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ALGEBRA
Manipulation of Algebraic Expressions
Here is a comprehensive list with definitions of common algebraic properties for REAL
numbers.
Reflexive Property: a = a
Transitive Property: if a=b and b=c, then a=c
Symmetric Property: if a = b, then b = a
Closure for Addition: if a and b are real numbers and a + b = c, then c is also Real.
Closure for Multiplication: if a and b are Real and ab = c, then c is also Real.
Substitution: if a + b = c and a = x, then x + b = c
Addition Property of Equality: if a = b, then a + c = b + c
Subtraction Property of Equality: if a = b, then a – c = b – c
Multiplication Property of Equality: if a = b, then ac = bc
Division Property of Equality: if a = b and c≠0, a÷c = b÷c
Commutative Property for Addition: a + b = b + a
Commutative Property for Multiplication: ab = ba
Associative Property for Addition: a + (b + c) = (a + b) +c
Associative Property for Multiplication: a(bc) = (ab)c
Identity Property for Addition: a + 0 = 0 + a = a
Identity Property for Multiplication: a•1 = 1•a = a
Distributive Property: a(b+c) = ab +ac
(b+c)a = ba + ca
0 is the Identity Element for Addition
1 is the Identity Element for Multiplication
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In addition to the properties, in the manipulation of algebraic expressions, it is important to
understand the correct order of operations. Often referred to as PEMDAS, the correct order
that multiple operations must be performed is: parenthetical expressions, exponential
expressions, multiplication or division in order from left to right, addition or subtraction in
order from left to right. It is extremely important to understand that multiplication and
division are equally important – sometimes multiplication is performed before the division
in a problem and sometimes the reverse is true; the order of the multiplication and division
depends upon which comes first in the problem. The same is true for addition and
subtraction. For example, in the problem: 3-2+6, the 2 is subtracted first and then the 6 is
added to the difference; in the problem 3+8-10, however, the addition is done first and then
the 10 is subtracted from the sum, 11.
Illustrates the Concept of a Function
“Function Machines” are pictorials used by students to illustrate what a function actually
does. Students put in a number, the “machine” manipulates the input according to a rule
and yields an “output” – a new number. Students are asked to determine the “rule” and
justify their thinking. For each input there is exactly one, unique value that is determined:
output.
In the diagram, three is input into the function machine. It is multiplied by 3 and then
increased by 2 and the result, 11, is output. Any number other than three would yield a
different output. There are, of course, other ways of representing functions. One way is by
using the following diagrams to show a 1 to 1 correspondence. For each input, exactly one
output is possible.
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NOT A FUNCTION
FUNCTION
In the diagram on the left, each value in the set of numbers on the left maps to exactly one
number in the right oval. The first set of ovals represents a function. The two ovals on the
right are NOT a function because the number -2 in the left oval maps to both the 0 and the 1
in the right oval. The value -2 does not map onto a single, unique number.
Functions can also be represented graphically. In a graph a function must pass what is
often called the “vertical line test.” This simply means that any value on the x-axis must
cross the graph in exactly one place. If you imagine a vertical line through each value on
the x-axis, that vertical line may not intersect the graph of a function in more than one
place.
NOT A FUNCTION
FUNCTION
The graph on the left is a function because every value of x is paired with only one value of
y. All vertical lines superimposed on this graph will intersect the parabola in only one
place. The graph on the right is NOT a function because a vertical line at x = 1, for example,
will intersect this graph in two places. This graph fails the vertical line test.
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Using Transformations to Illustrate Properties of Functions and
Relations
Literally, a transformation is a change in the form, shape, or location of a function.
Transformations can involve vertical or horizontal shifts or a change in the scale factor.
Transformations can also involve rotations, reflections over the x-axis, y-axis, the line x=y,
or some other line significant to the specific application.
In the graph above, the purple line is the graph of the equation y = x, the parent function.
The red line is the graph of y = x – 5, and represents the graph of the parent function
translated down (negative vertical shift) 5 units, while the green line is the graph y = -½ x,
the parent function with a scale factor of -½. The -½ causes the line to change to a
downward slope caused by the change in sign from positive to negative and become less
steep (more horizontal).
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Understanding the Concept of Linear Function
The root word of linear is, of course, the word line. The graphs of all linear equations are
lines and can be described by the equations: y = mx + b (slope-intercept form); Ax + By = C
(standard form); y – y 1 = m(x – x 1 ) (point-slope form). When graphing a real situation, it is
important to understand that not all parts of a graph may be reasonable solutions for a
specific situation. For example, if x represents the number of adults and y the number of
children and 4x + 2y = $500 models the amount of money received from a sale of tickets,
then it would not make sense for x and y to be anything but positive, whole integers. The
graph of 4x + 2y =500 goes through quadrants I, II, and IV, but only integer values in
quadrant I and on the x and y intercepts would be reasonable for this specific situation
since x and y represent the number of adults and the number of children.
It is also important to remember that parallel lines have the same slope; this is what keeps
the lines from ever intersecting. All lines with slopes of -2 are parallel. Perpendicular lines
have slopes that are negative reciprocals of each other. So, a line with a slope of 2 would be
perpendicular to any line with a slope of -1/2.
Understanding the Connection among Linear Functions, Proportions,
and Direct Variation
A linear function defined by an equation of the form y = kx is a direct variation and k is the
constant of variation or the constant of proportionality. Since k = y/x, the ratio y:x is
equivalent to the constant k. An equation that can be written in the form y = kx is also a
proportion. Two numbers are proportional if their ratio is constant. The equation y = kx is
𝒉
a proportion because 𝒌 = , which is a way of stating that the ratio of y to x is constant.
𝒙
The graphs of direct variations are always linear and always pass through the origin. Often
the idea of proportionality is limited to thinking about two ratios that are equal. But
proportional thinking is much broader and more encompassing. For example, the
circumference of a circle is proportional to its diameter because the ratio of circumference
to diameter is constant: π. Scale drawings are proportional to their actual object because
the ratio of the measurements on the actual object to the corresponding measurements on
the scale drawing is constant. The amount of sales tax is proportional to the cost of an item.
If the sales tax is 8.25%, the tax on that item will be a factor of 8.25% or .0825. Tax will
increase proportionately based on the cost of the item.
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Linear Functions in Sets of Data
It is often the case that when data points are plotted on a coordinate system, the points will
not lie on a straight line. However, the data might still look linear. Consider these three
graphs:
A.
B.
C.
While none of the graphs has data points that form a straight line, there is something
decidedly different about the data in graphs A and C that is very different from the data in
graph B. Graph B’s data is very random and one is unable to conclude anything about the
data because of its randomness. However graph A’s data seems to be more “linear” looking
and it seems that the data points cluster together and are on an upward trend. The same is
true for graph C with the exception being that the data seems to be on a downward trend.
Both graphs A and C lend themselves to finding a line of best fit to describe the data. To
find a line of best fit, a line is drawn through the “center” of the data with an eye on
distributing the data points pretty equally on either side of the line. Then the equation of
the line of fit is found using two points to determine the slope and y-intercept and the
equation is written. This is how it might work for graph C.
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A line has been drawn through the data points shown on graph C. Point A(4,7) and point
B(12,0) are two points on the line of fit. We will use these points to determine the slope;
use the slope and one of the points (Point B) to determine a value for b, the y-intercept;
then use these values to find our equation.
𝑚=
𝑏=
21
2
𝑦1 −𝑦2
𝑥1 −𝑥2
=
7−0
4−12
=
7
y=
−8
𝑦=
−7
8
𝑥+
−7
8
21
2
𝑥+𝑏
or
0=
−7
8
∗ 12 + 𝑏
7𝑥 + 8𝑦 = 84
Analyzing the Relationship between a Linear Equation and its Graph
Every linear equation represents a specific, unique set of data points that yield a specific,
unique graph of a line when plotted. Every line corresponds to exactly one linear equation
and every linear equation corresponds to a unique line. There are three ways to graph a
linear equation: 1) a set of points representing values of x and y that make the equation
true; 2) re-write the equation in point intercept form, y = mx + b, where m is the slope and
b is the y-intercept; locate the y-intercept on the y-axis and use the slope to locate at least
two additional points; draw a line through the points you have just plotted; 3) find the xintercept by setting y = 0 and solving for x; find the y-intercept by setting x = 0 and solving
for y; find one additional point by selecting a value for x and solving for y; draw a line
through the three points. Note that the third method is a special case of the first method.
The graph of y=2x-1 will be found by each of these methods:
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1. Set up a table of points by selecting an arbitrary value for x, substituting that value into
the equation and find the corresponding value for y. This gives you an ordered pair (x,y)
that can be plotted on a coordinate system, points connected, and a line determined.
2. The equation is already in slope intercept form. The slope: m =2 and the y-intercept: b =
-1. So, locate the intercept (0,-1) on the y-axis and use the slope to determine at least two
more points. Remember that the slope is defined as the amount of vertical change for each
unit moved to the right. So a slope of 2 means that for every one unit moved to the right, 2
units up (since the slope is positive) will also be required.
y = 2x – 1
m=2
b = -1
so (0,-1) is the yintercept
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3. Find both the x-intercept and y-intercept; plot these intercepts. If these two points are
plotted correctly, you can simply draw a line through them. Remember two points
determine a unique line. It is a “safety” measure to find a third point. The third point
should line up with the two intercepts; if does not, then a mistake has been made either in
determining and plotting the intercepts, or in finding the third point.
y = 2x - 1
When x = 0, y = 2(0) -1
y = -1 (0,-1)
When y = 0, 0 = 2x – 1
-2x = -1
x=½
(1/2, 0)
When x = 2, y = 2(2) -1
y=3
(2,3)
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Using Linear Functions, Inequalities, and Systems to Model Problems
Distance is a function of rate and time. If a car is traveling at a constant rate of 45 mph, the
distance traveled can be represented by the equation: d = 45t, where d is the distance
traveled and t is the time in hours. This specific relationship also happens to be a
proportion since the ratio of distance to time is constant: 45 mph. Not all linear
relationships are proportions. Suppose that a local charity wants to raise $25,000. They
decide to have a raffle for a trip to Disneyworld and sell tickets for $7.00 per person. If the
charity had to spend $3,500 for the trip expenses, how many tickets must they sell to
achieve their goal? This problem could be represented by the equation: 25000 + 3500 = 7t
or 25000 = 7t – 3500, where p is the profit and t, the number of tickets sold. Equations can
then be solved by graphing or by solving algebraically.
Suppose the question about the charities had stated that the charity wanted to raise at
least $25,000. Then the linear relationship would become the inequality: $25000 ≤ $7t $3500. To graph an inequality, first graph the line; the line will be dotted if inequality sign
is < or >. If the inequality is ≤ or ≥, the line will be solid. Shade above the line if the
inequality is > or ≥ and shade below the line if the inequality is < or ≤.
Methods and Representations of Linear Equations and Inequalities
Farmer Brown has both chickens and calves on his farm. He has a total of 75 animals. One
day he looked in the corral and discovered there were 180 feet. How many chickens and
how many calves did Farmer Brown have? This is a problem that requires two separate
but related equations: a system of equations. Let c = number of calves and k = number of
chickens; since calves have 4 feet and chicken have 2 feet, then:
c + k = 75
4c + 2k = 180
From this point the equations can be solved by three methods: 1) substitution, 2)
addition/subtraction, or 3) graphing. Let’s investigate each method.
1. If c+k = 75, then c = 75 – k. So, in the second equation we will substitute (75 – k)
for c and solve. This will give us:
4c + 2k = 180
4(75 – k) + 2k = 180
300 – 4k + 2k = 180
-2k = 180 – 300 = -120
k = 60
4c + 2(60) = 180
4c = 60
c = 15
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Farmer Brown had 60 chickens and 15 calves.
2.
c + k = 75 ….. multiply by -2 …. -2c - 2k = -150
4c + 2k = 180
add
4c + 2k = 180
2c
c
= 30
= 15
Substitute the 15 for the c in either equation and solve for k: 15 + k = 75; k = 60. Farmer
Brown had 60 chickens and 15 calves.
3. To solve by graphing, graph both equations on the same coordinate system:
4c + 2k = 180
c + k = 75
The two graphs intersect at the point: (c,k) or (15,60). This means there are 15 calves since
c (calves) was plotted on the x-axis and there are 60 chickens since k (chickens) was
plotted on the y-axis.
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The examples worked above were independent systems; independent linear systems have
one, unique solution. A dependent system has no unique solutions. When working with
linear equations, a dependent system results when the two equations are equivalent to
each other. An example of a dependent linear system would be: 2x – y =3 and 10x – 5y
=15. The second equation is the first equation with each term multiplied by 5. When
graphed, they would be coinciding or concurrent lines meaning that they have every point
in common and therefore have no unique solutions. When a system is made up of lines that
are parallel, the system will have no solution; there will not be any point that the lines have
in common. A system with no solution is inconsistent.
In some cases data might not be exactly linear but may be close enough to approximate a
linear relationship over certain limitations on the domain. In general, the domain that is
reasonable for a specific problem or situation is considerably smaller than the actual
domain of the function.
Roots of a Quadratic Function or Relation
Quadratic functions whose parent function is y = x2, are parabolas with a vertex at the
origin and opening upward. All other parabolas are transformations of this parent
parabola. The general form for a parabolic functions is: y = a(x-h)2 + k, vertex at (h,k) and
opening upward if a > 0 and opening downward if a < 0. All parabolas have either a
maximum (opening downward) or a minimum (opening upward) point.
There are several different ways to solve a quadratic equation:
1. Solve by factoring
2. Solve by completing the square
3. Solve by using the quadratic formula
4. Solve by graphing
All quadratics can be solved using the quadratic formula or graphing, but only a few can be
solved by factoring. Completing the square is an especially useful tool when working with
conic sections. In order to use the quadratic formula, the quadratic should be in standard
form: Ax2 +Bx + C = O. The quadratic formula is: 𝑥 =
−𝑏±√𝑏2 −4𝑎𝑐
2𝑎
. The expression under
the radical, √𝑏 2 − 4𝑎𝑐, is called the discriminant and can tell us several things about the
solution to the quadratic. If 𝑏 2 − 4𝑎𝑐 is positive, the quadratic will have two real roots; if it
is negative, there will be two complex roots; if the discriminant is equal to 0, there is one
solution, a double root.
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The ready availability of technology, specifically the graphing calculator, has made
graphing a quick way to solve quadratics. It takes a bit of time to understand domain and
range sufficiently to set the window properly, but the ease of graphing with the calculator
makes this method appealing to most students.
Basically, be able to recognize, write, and solve quadratic expressions from a variety of
applications. Be able to write a quadratic from a set of points, a geometric application, or
from a graph.
Solving Problems Involving Exponential Growth and Decay
Exponential functions have the general form y = abx , where x is a real number, a≠0, and
b≠1. When b >1, then the exponential function models a growth; when 0 > b > 1, then the
exponential function will model a decay. In both exponential growth and decay, the graph
will approach a line called the asymptote as x or y increases in absolute value.
For the function: f(x) = 2x, the graph would look like this:
Notice how the graph curves upward moving from left
to right. As the values for x get increasing small, the
graph approaches the x-axis which is the asymptote
with equation y = 0.
For the function: f(x) = (0.75)x, the graph would look like this:
Notice how the graph curves downward moving from
left to right. As the values for x get increasing large, the
graph approaches the x-axis which is the asymptote
with equation y = 0.
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Understanding the Connections among Proportions, Inverse Variation,
and Rational Functions
The relationship between direct variation and proportion was discussed, that information
will now be expanded with additional information on inverse variation and rational
𝑘
functions. If y varies inversely as x, then y = , x≠0, or xy = k. The graph of y = 1/x looks
𝑥
like this:
The graph has two branches, one in quadrant I, and the other in quadrant III. The vertical
asymptote is the y-axis (x=0) and the horizontal asymptote is the x-axis (y=0). The
asymptotes are the lines which the graph approaches but never reaches.
A rational function can be written as the ratio of two polynomial expressions: f(x) =
g(x)/p(x). g(x) and p(x) represent polynomials and p(x) cannot be equal to zero. Examples
of rational functions are:
𝟖𝒙+𝟏𝟓
f(x) =
𝟔𝒙+𝟓
, x≠
−𝟓
𝟔
and f(x) =
𝟐𝒙−𝟔
𝒙−𝟒
x≠4
The denominators determine the vertical asymptotes; set the denominators to zero and
solve to find out what these asymptotes are. The graph of the function on the left looks like
this:
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The dotted red line is the vertical asymptote, x= -5/6. The vertical asymptote is found by
setting the denominator, 6x+5, equal to 0, and solving for x. The green line is the horizontal
asymptote, y=4/3. The horizontal asymptote is found by analyzing f(x) as |𝑥| gets larger.
In this problem as larger and larger numbers are substituted for x, f(x) gets closer and
closer to 4/3. This graph represents a transformation of the first graph of the inverse
1
variation y = : a move to the left 5/6 of a unit and a move up of 4/3. The new origin is at (𝑥
5/6, 4/3).
The Effects of Transformations of a Nonlinear Function f (x)
If f(x) = 𝑥 2 then how would the graph of f(x+2) = (x+2)2 be different? Let’s take a close look
at this question.
The black graph is the graph of f(x)=x2. The
red graph is the graph of f(x+2) = (x+2)2. The
red graph represents a transformation of the
black graph by two units to the left. The graph
of f(x-2) would cause the parent function to
shift two units to the right.
Solving Systems of Quadratic Equations and Inequalities
The methods used to solve systems of quadratic equations and inequalities are the same as
those used to solve linear systems. These methods include addition/subtraction or
substitution and graphing. The solutions to systems of inequalities lie within the
overlapping regions.
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Let’s explore some examples. Solve the system: y = x2 + 6 and y = 3x2.
1)
By substitution: 3x2 = x2 + 6
2x2 = 6
x2 =3
2)
By graphing:
x = ±√3;
y = (±√3)2 + 6 = 9
The solutions to the system are marked by the intersection of the graphs of the equations
in the system. The points of intersection are marked and labeled. 1.73 represents the
approximation for √3.
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Using Properties, Graphs, and Applications of Nonlinear Relations
In previous subcategories, properties, graphs, and applications of polynomials, rational
functions, and exponential functions have been modeled and solved. Radical functions are
generally solved by raising each side of the equation to the power of the index of the
radicand. For example:
Solve: √𝑥 + 2 − 3 = 7
√𝑥 + 2 = 10
2
�√𝑥 + 2� = 102
𝑥 + 2 = 100
x = 98
A logarithm is an exponent. So the log b a = x means that bx =a. So, this is how to solve the
following problem for x:
log 3 27 = x
3x = 27
3x = 𝟑𝟑
x=3
Trig functions, sin, cos, and tan, are introduced with reference to a right triangle. These
functions are fundamentally the ratios that are created by the size of angles and the ratios
of the specific sides by which they are defined. Refer to the diagram below. Triangle ABC is
a rt. Triangle.
Trig Ratios
𝐬𝐬𝐬 𝚯 =
𝐜𝐜𝐜 𝚯 =
𝐭𝐭𝐭 𝚯 =
𝒃 𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐 𝒔𝒔𝒔𝒔
(
)
𝒄 𝒉𝒉𝒉𝒉𝒉𝒉𝒉𝒉𝒉𝒉
𝒂 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂 𝒔𝒔𝒔𝒔
(
)
𝒄 𝒉𝒉𝒉𝒉𝒉𝒉𝒉𝒉𝒉𝒉
𝒃 𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐 𝒔𝒔𝒔𝒔
(
𝒂 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂 𝒔𝒔𝒔𝒔
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Middle School Mathematics and the Concept of Limits in Sequences and
Series
A sequence is a list of numbers that follow a specific pattern. An example of a sequence is:
𝟏 𝟐 𝟑 𝟒
𝟗𝟗
𝒉
, , , ,…,
,…
𝟐 𝟑, 𝟒 𝟓
𝟏𝟎𝟎 𝒉 + 𝟏
In the sequence above each successive fraction gets larger and closer to 1. The limit of this
sequence is 1, the upper boundary of the sequence. As n gets infinitely larger the ratio of n
to n+1 gets closer to 1, though it will never reach nor pass 1.
A series is a sum of numbers. Convergent series approach a limit – an upper or lower
bound. Consider this series:
∑∞
𝒉=𝟏
𝟏
𝟐𝒉
=
𝟏
𝟐
+
𝟏
𝟒
+
𝟏
𝟖
+
𝟏
𝟏𝟔
+⋯
When you examine the series you can see that ½ the amount needed to make a whole is
added each time – beginning with ½ you would need another ½ to make a whole but the
next addend is only ¼. So now we have ½ + ¼ and we need ¼ more to make a whole – but
the next addend is only 1/8. So with each successive addend we get closer and closer to a
sum of 1, but we never will meet nor exceed that limit. The symbol, ∑ (sigma) represents a
sum or summation of values of n beginning with 1 and continuing infinitely.
In an earlier section, the concept of asymptotes was discussed. An asymptote is another
type of a limit because the graph will approach an asymptote but never reach nor intersect
it.
The rate of change of a function is its slope – the change in f(x) or y for a given change in x.
A secant line is a line that intersects a curve in two places. The two points of intersection
can be used to find the slope of the secant line. A tangent line intersects the curve in
exactly one point. The tangent line is usually drawn to critical points on the graph of a
curve, like maximum/minimum points, for example.
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Middle School Mathematics to the Area Under a Curve
In order to find the area under a curve, the area is divided into a series of rectangles with
equal widths and heights that match the limits of the curve as closely as possible.
The rectangles drawn in the figure above are bigger than what would normally be drawn.
But they will allow a clearer picture of exactly how this works. Suppose for a moment that
there is a need to know the area under the curve pictured, bordered by the x-axis. Each
rectangle has a width of 2 units. Beginning at the far left, the first rectangle has an area of 4
u2; the second, 8 u2; the next, 12 u2, the fourth, 12 u2, the fifth, 10 u2, the sixth has 8 u2, and
the final rectangle has an area of 4 u2. The total area would then be approximately
4+8+12+12+10+8+4 = 58 u2. The smaller the rectangle, the more accurate the
approximation will be. This is an illustration of the Riemann Sum.
In middle school math, many opportunities should be provided for finding approximate
areas of irregular shapes of all shapes, using grids or geoboards. These activities will afford
students opportunities to better understand the concept of area and to think through a
variety of ways to approximate those areas.
Some important concepts to understand here are the relationships between rates of change
(slope) and the graphs of linear functions. For example, what are the similarities and the
differences between the graphs of y = 2x + 1 and the graph of y = -2x +1? What happens to
the area of a square if the dimensions are doubled? What happens to the volume of a cube
if the dimensions are doubled?
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Practice Question
The following are 240Tutoring FTCE Middle Grades Mathematics 5-9 practice questions.
These and many others can be found in our FTCE Middle Grades Mathematics 5-9 Study
Guide. Please find the answers and explanations after the practice questions.
1. What is the next term in the sequence:
a10b0, a9b1, a8b2, a7b3, …
a.
b.
c.
d.
a6b3
a8b4
a6b4
a8b3
a.
b.
c.
d.
4x – 12
28 – 4x
-28 + 4x
12 – 4x
a.
b.
c.
d.
x = -2
x=2
x = 14
x=0
2. Simplify: 23 – 4(5-x):
3. Solve the following for x: 3(x + 2) = 4(x – 2)
4.
Which of the above is not a function?
a.
b.
c.
d.
I. y = x2
I, III
II, IV
III only
I only
II. y = 3x + 1
III. x = y2
IV. x = 3y -1
5. The linear function, f(x) = - ½ (3x -4), is graphed. Where will this graph cross the yaxis?
a. At y = 2
b. At y = -4
c. At y = 4
d. At y = -2
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6. Which of the following is the equation of a line that is perpendicular to the line in
this graph:
a.
b.
c.
d.
y = 1/3 x + 2
y = -3x – 1
y = 3x + 2
y = - 1/3 x +2
7. Which of the following is not a proportional relationship?
a. The ratio of the perimeter of a square to the length of one of its sides.
b. The amount of gasoline used on a trip and the length, in miles, of the trip.
c. The cost of several bunches of bananas that sell for $.50 a pound.
d. The cost of a gym membership that includes an initial membership fee
followed by a monthly fee based on frequency of use.
8. Which of the following is a true statement about the graph of: 4y = 8x – 5?
a. The graph would slope downward from left to right and cross the y-axis at 1.25.
b. The points (1, 0.75) and (-1, -3.25) are both points on the graph of this line.
c. The graph would cross the x-axis at 5/8 and would slant downward.
d. The slope of this line is ½ and the y-intercept is -5.
9. If a charity sold 2500 raffle tickets for $7 each, and their expenses totaled 1500, how
much profit did they actually make?
a. p = $7(2500) – 1500
b. p = 7t +2500 = 1500 or p = 7t+ 23,500
c. p = 7t + 4000
d. p = $7(2500) + 1500
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10. Which could be the graph of the inequality: -2y> 4x – 6?
A.
C.
B.
D.
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Answers and Explanations
1. Correct Answer C- This sequence is related to the binomial theorem. As the
sequence progresses the exponent on a decreases by one while the exponent on b
increases by one. So, after a7b3 would come: a 7-1 b 3+1 = a6b4.
2. Correct Answer A- In the problem 23 – 4(5 – x). The exponential term will be
evaluated first since the parenthetical expression cannot be simplified any further: 5
and x are unlike terms. So 23 = 8 and the expression now becomes: 8 – 4(5-x). Now,
distribute the 4 over the (5-x) and you have: 8 – (20 – 4x) = 8 – 20 + 4x = -12 + 4x.
Remember that -12 + 4x can be written as 4x – 12 (Commutative Property).
3. Correct Answer C- In solving the equation 3(x+2) = 4(x-2), first begin by distributing
the 3 on the left side of the equation and the 4 on the right side of the equation. This
gives us the following equation: 3x + 6 = 4x – 8 -> 3x = 4x – 14 -> subtract 6 from
both sides of the equation -> -x = -14- subtract 4x from both sides of the equation ->
x = 14 - multiply both sides by -1.
4. Correct Answer C- Choice III fails the vertical line text. For all values of x except 0,
there are two y-values. Equation I. y=x2 is a parabola that opens upward with a
vertex on (0,0). Both II and IV are linear equations. Only choice III is NOT a
function.
5. Correct Answer A- When the -1/2 is distributed over (3x – 4), the equation becomes:
−𝟑
𝟒
−𝟑
𝒇(𝒙) =
𝒙−
=
𝒙+𝟐
𝟐
−𝟐
𝟐
So, the y-intercept is: 2 and the graph will cross the y-axis at (0,2): y=2,
6. Correct Answer B- Lines that are perpendicular have slopes that are negative
reciprocals of each other. In this problem, the points (-6,0) and (0,2) are points on
the line. So the slope can be found by:
𝑦 −𝑦
0−2
−2
1
- 𝑚= 1 2=
= =
𝑥1 −𝑥2
−6−0
−6
3
This means that the slope of the perpendicular line would be -3 since the negative
reciprocal of 1/3 would be -3/1 or -3. Only choice B has -3 for its slope so choice B is
the only possible correct answer.
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7. Correct Answer D- (D) The cost will be $.50b. The common ratio is $.50, so this is
also a proportion. (A) A proportional relationship occurs when there is a constant
ratio of proportionality: common ratio. P = 4s; the common ratio is 4, so A is a
proportion. (B) The common ratio is whatever the mpg is for the specific car. Say
that ratio is x. Then, each mile you drive 1/x gallons of gas will be used. So, the
constant ratio is 1/x: amount of gas needed = 1/x(distance). (C) This is probably
not a proportional relationship because the fee is based on usage and there are also
additional charges, an initial fee for example. There is not a common ratio that can
be determined.
8. Correct Answer B- (B) (1,0.75) and (-1,-3.25). We can confirm this choice by
substituting each ordered pair into the equation: 0.75 = 2(1) – 5/4 = ¾ = 0.75. So
the first point is a correct solution to the equation. Substituting the values in the
second ordered pair we get: -3.25 = 2(-1) – 5/4 = -2 – 1.25 = -3.25. (A) By putting
the equation, 4y = 8x -5, in slope intercept form, we get: y = 2x – 5/4. This tells us
that the slope is 2 and the y-intercept is -5/4. (C) With a slope of 2, this would also
be false because the graph would slant upward, not downward. (D) By putting the
equation, 4y = 8x -5, in slope intercept form, we get: y = 2x – 5/4. This tells us that
the slope is 2 and the y-intercept is -5/4.
9. Correct Answer A- (A) The sale of 2500 raffle tickets at $7 each would bring in
7(2500) = $17500. But we must consider the expenses of $1500. So the profit
would be $17,500 - $1,500: p = $7(2500) - $1500. (B) This is incorrect because 7t
represents money while 2500 represents tickets. (C) This adds the number of
tickets sold, 2500, to the amount of money for expenses, $1500. This is incorrect
because these are unlike terms (money and tickets) and cannot be added. (D) This
adds the expenses to the money received – this does not make sense since one
amount is cash in and the other, cash out.
10. Correct Answer A- (A) To graph an inequality, first graph the linear equation with
which it is related. So, for the inequality, -2y>4x – 6, first graph the linear equation:
-2y = 4x -6 y = -2x+3. The inequality tells us that y is less than -2x +3 and this means
that the area below the graph of the line should be shaded. This makes the correct
choice A. One other important point about graphing inequalities to remember: if
the inequality is <or >, then the linear equation is graphed using a dotted line
because the actual points on the line are not in the solution set; if the inequality is
≤or≥, the linear equation is graphed using a dark, solid line, because the points on
the line are included in the solution set. (B) This equation would have a y-intercept
of -3, but according to our equation the y-intercept should be 3. (C) To graph an
inequality, first graph the linear equation with which it is related. So, for the
inequality, -2y>4x – 6, first graph the linear equation: -2y = 4x -6 y = -2x+3. The
inequality tells us that y is less than -2x +3 and this means that the area below the
graph of the line should be shaded. (D) This equation would have a y-intercept of -3,
but according to our equation the y-intercept should be 3.
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