MAP4C
Foundations for College
Mathematics, Grade 12, College
Preparation
Lesson 6
Exponents
MAP4C – Foundations for College Mathematics
Lesson 6
Lesson Six Concepts
¾ Evaluate simple numerical expressions involving rational exponents, without
using technology
¾ Evaluate numerical expressions involving negative exponents, with and without
using scientific calculators
¾ Simplify algebraic expressions involving integral exponents, using the laws of
exponents
Law #1: Multiplying Powers
When multiplying variables with the same base, we add the exponents together.
Example 1: Simplify the following:
a. 53 x 57
b. (-3)2 x (-3)
c. (x-3)(x11)
b. (-3)2 + 1
= (-3)3
c. x(-3) + 11
= x8
Solution
a. 53 + 7
= 510
Law #2: Dividing Powers
When dividing variables with the same base, we subtract the exponents.
Example 2: Simplify the following:
a.
12 5
12 3
b.
Solution
a. 125 - 3
= 122
y 15
y 14
c. ( −6) 9 ÷ ( −6) −4
b. y15 – 14
= y1 or y
c. (-6)9 – (-4)
= (-6)13
Law #3: Power to a Power
When we have a power to a power, we multiply the exponents together.
Example 3
Simplify the following:
a. (64)7
b. [(-2)5]4
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c. (s2)2
Page 2 of 45
MAP4C – Foundations for College Mathematics
Lesson 6
Solution
a. 64 x 7
= 628
b. (-2)5 x 4
= (-2)20
c. s2 x 2
= s4
Zero Exponents
Anything with an exponent of 0 equals 1.
***The Exception is: if the base is 0, then the number cannot be solved
Example 4
Simplify:
a. 20
b. (-4)0
⎛3⎞
c. ⎜ ⎟
⎝5⎠
b. 1
c. 1
0
d. 00
Solution
a. 1
d. cannot be solved
Negative Exponents
Anything with a negative exponent gets flipped and the negative exponent becomes
positive (this is called taking the negative reciprocal).
Simplify will always mean to put the power
into a single positive exponent.
Example 5
Simplify:
-1
1
21
a. 2
⎛3⎞
d. ⎜ ⎟
⎝4⎠
−2
b. 4
1
c. −6
3
⎛ 1⎞
b. ⎜ 3 ⎟
⎝4 ⎠
⎛4⎞
d. ⎜ ⎟
⎝3⎠
2
c. 36
-3
Solution
a.
Example 6
Simplify:
a.
n 4 × n −5
n6
b. (2a2)4 x 3(a4)2
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c.
(5x
3
)(
y 4 − 4x 2 y 6
(2xy)2
)
Page 3 of 45
MAP4C – Foundations for College Mathematics
Lesson 6
Solution
a. n4 + (-5) – 6
= n-7
1
n7
=
b. (24a2 x 4) x 3(a4 x 2)
c.
5( −4)x 3 + 2 y 4 + 6
22 x1x2 y1x2
− 20x 5 y 10
4x 2 y 2
= (16a8)(3a8)
=
= 16(3)a8 + 8
= −5x 5 −2 y10 −2
= 48a16
= −5x 3 y 8
Rational Exponents
Often values can have exponents that are fractions. Below are examples explaining
how this works.
Example 7
Simplify:
8
The numerator in the fraction acts the same
as any other power that is not in fraction form.
2
3
so the above is saying
82 = 64
The denominator in the fraction acts the root.
3
64 = 4
therefore
2
3
8 =4
Example 8
Simplify
a. 9
1
2
b. 4
3
2
c. 9
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−
1
2
d. 8
−
2
3
Page 4 of 45
MAP4C – Foundations for College Mathematics
Lesson 6
Solution
1
2
a. 9 = 2 91 = 2 9 = ±3
3
2
b. 4 = 2 4 3 = 2 64 = ±8
c. 9
d. 8
−
−
1
2
1
=
2
3
=
1
2
9
1
8
2
3
=
=
1
2
9
1
1
3
8
2
=±
1
3
=
1
3
64
=±
1
4
Support Questions
1.
2.
Evaluate:
a. (-11)0
b. 52
c. 81-1
d. 154320
e. 6-3
f. 2-3 x (16-7)0
g. 7-2
h. (-5)2
b. (-21)-2 x (-21)-3
c. (m-3)6
Simplify:
−6
⎛2⎞
⎛ 2⎞
a. ⎜ ⎟ × ⎜ ⎟
⎝5⎠
⎝5⎠
2
2
d. (3ab c)
3.
⎛ a −2
e. ⎜⎜ −2
⎝a
⎞
⎟⎟
⎠
−2
(7m n ) (− 3m n )
(3mn )(− m n )
2
f.
−5 2
4
−2
8
−4
Evaluate.
a. 25
3
2
−
e. 81
4.
−3
b. 36
3
4
f. 16
−
3
2
c. 16
5
4
3
4
d. 8
1
g. 5 2 + 25 2
−
1
4
3
4
h. 2 5 × 2 5
Simplify
1
1
a. (15625b 6 ) 6
b. (−8b 6 c 3 ) 3
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MAP4C – Foundations for College Mathematics
Lesson 6
Key Question #6
1.
2.
Evaluate:
a. (5)3
b. 70
c. 4-2
d. 10 0000
e. 5-4
f. 3-2 x (8-5)0
g. 8-1
h. (-2)5
b. 5-4 x 5-3
c. (nw2)-4
Simplify:
⎛2⎞
a. ⎜ ⎟
⎝3⎠
−3
⎛2⎞
÷⎜ ⎟
⎝3⎠
⎛ w1
e. ⎜⎜ −3
⎝w
3
d. (3abc)
3.
1
1
a. 49 2
b. 64 3
3
2
e. 8
−4
4 x 3 y −2
f.
2x −1y 4
c. 16
−
1
4
1
⎛ 1
⎞
f. ⎜⎜16 4 + 9 2 ⎟⎟
⎝
⎠
5
3
3
Simplify
2
a. (64n 9 ) 3
5.
⎞
⎟⎟
⎠
Evaluate.
d. 36
4.
−5
1
b. (5b 2 c 4 ) 3
1
3
The side length, s, of a cube is related to its volume, V, by the formula s = V . A
cube box when filled with materials has a volume of 729 cm 2 . What is the side
length of the cube box used?
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Page 6 of 45
MAP4C
Foundations for College
Mathematics, Grade 12, College
Preparation
Lesson 7
Linear, Quadratic and Exponential Functions
MAP4C – Foundations for College Mathematics
Lesson 7
Lesson Seven Concepts
¾
¾
¾
¾
¾
Comparing linear, quadratic and exponential functions
Understanding and graphing the y-intercept form of the line
Understanding the meaning of “a” in the equation y = a( x − h)2 + k
Understanding the meaning of “h” and “k” in the equation y = a( x − h)2 + k
Finding the axis of symmetry given the an equation in vertex form
y = a( x − h)2 + k
¾ Plotting and graphing quadratic equations
¾ Substitution into quadratic equations
¾ Introduction to graphing exponential functions
Linear Function
The equation of a line has 2 basic forms. One being the y-intercept form and the other
is called standard form.
y-intercept form takes the form y = mx + b
y-intercept form
It is called y-intercept form because we can use the y-intercept in the equation to help
us understand and graph the equation.
y = mx + b
“b’s” value is the spot
on the y axis where
the line intercepts the
y axis.
“m” value is the slope of
the line.
Example 1
State the slope and y-intercept for the following equations.
a. y = 3 x + 6
1
b. y = − x − 1
2
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Page 8 of 45
MAP4C – Foundations for College Mathematics
Lesson 7
Solution
a. y = 3 x + 6
slope = m =
rise 3
= =3
run 1
1
b. y = − x − 1
2
rise − 1
slope = m =
=
run
2
y-intercept = b = +6
y-intercept = b = -1
Example 2
Graph the following equations using the slope and y-intercept.
a. y = −2x + 3
Solution
a. y = −2x + 3
m = -2 =
b. y =
2
x −1
3
rise − 2
=
and b = +3
run
1
First plot the yintercept.
Starting at the y-intercept go down 2 and right 1 then
down 2 again and right one again and so on…Then draw
a line through the points plotted to create your line.
y = −2x + 3
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MAP4C – Foundations for College Mathematics
b. y =
Lesson 7
2
x −1
3
y=
2
x −1
3
Example 3
Find the point of intersection of these two equations.
(Solve the system of equations graphically)
y = −2x + 1
y = x+7
Solution
The point of intersection is (-2, 5) which solves the system of equations.
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Page 10 of 45
MAP4C – Foundations for College Mathematics
Lesson 7
(-2, 5) is the only two ordered pairs that will satisfy both equations.
y = −2x + 1
5 = −2( −2) + 1
5=5
y = x+7
5 = −2 + 7
5=5
Support Questions
1.
State the slope and y-intercept of the equations below, then properly graph and
label the equations using the slope and y-intercept.
a. y = −4 x + 3
2.
b. y = x − 4
2
c. y = − x + 7
3
d. y =
3
x −1
5
Solve for the system of equations by graphing. (Find the point of intersection)
a.
y = −3 x + 4
y = −2x + 5
2
x+6
b.
3
y =x+4
y=
c.
y = x −1
y = 4x + 2
Quadratic Functions
Opening of a Parabola y = a( x − h) + k
2
When the equation of a parabola has its “a” value as positive then the parabola
opens up.
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MAP4C – Foundations for College Mathematics
Lesson 7
When the equation of a parabola has its “a” value as negative then the parabola opens
down.
Example 1
Using the value of “a” in the equation given, state whether its parabola would open up or
down.
y = −2( x − 3)2 + 5
Example 2
Using the value of “a” in the equation given, state whether its parabola would open up or
down.
y = ( x − 3) 2 + 5
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MAP4C – Foundations for College Mathematics
Lesson 7
Support Questions
3.
For each of the following equations state whether the parabola would open up or
down.
a. y = 3( x − 1) 2 + 2
d. y = − x 2
b. y = −( x − 3) 2
e. y = −1( x − 2) 2 + 1
c. y = x 2 + 5
f. y = 3( x − 4) 2 − 2
Vertex of a Parabola y = a( x − h) + k
2
When an equation of a parabola is in vertex form y = a( x − h)2 + k the values of h and k
become the coordinates for the vertex of the equation.
Example 1
State the coordinates of the vertex of the parabola given by the following equation:
y = −2( x − 3)2 + 5
Example 2
State the coordinates of the vertex of the parabola given by the following equation:
y = ( x + 2) 2 − 4
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MAP4C – Foundations for College Mathematics
Lesson 7
Support Questions
4.
For each of the following equations state the coordinates of the vertex for its
parabola.
a. y = 3( x − 1) 2 + 2
d. y = − x 2
b. y = −( x − 3) 2
e. y = −1( x − 2) 2 + 1
c. y = x 2 + 5
f. y = 3( x − 4) 2 − 2
Equation of the Axis of Symmetry for a Parabola
As stated in the previous lesson the equation of the axis of symmetry is the equation for
the vertical line that divides the parabola exactly in half vertically (up and down).
The equation is always the “x” value in the vertex coordinate or represented by the “h” in
the vertex form of the equation y = a( x − h)2 + k .
Example 1
What is the equation of the axis of symmetry for the following equation?
y = −2( x − 3) 2 + 5
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MAP4C – Foundations for College Mathematics
Lesson 7
Example 2
What is the equation of the axis of symmetry for the following equation?
y = −x 2
Support Questions
5.
For each of the following equations state the equation of the axis of symmetry for
its parabola.
a. y = 3( x − 1) 2 + 2
b. y = −( x − 3) 2
c. y = − x 2 + 5
Graphing of a Parabola y = a( x − h) + k using a table of values.
2
The key to completing a table of values and plotting its coordinates is doing the order of
operations correctly.
BEDMAS
(Brackets, Exponents, Division, Multiplication, Addition, Subtraction)
Do the brackets first
Then exponents
Then any multiplication and/or division
Then any addition and/or subtraction
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MAP4C – Foundations for College Mathematics
Lesson 7
Example 1
Complete the table of values below and then graph.
y = ( x − 1) 2 + 2
Solution
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MAP4C – Foundations for College Mathematics
Lesson 7
Support Questions
6.
For each of the following equations complete a table of values for x values from –
2 to 2 and graph.
a. y = 3( x − 1) 2 + 2
b. y = −( x − 3) 2
c. y = − x 2 + 5
Exponential Functions
Exponential functions were used in the finance section of this course. An exponential
function results in either a rapid increase or decrease of the initial value.
Example 1
Graph the exponential function y = 2 x for integer values of x from 0 to 5.
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MAP4C – Foundations for College Mathematics
Lesson 7
Solution
x y
0 20 = 1
1 21 = 2
2 22 = 4
3 23 = 8
4 2 4 = 16
5 2 5 = 32
Support Questions
7.
Which of the following represent exponential functions? Explain.
a. y = 3 x
c. 4 x 2 − 3 x + 3
b. y = 3 x
d. y =
1
x+4
2
e. y = 4(5) n
8.
Graph the exponential function y = 2(3) x where 0 ≤ x ≤ 5 .
Key Question #7
1.
State the slope and y-intercept of the equations below, then properly graph and
label the equations using the slope and y-intercept.
a. y = 3 x − 2
2.
b. y = − x + 4
1
c. y = − x + 1
2
d. y =
2
x −3
3
Solve for the system of equations by graphing. (Find the point of intersection)
a.
y =x+4
y = −x + 5
1
x −1
b.
2
y =x+4
y=
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c.
y = x +1
y = 2x − 5
Page 18 of 45
MAP4C – Foundations for College Mathematics
Lesson 7
Key Question #7 (continued)
3.
For each of the following equations state whether the parabola would open up or
down.
a. y = −2( x + 1) 2 + 3
4.
b. y = ( x − 1) 2 + 2
c. y = − x 2 + 3
For each of the following equations state the equation of the axis of symmetry for
its parabola.
a. y = −2( x + 1) 2 + 3
6.
c. y = − x 2 + 3
For each of the following equations state the coordinates of the vertex for its
parabola.
a. y = −2( x + 1) 2 + 3
5.
b. y = ( x − 1) 2 + 2
b. y = ( x − 1) 2 + 2
c. y = − x 2 + 3
For each of the following equations complete a table of values for x values from –
2 to 2 and graph.
a. y = −( x + 1) 2 + 3
b. y = ( x − 1) 2 + 2
c. y = − x 2 + 3
7.
The path of a cliff diver as he dives into a lake, is given by the equation
y = −( x − 10) 2 + 75 , where y metres is the diver’s height above the water and, x
metres is the horizontal distance travelled by the diver. What is the maximum
height the diver is above the water?
8.
Graph the exponential function y = 3(2) x where 0 ≤ x ≤ 5 .
9.
Complete the table below and graph each function on the same set of axis.
Which of the function grows the most rapidly? Which of the functions grows the
slowest?
x
y = 3x
y = x3
y = 3x
1
2
3
4
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MAP4C
Foundations for College
Mathematics, Grade 12, College
Preparation
Lesson 8
Scatter Plots
MAP4C – Foundations for College Mathematics
Lesson 8
Lesson Eight Concepts
¾ Introduction to scatter plots
¾ Creating scatter plots
¾ Positive, negative and no correlation
Scatter plots
A scatter plot is a graph of data that is a series of points.
Example 1
Study for that Test
Here a set of data
was plotted. Hours
studying vs. Percent
of Exam
Percent on Exam
100
80
60
40
20
0
0
1
2
3
4
5
6
7
Hours of study
Data in a scatter plot can have a positive/negative or no correlation.
Negative Correlation
No Correlation
120
100
100
80
80
60
60
40
40
20
20
0
0
0
2
4
6
8
6
8
0
2
4
6
8
Positive Correlation
80
60
40
20
0
0
2
4
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MAP4C – Foundations for College Mathematics
Lesson 8
Example
Make a scatter plot for the men’s times. Plot the year on the x-axes and the times on
the y-axes.
Year
1981
1982
1983
1984
1985
Time
11.82
11.83
11.74
11.64
11.65
Year
1986
1987
1988
1989
1990
Time
11.42
11.51
11.46
11.38
11.36
Year
1991
1992
1993
1994
1995
Time
11.31
11.21
11.09
11.01
10.94
Solution
Winning Times for 100m
Time (seconds)
12
11.8
11.6
11.4
11.2
11
10.8
1980
1982
1984
1986
1988
1990
1992
1994
1996
Year
Support Questions
1.
State whether the scatter plot has a positive, negative or no correlation.
a.
b.
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c.
Page 22 of 45
MAP4C – Foundations for College Mathematics
Lesson 8
Support Questions (continued)
2.
The table below shows the winning times for the 800-m race at the Olympic
Summer Games. Construct a scatter plot.
Year
1960
1964
1968
1972
1976
1980
1984
1988
Men’s
Time
106.3
105.1
104.3
105.9
103.5
105.4
103
103.45
Key Question #8
1.
State whether the scatter plot has a positive, negative or no correlation.
b.
a.
2.
c.
The table below shows the winning heights in a men’s high jump competition.
Year
1912
1932
1952
1972
1982
2002
Winning Country
Canada
United States
England
United States
United States
Australia
Jump in Height (m)
1.93
1.96
1.99
2.16
2.23
2.41
Create a properly labelled Scatter Plot using the data given in the table.
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MAP4C – Foundations for College Mathematics
Lesson 8
Key Question #8 (continued)
3.
The table below shows the percent to high school girls who smoked more than
one cigarette during the previous year. Graph the data in a scatter plot.
Year
Percent
1981
34.3
1983
29.5
1985
25.9
1987
25.1
1989
24.4
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1991
22
1993
21.3
1995
21.6
1997
20.4
Page 24 of 45
MAP4C
Foundations for College
Mathematics, Grade 12, College
Preparation
Lesson 9
Line of Best Fit and Extrapolation
MAP4C – Foundations for College Mathematics
Lesson 9
Lesson Nine Concepts
¾ Determining the equation of best fit
¾ Extrapolation
¾ Interpolation
Line of Best Fit and Extrapolation
Line of Best Fit
Line of best fit is a line that passes as close as possible to a set of plotted points.
Example
Find the line of best fit for the data in the previous example.
Solution
This is the yintercept.
Winning Times for 100m
Time (seconds)
12
11.8
11.6
11.4
11.2
11
10.8
1980
1982
1984
1986
1988
1990
1992
1994
1996
Year
First pick two points that represent the general center of the points plotted.
Then draw the line of best fit through those points generally dividing the plotted point
even on both sides.
Example
What is the approximate equation for the line of best fit in the previous example?
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Page 26 of 45
MAP4C – Foundations for College Mathematics
Lesson 9
Solution
Choose the coordinates of the two previously chosen points used to draw the line of
best fit. (1985, 11.7) and (1989, 11.4)
slope = m =
m=
y 2 − y 1 11.4 − 11.7
=
x 2 − x1 1989 − 1985
− 0. 3
= −0.075
4
y − intercept = b = +12
Therefore the equation of the line of best fit is y = -0.075x +12.
Extrapolation and Interpolation
Extrapolation is extending a graph to estimate the values that are beyond the table of
values.
Interpolation is using a graph to estimate the value that are not in the table of values
but are within the range of the lowest and largest values presented in the table of
values.
Example 1
a. Using extrapolation what would be the approximate distance for the discus
will be thrown during the 2004 Summer Olympics?
b. Using Interpolation how far was the discus thrown in 1952?
Distance (m)
Women's Olympic Discus Records
80
70
60
50
40
30
20
10
0
1940
1950
1960
1970
1980
1990
2000
2010
2020
Year
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Page 27 of 45
MAP4C – Foundations for College Mathematics
Lesson 9
Solution
a. Using extrapolation, what would be the approximate distance for the discus
will be thrown during the 2004 Summer Olympics?
approximately 78 meters.
Women's Olympic Discus Records
80
Draw a vertical line
from 2004 until you hit
the line of best fit then
run horizontally until
you hit the value on
the y axis
Distance (m)
70
60
50
40
30
20
10
0
1940
1950
1960
1970
1980
1990
2000
2010
2020
Year
b. Using Interpolation, how far was the discus thrown in 1952?
Approximately 51 metres
Women's Olympic Discus Records
80
Distance (m)
70
Draw a vertical line
from 1952 until you
hit the line of best
fit then run
horizontally until
you hit the value
on the y axis
60
50
40
30
20
10
0
1940
1950
1960
1970
1980
1990
2000
2010
2020
Year
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Page 28 of 45
MAP4C – Foundations for College Mathematics
Lesson 9
Support Questions
1.
The table below shows the winning times for the 100-m Men’s Freestyle
Swimming at the Olympic Summer Games.
Year
1960
1964
1968
1972
1976
1980
1984
1988
Men’s
Time
61.2
59.5
60
58.6
55.7
54.8
55.9
54.9
a. Construct a scatter plot and line of best fit for the data.
b. What is the equation of the line of best fit?
c. What is the approximate winning time in the year 2020?
d. What approximate year does the time drop below 55 seconds?
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Page 29 of 45
MAP4C – Foundations for College Mathematics
Lesson 9
Key Question #9
1.
The table below shows Varsity Football Home Game Attendance.
Game
1
2
3
4
5
6
7
Attendance
222
285
399
641
529
952
1171
a. Create a properly labelled scatter plot using the data given in the table.
b. Draw a line of best fit.
c. What is the approximate slope of the line of best fit?
d. Using extrapolation, what would the approximate attendance be for game 9?
Is this a safe assumption? Explain.
2.
The table below shows live bacteria count from a science experiment.
Temperature
(C)
Bacteria
(000s)
20
22
24
26
28
30
32
34
36
2.1
4.3
5.2
6.1
6.7
7.6
10.3
8.2
14.1
a. Graph the data in a scatter plot.
b. Determine the equation of the line of best fit.
c. Use your equation to predict the bacteria count when the temperature
reaches 42 °C
d. What is the approximate degree, by using extrapolation, to find when the
bacteria count when it reaches 18 000?
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MAP4C
Foundations for College
Mathematics, Grade 12, College
Preparation
Lesson 10
Uses and Misuses of Sample Data
MAP4C – Foundations for College Mathematics
Lesson 10
Lesson Ten Concepts
¾
¾
¾
¾
¾
Use of scatter plots
Use of correlation coefficient
Recognizing lack of data
Recognizing outliers in a scatter plot
Use of extrapolation
Uses and Misuses of Sample Data
Lack of Data
Example 1
Describe the anomaly in the scatter plot given below:
Solution
The scatter plot above does not have enough data to analyse. There is no minimum
that is correct however the more the better.
Low correlation/ Low reliability
Example 2
Describe the anomaly in the scatter plot given below:
No Correlation
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MAP4C – Foundations for College Mathematics
Lesson 10
Solution
In this example the correlation value is low, meaning that the points in the scatter plot
are all over and not close to the line of best fit. The closer the points to the line of best
fit the more accurate the line of best fit is and any conclusions made using the line of
best fit.
Outliers
Example 3
Describe the anomaly in the scatter plot given below:
Solution
The point in the upper left corner distorts the data and causes the line of best fit to shift
slightly towards the outlier causing the distortion.
Support Questions
1.
Describe the misuse of the line of best fit in each graph below.
2.
Give an example of a data set that may be influenced by each factor.
a. religion
b. society
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c. sports
Page 33 of 45
MAP4C – Foundations for College Mathematics
Lesson 10
Support Questions (continued)
3.
The number of forest fires in Esexx province is shown below:
Year
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
Forest Fires in Esexx JuneAug
212
178
287
77
168
318
201
112
322
240
a. Construct a scatter plot and determine whether a trend might exist.
b. Is it possible to extrapolate the number of forest fires to expect in the future?
Explain.
4.
The Ministry of Natural Resources monitors reforestation in Canada. The table
below shows the area reforested by seed and seedlings over a six year period;
Year
1991
1992
1993
1994
1995
1996
Reforestation (ha)
509 675
465 547
453 701
488 927
459 429
450 173
a. Create a scatter plot and line of best fit on the information given above.
b. How strong is the reliability of the line of best fit? Explain.
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MAP4C – Foundations for College Mathematics
Lesson 10
Key Question #10
1.
Describe the misuse of the line of best fit in each graph below.
2.
Give an example of a data set that may be influenced by each factor.
a. politics
3.
b. family
c. education
David is running the 100 m for a high school track and field competition. David’s
time trials had him record the time for 9 runs. His times are recorded in the table
below.
Trial Number 1
2
3
4
5
6
7
8
9
Time (s)
13.2 13.16 12.89 12.91 13.22 15.8 12.93 13.01 12.99
a. Determine a line of best fit for this data
b. Are there any outliers in this data, and describe how it/they might have
occurred.
c. Last year the qualifying time to make the provincial championships was
13.08. Using the model created above do you think David will likely qualify for
the championships?
4.
The regular season points earned by the President’s Trophy winners over the
last 7 years is shown below:
Year
99-00
00-01
01-02
02-03
03-04
05-06
06-07
5.
points
105
102
118
115
112
108
113
a. Construct a scatter plot and determine whether a
trend might exist.
b. Draw a line of best fit on your scatter plot.
c. Is it possible to extrapolate the number of points to
predict the winning points for the next 4 years?
Explain.
List the 3 anomalies that have been discussed in this lesson that can distort data
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MAP4C
Foundations for College
Mathematics, Grade 12, College
Preparation
Answers to Support Questions
MAP4C – Foundations for College Mathematics
Support Question Answers
Lesson 6:
1.
a. 1
b. 25
-3=
e. 6
1
1
=
3
216
6
c.
f. 2-3 x (16-7)0 =
1
81
1
1
×1 =
3
8
2
d. 1
g. 7-2=
1
1
=
2
49
7
h. 25
−6
2.
⎛2⎞
⎛2⎞
a. ⎜ ⎟ × ⎜ ⎟
⎝5⎠
⎝5⎠
−3
⎛ 2⎞
=⎜ ⎟
⎝5⎠
( −6)+( −3)
c. (m-3)6 = m ( −3 )( 6 ) = m −18 =
−2
⎛ a ( −2 )( −2 ) ⎞ a 4
= ⎜⎜ ( −2 )( −2 ) ⎟⎟ = 4 = 1
⎝a
⎠ a
−5 2
4
−4
)
= (5 ) = 125
3
2
( 36 )
3
= (6 ) = 216
3
( 16 )
3
= (2) = 8
2
4
3
=
25
2
4
1
( 8)
4
3
−
e. 81
3
4
3
=
( 81)
4
3
4
8
)
=
( 49m 4 n −10 )(m 5 n11 )
= 49m 9 n
1
3
3
=
1
−10
−1 −3
(
c. 16 4 =
−
8
3
2
b. 36 =
d. 8
4
−2
a. 25 =
1
1
=−
5
4084101
( −21)
1
m18
(7m n ) (− 3m n ) = (49m n )(−3m n
(3mn )(− m n )
− 3m n
2
3.
2−9 59 1953125
357
= −9 = 9 =
= 3814
5
2
512
512
= (3ab 2 c )(3ab 2 c ) = 9a 2 b 4 c 2
d. (3ab2c)2
f.
−9
= ( −21)( −2 )+( −3 ) = ( −21) −5 =
b. (-21)-2 x (-21)-3
⎛ a −2 ⎞
e. ⎜⎜ −2 ⎟⎟
⎝a ⎠
⎛ 2⎞
=⎜ ⎟
⎝5⎠
3
1
=
(2)
4
=
1
(3)
3
1
16
=
1
27
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MAP4C – Foundations for College Mathematics
f. 16
−
5
4
=
1
( 16 )
5
4
=
1
(2)
5
=
Support Question Answers
1
32
1
g. 5 2 + 25 2 = 25 + 25 = 25 + 5 = 30
1
4
1 4
+
5
h. 2 5 × 2 5 = 2 5
4.
= 21 = 2
1
6
a. (15625b ) = 6 15625b 6 = 5b
6
1
b. ( −8b 6 c 3 ) 3 = 3 − 8b 6 c 3 = −2b 2 c
Lesson 7:
2.
a. slope = −4, y- int = 3
2
c. slope = − , y- int = 7
3
a.
3.
a. “a” is a positive number (+3) so opens up
1.
b.
b. slope = 1, y- int = −4
3
d. slope = , y- int = −1
5
c.
b. “a” is a negative number (-nothing equates to -1) so opens down
c. “a” is a positive number (nothing equates to +1) so opens up
d. “a” is a negative number (-nothing equates to -1) so opens down
e. “a” is a negative number (-1) so opens down
f. “a” is a positive number (+3) so opens up
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MAP4C – Foundations for College Mathematics
Support Question Answers
4.
a. (1, 2)
b. (3, 0)
c. (0, 5)
e. (2, 1)
5.
a. x = 1
b. x = 3
c. x = 0
6.
a.
d. (0, 0)
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f. (4, -2)
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MAP4C – Foundations for College Mathematics
6.
Support Question Answers
b.
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MAP4C – Foundations for College Mathematics
6,
c.
7.
b and e (both represent a rapid increase or decrease)
Support Question Answers
8.
x
0
1
2
3
4
5
y
2
6
18
54
162
486
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MAP4C – Foundations for College Mathematics
Support Question Answers
Lesson 8:
1.
a. no correlation
b. positive correlation
c. negative correlation
2.
Lesson 9:
1.
a.
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MAP4C – Foundations for College Mathematics
Support Question Answers
b.
(1964,59.5) and (1984,55.9)
slope = m =
m=
y 2 − y 1 55.9 − 59.5
=
x 2 − x 1 1984 − 1964
− 3 .6
= −0.18
20
y − intercept = b = +61
Therefore the equation of the line of best fit is y = -0.18x +61.
c.
≈ 49 .5s
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MAP4C – Foundations for College Mathematics
d.
Support Question Answers
about 1990
Lesson 10:
1.
a. No or little correlation. Points are too spread out and line of best fit has little
reliability.
b. The point in the upper left corner is an outlier and distorts the date and the line
of best fit.
c. No enough data to support the reliability of the line of best fit.
2.
a. Answers vary: A survey asking whether stores should be open on Sundays.
b. Answers vary: The amount of recycled materials sent to recycling.
c. Answers vary: A survey asking what individuals like to do with their spare
time.
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MAP4C – Foundations for College Mathematics
3.
Support Question Answers
a.
appears to be no trend existing
b. Since there is no trend it would be highly inaccurate when making any
conclusions based on extrapolating any line of best fit.
4.
a.
b. Weak reliability: few data points and what data that is present is fairly spread
out.
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