residual for each point. Recall that residuals have the
same units as the dependent variable of the data.
If a linear model fits the data well, no interesting pattern will be made by the residuals.
That is because a line that fits the data well just goes through the “middle” of all the
data.
A residual plot can be used as evidence that the description of the form of a linear
association has been made appropriately.
6.2.4 Correlation Coefficient
The correlation coefficient, r, is a measure of how much or how little data is scattered
around the LSRL; it is a measure of the strength of a linear association. The correlation
coefficient can take on values between –1 and 1. If r = 1 or r = –1 the association is
perfectly linear. There is no scatter about the LSRL at all. A positive correlation
coefficient means the trend is increasing (slope is positive), while a negative
25
correlation means the opposite. A correlation coefficient of zero means the slope of the
LSRL is horizontal and there is no linear association whatsoever between the
variables.
The correlation coefficient does not have units, so it is a useful way to compare scatter
from situation to situation no matter what the units of the variables are. The correlation
coefficient does not have a real-world meaning other than as an arbitrary measure of
strength.
The value of the correlation coefficient squared, however, does have a real-world
meaning. R-squared, the correlation coefficient squared, is written as R² and expressed
as a percent. Its meaning is that R²% of the variability in the dependent variable can be
explained by a linear relationship with the independent variable.
For example, if the association between the amount of fertilizer and plant height has
correlation coefficient r = 0.60, we can say that 36% of the variability in plant height
can be explained by a linear relationship with the amount of fertilizer used. The rest of
the variation in plant height is explained by other variables: amount of water, amount
of sunlight, soil type, and so forth.
The correlation coefficient, along with the interpretation of R², is used to describe
the strength of a linear association.
6.2.5 Non-Linear Models
Sometimes a non-linear model best fits the data, and therefore makes better predictions,
than a linear model. This is usually made apparent by comparing the residual plots of
various models.
A good model should also be representative of the physical situation. In
problem 6-107 an exponential model made physical sense because we were measuring
decay over time. A quadratic model would not have been very satisfactory because of
its U-shape. On the other hand, if we were making predictions about the path of a
rocket, a quadratic model of the data would make a lot of sense because gravity has a
quadratic relationship with height. If we were modeling a relationship between volume
and length, a power model would be appropriate, because volume is related to length
1
y  x3
2
2 .
by a power of 3. A power model has exponents in it, for example y  5 x or
The most common models are:
Linear y = mx + b
Exponential y = abx
Power y = axb
Quadratic y = ax2 + bx + c
26
Prob #
Date
Learning Log Entry
6-15
Residuals
6-54
Residual Plots
6-70
Correlation Coefficient, r
27
Prob #
6-98
Date
Learning Log Entry
Completely Describing Association
7.1.1 Graphs with Asymptotes
A mathematically clear and
complete definition of an
asymptote requires some ideas
from calculus, but some
examples of graphs with
asymptotes might help you
recognize them when they occur.
In the following examples, the
dotted lines are the asymptotes,
and their equations are given. In
the two lower graphs, the yaxis, x = 0, is also an asymptote.
As you can see in the examples above, asymptotes can be diagonal lines or even
curves. However, in this course, asymptotes will almost always be horizontal or
vertical lines. The graph of a function has a horizontal asymptote if, as you trace
along the graph out to the left or right (that is, as you choose x-coordinates farther and
farther away from zero, either toward infinity or toward negative infinity), the distance
between the graph of the function and the asymptote gets closer to zero.
A graph has a vertical asymptote if, as you choose x-coordinates closer and closer to a
certain value, from either the left or right (or both), the y-coordinate gets farther away
from zero, either toward infinity or toward negative infinity.
7.1.3 Compound Interest
A bank can pay simple interest in which case the amount in the bank grows linearly.
For example, 3% simple interest compounded annually on an initial investment of
$2500 would grow in a sequence with a common difference: 0.03(2500) = $75.The
equation and table follow:
t(n) = 2500 + 75n
28
If the bank compounds interest, the relationship is exponential. For example, 3%
annual interest, compounded annually, would have a multiplier of 1.03 every year. The
equation and table using the example above are:
t(n) = 2500 · 1.03n
If the bank compounds monthly, the 3% annual interest becomes
3%
 0.25%
12 months / year
per month, and the multiplier becomes 1.0025. The equation and table for the first ten
years follows:
t(m) = 2500 · 1.0025m
7.1.5 Step Functions
A step function is a special kind of piecewise function ( a
function composed of parts of two or more functions). A step
function has a graph that is a series of line segments that often
looks like a set of steps. Step functions are used to model realworld situations where there are abrupt changes in the output
of the function.
The endpoints of the segments on step functions are either
open circles (indicating this point is not part of the segment) or filled-in circles
(indicating this point is part of the segment).
The graph below models a situation in which a tour bus company has busses that can
each hold up to 20 passengers with 5 available busses.
7.2.2 Negative and Fractional Exponents
For all x not equal to zero:
x0 = 1
Examples:
For positive values of x :
7.2.3 Equations for Sequences
Arithmetic Sequences
The equation for an arithmetic sequence is: t(n) = mn + b or an = mn + a0 where n is the
term number, m is the sequence generator (the common difference), and b or a0 is the
zeroth term. Compare these equations to a continuous linear function f(x)
= mx + b where m is the growth (slope) and b is the starting value (y-intercept).
29
For example, the arithmetic sequence 4, 7, 10, 13, … could be represented by t(n) =
3n + 1 or by an = 3n + 1. (Note that “4” is the first term of this sequence, so “1” is the
zeroth term.)
Another way to write the equation of an arithmetic sequence is by using the first term
in the equation, as in an = m(n – 1) + a1, where a1 is the first term. The sequence in the
example could be represented by an = 3(n – 1) + 4.
You could even write an equation using any other term in the sequence. The equation
using the fourth term in the example would be an = 3(n – 4) + 13.
Geometric Sequences
The equation for a geometric sequence is: t(n) = abn or an = a0 · bn where n is the term
number, b is the sequence generator (the multiplier or common ratio), and a or a0 is the
zeroth term. Compare these equations to a continuous exponential function f(x)
= abx where b is the growth (multiplier) and a is the starting value (y-intercept).
For example, the geometric sequence 6, 18, 54, … could be represented by t(n) = 2 ·
3n or by an = 2 · 3n.
You can write a first term form of the equation for a geometric sequence as
well: an = a1 · bn–1. For the example, first term form would be an = 6 · 3n–1.
Prob #
Date
Learning Log Entry
7-6
Investigating y=bx
7-22
Multiple Representations Web for Exponential Functions
TABLE
EQUATION
GRAPH
SITUATION
30
Prob #
Date
Learning Log Entry
7-60
Graph  Equation for Exponential Functions
7-72
Important Ideas about Exponential Functions
7-86
Zero, Negative, and Fractional Exponents
31
8.1.1 More Vocabulary for Expressions
Since algebraic expressions come in several different forms, there are special words
used to help describe these expressions. For example, if the expression can be written
in the form ax2 +bx + c and if a is not 0, it is called a quadratic expression. Review
the examples of quadratic expressions below.
Examples of quadratic expressions:
The way an expression is written can also be named. When an expression is written in
product form, it is said to be factored. When factored, each of the expressions being
multiplied is called a factor. For example, the factored form of x2 − 15x + 26 is (x −
13)(x − 2), so x − 13 and x − 2 are each factors of the original expression.
Finally, if the expression is a polynomial (see Math Notes box in Lesson 3.2.3) the
number of terms can help you name the polynomial. If the polynomial has one term, it
is called amonomial, while a polynomial with two terms is called a binomial. If the
polynomial has three terms, it is called a trinomial. Review the examples below.
Examples of monomials:
Examples of binomials:
Examples of trinomials:
15y 2 and 2
16m  25 and 7h9 
1
2
12  3k  5k and x  15x  26
3
2
8.1.2 Diagonals of Generic Rectangles
Why does Casey’s pattern from problem 8-4 work? That is, why does the product of
the terms in one diagonal of a 2-by-2 generic rectangle always equal the product of the
terms in the other diagonal?
Examine the generic rectangle at right for
(a + b)(c + d). Notice that each of the resulting
diagonals have a product of abcd. Thus, the
product of the terms in the diagonals are equal.
8.1.3 Standard Form of a Quadratic Expression
A quadratic expression in the form ax2 + bx + c is said to be in standard form. Notice
that the terms are in order from greatest exponent to least.
Examples of quadratic expressions in standard form: 3m2 + m − 1, x2 − 9, and 3x2 +
5x. Notice that in the second example, b = 0, while in the third example, c = 0.
8.1.4 Factoring Quadratic Expressions
Review the process of factoring quadratics developed in problem
8-13 and outlined below. This example demonstrates how to factor
3x2 + 10x + 8.
1.
Place the x2 and constant terms of the quadratic
expression in opposite corners of a generic rectangle.
Determine the sum and product of the two remaining
32
2.
3.
4.
corners: The sum is simply the x-term of the quadratic expression, while the
product is equal to the product of the x2 and constant terms.
Place this sum and product into a Diamond Problem and
solve it.
Place the solutions from the Diamond Problem into the
generic rectangle and find the dimensions of the generic
rectangle.
Write your answer as a product: (3x + 4)(x + 2).
8.2.2 Zero Product Property
When the product of two or more numbers is zero, one of those numbers must be zero.
This is known as the Zero Product Property. If the two numbers are represented
by a and b, this property can be written as follows:
If a and b are two numbers where a · b = 0, then a = 0 or b = 0.
For example, if (2x − 3)(x + 5) = 0, then 2x − 3 = 0 or x + 5 = 0. Solving yields the
solutions x 
3
or x = −5. This property helps you solve quadratic equations when the
2
equation can be written as a product of factors.
8.2.4 Forms of a Quadratic Function
There are three main forms of a quadratic function: standard form, factored form, and
graphing form. Study the examples below. Assume that a ≠ 0 and that the meaning
of a, b, and c are different for each form below.
Standard form: f(x) = ax2+ bx + c. The y-intercept is (0, c).
Factored form: f(x) = a(x + b)(x + c). The x-intercepts are (–b, 0) and(–c, 0).
Graphing form (vertex form): f(x) = a(x – h)2+ k. The vertex is (h, k).
Prob #
8-5
Date
Learning Log Entry
Diagonals of a Generic Rectangle
33
Prob #
Date
Learning Log Entry
8-28
Factoring Quadratics
8-48
Factoring Shortcuts
8-57
Quadratic Web
TABLE
EQUATION
GRAPH
SITUATION
34
Prob #
Date
Learning Log Entry
8-66
Zero Product Property
8-82
Strategies for Finding x-intercepts
9.1.1 Zeros and Roots of Quadratics
A root or zero of a quadratic expression is a value of x that makes the
expression equal to zero. For example, the roots or zeros of the quadratic
expression x2 − 2x − 8 are the solutions to the equation x2 − 2x − 8= 0
The x-intercepts of any quadratic function are roots. You find
the x-intercepts by setting the function equal to zero and
solving for x.
For example, the quadratic function f(x) = x2 − 2x − 8= (x +
2)(x − 4) is graphed at right. The x-intercepts are at (0, –2) and
(0, 4). The roots or zeros are –2 and 4 because the solutions to
the equation x2 − 2x − 8 = 0 are –2 and 4.
35
9.1.2 The Quadratic Formula
2
Why is x  b  b  4ac
2a
formula is shown below.
1.
a solution of ax2 + bx + c = 0? One way to derive this
Begin with the quadratic equation in
standard form.
ax2 + bx + c = 0
4a(ax2 + bx + c) = 4a(0)
2.
Multiply each side by 4a.
4a2x2 + 4abx + 4ac = 0
3.
2
Add b − 4ac to each side in order to get a
factorable quadratic on the left.
4.
The left side can be factored as (2ax + b)2,
which is demonstrated in the generic
rectangle shown at right.
5.
Take the square root of each side. Since a
square root refers to thepositive root, the
absolute value of 2ax + b is used. Then by
“looking inside” there are two possible
values for 2ax + b:  b2  4ac and
4a2x2 + 4abx + b2 = b2 − 4ac
(2ax + b)2 = b2 − 4ac
 b2  4ac .
6.
Now continue to solve for x by
subtracting b from both sides and dividing
by 2a. Notice that a cannot equal zero or
else you will get an error! However, if a =
0, then this equation would not be quadratic
and you would not use this formula.
are solutions of the equation ax2 + bx + c = 0.
Thus,
9.1.3 Solving a Quadratic Equation
So far in this course, you have learned two algebraic methods to solve a quadratic
equation of the form ax2 + bx + c = 0.
Example 1: Solve 3x2 + x − 14 = 0 for x using the Zero Product Property.
Solution: First, factor the quadratic so it is written as a product: (3x + 7)(x − 2) = 0. (If
factoring is not possible, one of the other methods of solving must be used.) The Zero
Product Property states that if the product of two terms is 0, then at least one of the
factors must be 0. Thus, 3x + 7 = 0 or x − 2 = 0. Solving these equations for x reveals
that x =
7
or that x = 2.
3
Example 2: Solve 3x2 + x − 14 = 0 for x using the Quadratic Formula.
Solution: This method works for any quadratic. First, identify a, b, and c. a equals
the number of x2 terms, b equals the number of x terms, and c equals the constant. For
3x2 + x − 14 = 0, a = 3, b = 1, and c = −14. Substitute the values of a, b, and c into the
36
Quadratic Formula and evaluate the expression twice: once with addition and once
with subtraction. Examine this method below:
Example 3: Solve x2 + 5x + 4 = 0 by completing the square.
Solution: This method works most efficiently when the coefficient of x2 is 1. Rewrite
the equation as x2 + 5x = −4. Rewrite the left side as an incomplete square:
Take the square root of both sides, x + 2.5 = ±1.5. Solving for x reveals
that x = −1 or x = −4.
9.1.4 Simplifying Square Roots
Before calculators were universally available, people who wanted to use approximate
decimal values for numbers like 45 had a few options:
1. Carry around copies of long square-root tables.
2. Use Guess and Check repeatedly to get desired accuracy.
3. “Simplify” the square roots. A square root is simplified when there are no
more perfect square factors (square numbers such as 4, 25, and 81) under the
radical sign.
Simplifying square roots was by far the fastest method. People factored the number as
the product of integers hoping to find at least one perfect square number. They
memorized approximations of the square roots of the integers from one to ten. Then
they could figure out the decimal value by multiplying these memorized facts with the
roots of the square numbers. Here are some examples of this method.
Example 1: Simplify
45 .
First rewrite 45 in an equivalent factored form so
that one of the factors is a perfect square. Simplify
the square root of the perfect square. Verify with
your calculator that both 3 5 and 45 ≈ 6.71.
Examine Example 2 and Example 3 at right. Note
that in Example 3, 72 was rewritten as 36 2 ,
rather than as 9 8 or 4 18 , because 36 is the
largest perfect square factor of 72. However, since
4 18  2 9 2  2 9 2  2 3 2  6 2 and
9 8  3 4 2  3 4 2  3 2 2  6 2 , you can
still get the same answer if you simplify it using
different methods.
When you take the square root of an integer that is not
a perfect square, the result is a decimal that never
repeats itself and never ends. It is a number that
cannot be written as a fraction using integers. This
result is called an irrational number. The irrational
37
numbers and the rational numbers together form
the real numbers.
Generally, since it is now the age of technology, when
a decimal approximation of an irrational square root
is desired, a calculator is used. However for an exact
answer, called exact form or radical form, the
number must be written using the
symbol.
9.2.1 Inequality Symbols
Just as the symbol “=” is used to represent
that two quantities are equal in
mathematics, the inequality symbols at
right are used when describing the
relationships between quantities that are
not necessarily equal.
< less than
< less than or equal to
> greater than
> greater than or equal to
When graphing an inequality on a number
line, such as x ≥ −1, a filled circle (point)
indicates that the value is a solution of the
inequality, as shown at right.
An open circle indicates that the value is
not part of the solution, as in x < –3, as
shown at right.
9.3.1 Curve Fitting an Exponential Function
You can find an exponential function that goes through two points (if both points are
above the x-axis). Recall that an exponential function with an asymptote of the x-axis
has an equation of the form y = abx.
To find an exponential function that goes through two given points, create a system of
equations by substituting one (x, y) point into y = abx, then substituting the other point.
Rewrite both equations in “a =” form. Solve the system with the equal values method
to find a and b and now you can write the equation.
For example, find an exponential function that passes through (2, 14) and (5, 112).
Create a system of equations by substituting (x, y) = (2, 14) into y = abx, and then
substituting again with (x, y) = (5, 112):
14
112
14 = ab2 and 112 = ab5
Rewrite as a  2
and a  5 .
b
b
Use the equal values method to find b:
38
The equation of the exponential function that passes through the two given points is
y = 3.5 · 2x.
9.3.2 Solving One-Variable Inequalities
To solve a one-variable inequality, first treat the problem as if it were an equality. The
solution to the equality is called the boundary point. For example, x = 12 is the
boundary point for the inequality 10 − 2(x − 3) > −8, as shown below.
Problem: 10 − 2(x − 3) > −8
10 − 2(x − 3) = −8
10 − 2x + 6 = −8
−2x + 16 = −8
−2x = −24
x = 12
First change the problem to an
equality and solve for x:
Since the original inequality is true
when x = 12, place your boundary
point on the number line as a solid
point. Then test one value on either
side in the original inequality to
determine which set of numbers
makes the inequality true.
Therefore, the solution is x < 12.
Test:
x=8
10 − 2(8 − 3) > −8
10 − 2(5) > −8
0 > −8
TRUE!
Test:
x = 15
10 − 2(15 − 3) > −8
10 − 2(12) > −8
−14 > 17
FALSE!
When the inequality is < or >, the
boundary point is not included in the
answer. On a number line, this
would be indicated with an open
circle at the boundary point.
9.4.1 Graphing Inequalities with Two Variables
To graph solve an inequality with two variables, first graph the boundary line or curve.
If the inequality does not include an equality (that is, if it is > or < rather than > or <),
then the graph of the boundary is dashed to indicate that it is not included in the
solution. Otherwise, the boundary is a solid line or curve.
Once the boundary is graphed, choose a point that does not lie on the boundary to test
in the inequality. If that point makes the inequality true, then the entire region in which
that point lies is a solution. Examine the two examples below. There are infinite
solutions to each of the inequalities. The shaded portion of the graph is a diagram of
all of the solutions.
39
Prob #
Date
Learning Log Entry
9-16
Quadratic Formula
9-35
Choosing a Strategy to Solve Quadratics
9-69
Graphing Linear Inequalities
40
Prob #
Date
Learning Log Entry
9-104
Graphing Systems of Inequalities
10.2.1 Equivalent Equations
Two equations are equivalent if all of their solutions are the same. There are
several ways to change one equation into a different, equivalent equation.
Common ways include: adding the same number to both sides, subtracting the same
number from both sides, multiplying both sides by the same number, dividing both
sides by the same (non-zero) number, and rewriting one or both sides of the equation.
For example, the equations below are all equivalent to 2x + 1 = 3:
20x + 10 = 30 2(x + 0.5) = 3
0.002x + 0.001 = 0.003
10.2.2 Solving Equations with Fractions (also known as Fraction
Busters)
Example: Solve
x
3

x
5
 2 for x.
This equation would be much easier to solve if it had
no fractions. Therefore, the first goal is to find an
equivalent equation that has no fractions.
To eliminate the denominators, multiply both sides
of the equation by the common denominator. In this
example, the lowest common denominator is 15, so
multiplying both sides of the equation by 15
eliminates the fractions. Another approach is to
multiply both sides of the equation by one
denominator and then by the other.
Either way, the result is an equivalent equation
without fractions:
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5x + 3x = 30
8x = 30
The number used to eliminate the denominators is
called a fraction buster. Now the equation looks
like many you have seen before, and it can be solved
in the usual way.
Once you have found the solution, remember to
check your answer.
10.2.3 Methods to Solve Single-Variable Equations
Here are three different approaches you can take to solve a one-variable equation:
Rewriting: Use algebraic techniques to rewrite the equation.
This will often involve using the Distributive Property to get rid
of parentheses. Then solve the equation using solution methods
you know.
Looking inside: Choose a part of the equation that includes the
variable and is grouped together by parentheses or another
symbol. (Make sure it includes all occurrences of the variable!)
Ask yourself, “What must this part of the equation equal to make
the equation true?” Use that information to write and solve a
new, simpler equation.
Undoing: Start by undoing the last operation that was done to
the variable. This will give you a simpler equation, which you
can solve either by undoing again or with some other approach.
10.2.4 Forms of a Quadratic Equation
There are three main forms of a quadratic function: standard form, factored form, and
graphing form. (See the Math Notes box in Lesson 8.2.4) Similarly, there are three
forms of a single-variable quadratic equation.
Standard form: Any quadratic equation written in the form ax2 + bx + c = 0.
Factored form: Any quadratic function written in the form a(x + b)(x + c) = 0.
Perfect Square form: Any quadratic function written in the form (ax − b)2 = c2.
Notice that when the expression on the left side of the equation below is built with
tiles, it forms a perfect square, as shown at right.
(2x + 3)2 = 5
Solutions to a quadratic equation can be written in exact form (radical form) as in:
or solutions can be estimated in approximate decimal form:
x = −0.38 or x = −2.62
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10.2.5 The Number System
The collection (set) of all numbers is organized into categories.
10.2.6 Solving Absolute-Value Equations
To solve an equation with an absolute value algebraically, first
“isolate” the absolute value on one side of the equation.
Determine the possible values of the quantity inside the absolute
value. For example, if 2 x  3  7 , then the quantity (2x +
3) must equal 7 or –7. With these two values, set up
new equations and solve as shown below.
Note that distributing over an absolute value is not
allowed. For example,
Prob #
10-66
Date
Learning Log Entry
Number of Solutions
43
Prob #
Date
Learning Log Entry
10-80
The Number System
10-102
Intercepts and Intersections
10-136
Solving Inequalities with Absolute Value
44
11.1.2 Number of Solutions to a Quadratic Equation
The solutions to a quadratic equation in standard form, ax2 + bx + c = 0,
2
are x  b  b  4ac .
2a
Therefore you can tell how many solutions a quadratic equation in standard form has
by looking at the value of the expression b2– 4ac. The expression b2– 4ac is called
the discriminant.



If b2– 4ac is negative, there are no real solutions, because the square roots of
negative numbers are not real numbers. However, there are two solutions that
contain imaginary numbers whenb2– 4ac is negative.
If b2– 4ac is zero, there is only one solution, because  0 simplifies only to
0.
If b2– 4ac is positive, there are two solutions involving  b2  4ac .
If a quadratic is in perfect square form, (ax – b)2 = c, you may solve by taking the
square root of both sides: ax  b  c . The number of solutions is determined by c:



If c is negative, there are no real solutions.
If c is zero, there is one solution.
If c is positive, there are two solutions.
If a quadratic is in factored form, a(x + b)(x + c) = 0, the Zero Product Property usually
gives two solutions: x + b = 0 and x + c = 0. However, if b = c then there is only one
solution.
11.2.1 Interquartile Range and Boxplots
Quartiles are points that divide a data set into four equal parts (and thus, the use of the
prefix “quar” as in “quarter”). One of these points is the median. The first quartile
(Q1) is the median of the lower half, and the third quartile (Q3) is the median of the
upper half.
To find quartiles, the data set must be placed in order from smallest to largest. Note
that if there are an odd number of data values, the median is not included in either half
of the data set.
Suppose you have the data set: 22, 43, 14, 7, 2, 32, 9, 36, and 12.
The interquartile range
(IQR) is the difference between
the third and first quartiles. It is
used to measure the spread (the
variability) of the middle fifty
percent of the data. The
interquartile range is 34 –
8 = 26.
A boxplot (also known as a box-and-whisker plot) displays a five number summary of
data: minimum, first quartile, median, third quartile, and maximum. The box contains
“the middle half” of the data and visually displays how large the IQR is. The right
segment represents the top 25% of the data and
the left segment represents the bottom 25% of
the data. A boxplot makes it easy to see where
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the data are spread out and where they are concentrated. The wider the box, the more
the data are spread out.
11.2.2 Describing Shape (of a Data Distribution)
Statisticians use the words below to describe the shape of a data distribution.
Outliers are any data values that are far away from the bulk of
the data distribution. In the example at right, data values in the
right-most bin are outliers. Outliers are marked on a modified
boxplot with a dot.
11.2.3 Describing Spread (of a Data Distribution)
A distribution of data can be summarized by describing its center, shape, spread, and
outliers. You have learned three ways to describe the spread.
Interquartile Range (IQR)
The variability, or spread, in the distribution can be numerically summarized with the
interquartile range (IQR). The IQR is found by subtracting the first quartile from the
third quartile. The IQR is the range of the middle half of the data. IQR can represent
the spread of any data distribution, even if the distribution is not symmetric or has
outliers.
Standard Deviation
Either the interquartile range or standard deviation can be used to represent the spread
if the data is symmetric and has no outliers. The standard deviation is the square root
of the average of the distances to the mean, after the distances have been made positive
by squaring.
For example, for the data 10 12 14 16 18 kilograms:

The mean is 14 kg.

The distances of each data value to the mean are –4, –2, 0, 2, 4 kg.

The distances squared are 16, 4, 0, 4, 16 kg2.

The mean distance-squared is 8 kg2.

The square root is 2.83. The standard deviation is 2.83 kg.
Range
The range (maximum minus minimum) is usually not a very good way to describe the
spread because it considers only the extreme values in the data, rather than how the
bulk of the data is spread.
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Prob #
Date
Learning Log Entry
11-3
Transforming Functions
11-19
Finding and Checking Inverses
11-27
Representing Data Graphically
47
Prob #
Date
Learning Log Entry
11-45
Describing Single Variable Data
11-66
Standard Deviation
48
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