1-5 Parent Functions and Transformations
Describe the following characteristics of the
graph of each parent function: domain, range,
intercepts, symmetry, continuity, end behavior,
and intervals on which the graph is
increasing/decreasing.
1. f(x) = [[x]]
The graph is a horizontal line for all non-integer
values of x so the graph is constant for {x | x
}.
The graph increases by 1 for every integer value of
x, so the graph increases for {x | x
}.
SOLUTION:
2. f(x) =
SOLUTION:
The graph is continuous for all values of x, so D = {x
|x
}.
The only y-values on the graph are integer values,
and all of the integers are included, so R = {y | y
}.
The graph intersects the y-axis at (0, 0), so there is a
y-intercept at (0, 0).
The graph intersects the x-axis over [0, 1), so there
are x-intercepts for {x | 0 ≤ x < 1, x
}.
There are no mirror images with respect to the origin,
the axes, or any other lines, so the graph has no
symmetry.
For every integer value of x, the limits if the function
are different values from the left and the right, so the
graph has a jump discontinuity for {x | x
}.
The graph approaches negative infinity as x
approaches negative infinity so
=−
The graph approaches infinity as x approaches
negative infinity so
= .
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.
The graph is continuous for all values of x except
when x = 0, so D = {x | x ≠ 0, x
.
The range includes all values of y except when x = 0,
so R = {y | y ≠ 0, y
}.
The graph does not cross the x- or y-axis and thus
has no intercepts.
The graph has mirror images over the origin, thus he
graph is symmetric with respect to the origin.
The graph has an infinite discontinuity at x = 0.
The graph approaches zero as x approaches negative
infinity so
= 0.
The graph approaches zero as x approaches positive
infinity so
= 0.
The graph is decreasing on (−
, 0) and (0,
).
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1-5 Parent Functions and Transformations
3. f(x) = x3
4. f(x) = x4
SOLUTION:
SOLUTION:
The graph is continuous for all values of x, so D = {x
|x
}.
The range includes all values of y, so R = {y | y
}.
The range includes all values of y ≥ 0, so R = {y | y ≥
0, y
}.
The graph intersects the y-axis at (0, 0), so there is a
y-intercept at (0, 0).
The graph is mirrored over the origin, so it is
symmetric with respect to the origin.
The graph approaches infinity as x approaches
positive infinity so and
= .
The graph is increasing on (−
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,
).
The graph intersects the y-axis at (0, 0), so there is a
y-intercept at (0, 0).
The graph is mirror over the y-axis, thus it is
symmetric with respect to the y-axis.
The graph is continuous.
The graph approaches negative infinity as x
approaches negative infinity so
=−
The graph is continuous for all values of x, so D = {x
|x
},
.
The graph is continuous.
The graph approaches infinity as x approaches
negative infinity so
.
The graph approaches infinity as x approaches
infinity so
.
The graph is decreasing on (−
on (0, ).
, 0) and increasing
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1-5 Parent Functions and Transformations
5. f(x) = c
6. f(x) = x
SOLUTION:
SOLUTION:
The graph is continuous for all values of x, so D = {x
|x
},
The graph is continuous for all values of x, so D = {x
|x
}.
The only y value in the domain is c. Thus, R = {y | y
= c, c
}.
The range includes all values of y, so R = {y | y
}.
If c = 0, all real numbers are x-intercepts. If c ≠ 0,
there are no x-intercepts.
The graph intersects the y-axis at (0, 0), so there is a
y-intercept at (0, 0).
The graph intersects the y-axis at (0, c), so there is a
y-intercept at (0, c).
The graph is symmetric with respect to the origin.
Therefore, it is odd.
If c 0, the graph is symmetric with respect to the
y-axis. If c = 0, the graph is symmetric with respect
to the x-axis, y-axis, and origin.
The graph is continuous.
The graph approaches negative infinity as x
approaches negative infinity so
The graph is continuous.
The graph approaches infinity as x approaches
infinity so
.
The graph approaches c as x approaches negative
infinity so
.
The graph is increasing for (−
,
.
).
The graph approaches c as x approaches infinity so
.
The graph is constant on (−
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,
).
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1-5 Parent Functions and Transformations
Use the graph of f(x) =
function.
to graph each
–4
SOLUTION:
7. g(x) =
g(x) = f(x + 6) − 4.
SOLUTION:
Therefore, g(x) is the graph of f(x) =
6 units left and 4 units down.
g(x) = f(x − 4).
Therefore, g(x) is the graph of f(x) =
4 units to the right.
9. g(x) =
translated
translated
10. g(x) =
+3
SOLUTION:
8. g(x) =
g(x) = f(x − 7) + 3.
Therefore, g(x) is the graph of f(x) =
7 units right and 3 units up.
SOLUTION:
g(x) = f(x + 3).
Therefore, g(x) is the graph of f(x) =
3 units to the left.
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translated
translated
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1-5 Parent Functions and Transformations
Use the graph of f(x) =
to graph each
function.
11. g(x) =
13. g(x) =
+8
SOLUTION:
+4
g(x) = f(x − 6) + 8.
SOLUTION:
Therefore, g(x) is the graph of f(x) =
g(x) = f(x) + 4.
units right and 8 units up.
Therefore, g(x) is the graph of f(x) =
translated 6
translated 4
units up.
14. g(x) =
–4
SOLUTION:
12. g(x) =
–6
g(x) = f(x + 7) − 4.
SOLUTION:
Therefore, g(x) is the graph of f(x) =
g(x) = f(x) − 6.
units left and 4 units down.
Therefore, g(x) is the graph of f(x) =
translated 7
translated 6
units down.
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1-5 Parent Functions and Transformations
Describe how the graphs of f(x) = [[x]] and g(x)
are related. Then write an equation for g(x).
16.
15.
SOLUTION:
There are several important characteristics for f(x)
= [[x]]. First determine if the graph increasing from
left to right. Identify if the graph has open dots on
the left or right. Determine the length of each
horizontal line. Also identify how far a horizontal
segment on the x-axis or y-axis is from the origin.
The graph of g(x) is increasing from left to right
which is the same as f(x) = [[x]].
The graph of g(x) has closed dots on the left and
open on the right which is the same as f(x) = [[x]].
Each horizontal line is 1 unit which is the same as
f(x) = [[x]].
The horizontal bar on the y-axis is shifted five units
down or the horizontal bar on the x-axis is shifted
five units to the right. Thus the graph of g(x) is the
SOLUTION:
There are several important characteristics for f(x)
= [[x]]. First determine if the graph increasing from
left to right. Identify if the graph has open dots on
the left or right. Determine the length of each
horizontal line. Also identify how far a horizontal
segment on the x-axis is from the origin.
The graph of g(x) is increasing from left to right
which is the same as f(x) = [[x]].
The graph of g(x) has closed dots on the left and
open on the right which is the same as f(x) = [[x]].
Each horizontal line is 1 unit which is the same as
f(x) = [[x]].
The horizontal bar on the x-axis is shifted three units
to the left. Thus, the graph of g(x) is the graph of
f(x) translated 3 units to the left when g(x) = [[x +
3]], or translated 3 units up when g(x) = [[x]] + 3.
graph of f(x) translated 5 units to the right when g(x)
= [[x – 5]], or translated 5 units down when g(x) =
[[x]] − 5.
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1-5 Parent Functions and Transformations
18.
17.
SOLUTION:
There are several important characteristics for f(x)
= [[x]]. First determine if the graph increasing from
left to right. Identify if the graph has open dots on
the left or right. Determine the length of each
horizontal line. Also identify how far a horizontal
segment on the x-axis or y-axis is from the origin.
SOLUTION:
There are several important characteristics for f(x)
= [[x]]. First determine if the graph increasing from
left to right. Identify if the graph has open dots on
the left or right. Determine the length of each
horizontal line. Also identify how far a horizontal
segment on the x-axis or y-axis is from the origin.
The graph of g(x) is decreasing from left to right
which is the opposite or a reflection of f(x) = [[x]].
The graph of g(x) is decreasing from left to right
which is the opposite so the graph g(x) is reflected
from f(x) = [[x]].
The graph of g(x) has closed dots on the left and
open on the right which is the opposite as f(x) =
[[x]].
Each horizontal line is 1 unit which is the same as
f(x) = [[x]].
The horizontal bar on the x-axis is shifted five units
to the right. Thus the graph of g(x) is the graph of
f(x) reflected in the y-axis and translated 5 units
right when g(x) = [[5 – x]], or reflected in the yaxis and translated 5 units up when g(x) = [[−x]] +
5.
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The graph of g(x) has open dots on the left and
closed on the right which is the opposite of f(x) =
[[x]].
Each horizontal line is 1 unit which is the same as
f(x) = [[x]].
The horizontal bar on the x-axis is shifted two units
to the left. The graph of g(x) is the graph of f(x)
reflected in the y-axis and translated 2 units to the
left when g(x) = [[−x – 2]], or reflected in the yaxis and translated 2 units down when g(x) = [[−x]]
− 2.
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1-5 Parent Functions and Transformations
19. PROFIT An automobile company experienced an
Describe how the graphs of f(x) = |x| and g(x)
are related. Then write an equation for g(x).
unexpected two-month delay on manufacturing of a
new car. The projected profit of the car sales before
the delay p(x) is shown below. Describe how the graph
of p(x) and the graph of a projection including the
delay d(x) are related. Then write an equation for d(x).
20.
SOLUTION:
SOLUTION:
Since there is a two-month delay, the graph of g(x) is
the graph of p(x) translated 2 units (months) to the
right. The equation for d(x) can be written by
replacing x with x − 2 in p(x). So, d(x) = 10(x – 2)3 –
70(x –
2)2 + 150(x
– 2) – 2.
The central characteristic of f(x) = x is the point
where the two lines meet. For our purposes here, it
can be considered as a vertex or a critical point. This
point is at (0, 0) for the parent function. Identifying
where it shifts will help you identify g(x). Note that
these are translations only. For reflections and
dilations, we will have to consider more aspects of
the graph.
The vertex is at (8, 0), so the vertex is translated 8
unit to the right. Therefore, the graph of f(x) is also
translated 8 unit to the right.
Now we need to identify an equation for g(x).
The x-coordinate tells us what changed inside the
absolute value symbols. Treat this like a zero for a
linear equation. If the coordinate is 8, the expression
inside the absolute value should be x − 8.
Thus, the graph of g(x) is the graph of f(x) translated
8 units to the right; g(x) = | x – 8 |.
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1-5 Parent Functions and Transformations
21.
22.
SOLUTION:
SOLUTION:
The central characteristic of f(x) = x is the point
where the two lines meet. For our purposes here, it
can be considered as a vertex or a critical point. This
point is at (0, 0) for the parent function. Identifying
where it shifts will help you identify g(x). Note that
these are translations only. For reflections and
dilations, we will have to consider more aspects of
the graph.
The central characteristic of f(x) = x is the point
where the two lines meet. For our purposes here, it
can be considered as a vertex or a critical point. This
point is at (0, 0) for the parent function. Identifying
where it shifts will help you identify g(x). Note that
these are translations only. For reflections and
dilations, we will have to consider more aspects of
the graph.
The vertex is at (0, −5), so the vertex is translated 5
units down. Therefore, the graph of f(x) is also
translated 5 units down.
The vertex is at (−4, −8), so the vertex is translated 4
unit to the left and 8 units down. Therefore, the graph
of f(x) is also translated 4 unit to the left and 8 units
down.
Now we need to identify an equation for g(x).
Now we need to identify an equation for g(x).
The y-coordinate tells us what was added outside of
the absolute value symbols. It describes the vertical
shift from the origin. The y-coordinate is −5, so we
need to subtract 5.
Thus, the graph of g(x) is the graph of f(x) translated
5 units down; g(x) = | x | − 5.
The x-coordinate tells us what changed inside the
absolute value symbols. Treat this like a zero for a
linear equation. If the coordinate is −4, the expression
inside the absolute value should be x + 4.
The y-coordinate tells us what was added outside of
the absolute value symbols. It describes the vertical
shift from the origin. The y-coordinate is −8, so we
need to subtract 8.
Thus, the graph of g(x) is the graph of f(x) translated
4 units to the left and 8 units down; g(x) = | x + 4 | −
8.
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1-5 Parent Functions and Transformations
Identify the parent function f(x) of g(x), and
describe how the graphs of g(x) and f(x) are
related. Then graph f(x) and g(x) on the same
axes.
24. g(x) = 3| x | – 4
SOLUTION:
23.
SOLUTION:
The central characteristic of f(x) = x is the point
where the two lines meet. For our purposes here, it
can be considered as a vertex or a critical point. This
point is at (0, 0) for the parent function. Identifying
where it shifts will help you identify g(x). Note that
these are translations only. For reflections and
dilations, we will have to consider more aspects of
the graph.
g(x) = 3f(x) − 4, so the graph of g(x) is the graph of
f(x) = |x| expanded vertically and translated 4 units
down. The expansion is represented by the
coefficient of 3 on the outside of f(x). The translation
down is represented by the subtraction of 4 on the
outside of f(x).
The vertex is at (1, −2), so the vertex is translated 1
unit to the right and 2 units down. Therefore, the
graph of f(x) is also translated 1 unit to the right and
2 units down.
Now we need to identify an equation for g(x).
25. g(x) = 3
SOLUTION:
The x-coordinate tells us what changed inside the
absolute value symbols. Treat this like a zero for a
linear equation. If the coordinate is 1, the expression
inside the absolute value should be x − 1.
The y-coordinate tells us what was added outside of
the absolute value symbols. It describes the vertical
shift from the origin. The y-coordinate is −2, so we
need to subtract 2.
g(x) = 3f(x + 8), the graph of g(x) is the graph of
f(x) translated 8 units to the left and expanded
vertically. The translation left is represented by the
addition of 8 on the inside of f(x). The expansion is
represented by the coefficient of 3 on the outside of
f(x).
g(x) = | x − 1 | − 2
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1-5 Parent Functions and Transformations
26. g(x) =
28. g(x) = −5[[x – 2]]
SOLUTION:
SOLUTION:
g(x) = 4f(x + 1), so the graph of g(x) is the graph of
f(x) translated 1 unit to the left and expanded
vertically. The translation left is represented by the
addition of 1 on the inside of f(x). The expansion is
represented by the coefficient of 4 on the outside of
f(x).
g(x) = −5f(x − 2), so the graph of g(x) is the graph of
f(x) translated 2 units to the right, expanded
vertically, and reflected in the x-axis. The translation
right is represented by the subtraction of 2 on the
inside of f(x). The expansion is represented by the
coefficient of 5 on the outside of f(x).The reflection
is represented by the negative coefficient on the
outside of f(x).
27. g(x) = 2[[x – 6]]
SOLUTION:
g(x) = 2f(x − 6), so the graph of g(x) is the graph of
f(x) translated 6 units to the right and expanded
vertically. The translation left is represented by the
subtraction of 6 on the inside of f(x). The expansion
is represented by the coefficient of 2 on the outside
of f(x).
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29. g(x) = −2| x + 5|
SOLUTION:
g(x) = −2f(x + 5), so g(x) is the graph of f(x)
translated 5 units to the left, expanded vertically, and
reflected in the x-axis. The translation left is
represented by the addition of 5 on the inside of f(x).
The expansion is represented by the coefficient of 2
on the outside of f(x). The reflection is represented
by the negative coefficient on the outside of f(x).
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1-5 Parent Functions and Transformations
30. g(x) =
Graph each function.
+7
32.
SOLUTION:
g(x) =
f(x) + 7, so g(x) is the graph of f(x)
compressed vertically and translated 7 units up. The
SOLUTION:
compression is represented by the coefficient of
on the outside of f(x). The translation up is
represented by the addition of 7 on the outside of
f(x).
On the interval [−∞, −2), graph y = −x2.
On the interval [−2, 7), graph y = 3.
On the interval [7, ∞), graph y = (x − 5)2.
Multiple points must be found for x = −2 and x = 7
because of the domain intervals.
31. g(x) =
SOLUTION:
g(x) =
f(x + 3), so g(x) is the graph of f(x)
translated 3 units to the left and compressed
vertically. The translation left is represented by the
addition of 3 on the inside of f(x). The compression is
represented by the coefficient of
Since f(−2) = 3 and f(7) = 6, draw dots at (−2, 3) and
(7, 6).
f(−2) ≠ −4 and f(7) ≠ 3, so the points (−2, −4) and (7,
3) are not included in the graph. Draw circles at
these points.
on the outside of
f(x).
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1-5 Parent Functions and Transformations
33.
34.
SOLUTION:
SOLUTION:
On the interval [−∞, −6), graph y = −x + 4.
On the interval [−6, 4), graph y = .
On the interval [4, ∞), graph y = 6.
On the interval [−∞, −5), graph y = 74.
On the interval [−2, 2], graph y = x3.
On the interval (3, ∞), graph y =
.
Multiple points must be found for x = −6 and x = 4
because of the domain intervals.
Since f(−2) = −8 and f(2) = 8, draw dots at (−2, −8)
and (2, 8).
Since f(−6) =
f(−5) ≠ 4 and f(3) ≠
and f(4) = 6, draw dots at
and (4, 6).
, so the points (−5, 4) and (3,
) are not included in the graph. Draw circles at
these points.
f(−6) ≠ −2 and f(4) ≠ , so the points (−6, −2) and
are not included in the graph. Draw circles at
these points.
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1-5 Parent Functions and Transformations
36.
35.
SOLUTION:
SOLUTION:
On the interval (−∞, −4), graph y = 2.
On the interval (−∞, −3), graph y = |x − 5|.
On the interval [−1, 3), graph y = 4x − 3.
On the interval [4, ∞), graph y =
.
Since f(−1) = −7 and f(4) = 2, draw dots at (−1, −7)
and (4, 2).
f(−3) ≠ −8 and f(3) ≠ 9, so the points (−3, −8) and (3,
9) are not included in the graph. Draw circles at
these points.
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On the interval [−1, 1), graph y = x4 − 3x2 + 5.
On the interval [3, ∞), graph y =
.
Since f(−1) = 9 and f(3) = 4, draw dots at (−2, 3) and
(7, 6).
f(−4) ≠ −2 and f(1) ≠ 3, so the points (−4, −2) and (1,
3) are not included in the graph. Draw circles at
these points.
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1-5 Parent Functions and Transformations
left end of each segment.
Plot each of the following segments.
37.
SOLUTION:
On the interval (−∞, −1), graph y = −3x − 1.
On the interval (−1, 3], graph y = 0.5x + 5.
On the interval (3, ∞), graph y = −|x − 5| + 3.
x-coordinates
dot
circle
0
3
3
7
7
11
11
13
13
14
14
18
18
19
19
20
20
y-coordinates
0.25
0.29
0.32
0.33
0.34
0.37
0.39
0.41
0.42
Multiple points must be found for x = −1 and x = 3
because of the domain intervals.
Since f(−1) = −2 and f(3) = 6.5, draw dots at (−1, −2)
and (3, 6.5).
f(−1) ≠ 4.5 and f(3) ≠ 1, so the points (−1, 4.5) and
(3, 1) are not included in the graph. Draw circles at
these points.
38. POSTAGE The cost of a first-class postage stamp
in the U.S. from 1988 to 2008 is shown in the table
below. Use the data to graph a step function.
SOLUTION:
Let x = 0 represent 1988. The price changes at the
beginning of the year, so the dots will appear on the
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39. BUSINESS A no-contract cell phone company
charges a flat rate for daily access and $0.10 for
each minute. The cost of the plan can be modeled by
c(x) = 0.1[[x]] + 1.99, where x is the number of
minutes used.
a. Describe the transformation(s) of the parent
function f(x) = [[x]] used to graph c(x).
b. The company offers another plan in which the
daily access rate is $2.49, and the per-minute rate is
$0.05. What function d(x) can be used to describe
the second plan?
c. Graph both functions on the same graphing
calculator screen.
d. Would the cost of the plans ever equal each other?
If so, at how many minutes?
SOLUTION:
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1-5 Parent Functions and Transformations
a. c(x) = a × f(x) + b. The value of a is less than 1,
so the graph of f(x) compressed vertically. The value
of b is 1.99, so the graph of f(x) is translated 1.99
units up.
horizontal distance in feet such that x = 0
corresponds to the initial point.
b. The per-minute rate is the value of a and the
access rate is b in c(x) = a × f(x) + b. Therefore,
d(x) = 0.05[[x]] + 2.49.
c.
a. Describe the transformation(s) of the parent
d. Yes; the plans will equal each other at 10 minutes.
Use the Intersect function of the calculator to find
the intersection of the graphs. Notice that the
intersection is the segment from x = 10 to x = 11.
function f(x) = x2 used to graph g(x).
b. If a second golfer hits a similar shot 30 feet
farther down the fairway from the first player, what
function h(x) can be used to describe the second
golfer’s shot?
c. Graph both golfers’ shots on the same graphing
calculator screen.
d. At what horizontal and vertical distances do the
paths of the two shots cross each other?
SOLUTION:
a. Rewrite the function in the form f(x) = a(x − h)2
+ k.
0.176x – 0.0004x2
= –0.0004x2 + 0.176x
= –0.0004(x2 – 440x)
[0, 20] scl: 2 by [0, 5] scl: 0.5
Creating a table of values will also confirm when
c(x) = d(x).
= –0.0004(x2 – 440x + 48,400) – (–0.0004)(48,400)
g(x) = −0.0004(x − 220)2 + 19.36
So, the graph of g(x) is the graph of f(x) translated
220 units to the right, compressed vertically, reflected
in the x-axis, and translated 19.36 units up.
b. This is a shift right, so h(x) = g(x − 30) or h(x) =
−0.0004(x − 250)2 + 19.36.
c.
40. GOLF The path of a drive can be modeled by the
function shown, where g(x) is the vertical distance in
feet of the ball from the ground and x is the
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1-5 Parent Functions and Transformations
Use the graph of f(x) to graph g(x) = |f(x)| and
h(x) = f(|x|).
41. f(x) =
SOLUTION:
|f(x)| replaces all of the negative y-values with the
d. Use the Intersectionfeature of your calculator.
corresponding positive y-values. If f(–3) =
, then
|f(–3)| = .
To graph g(x) = |f(x)|, reflect the range with respect
to the x-axis for all elements of the domain where
f(x) is less than zero.
f(|x|) replaces all of the negative x-values with the
The shots will cross paths at a horizontal distance of
235 feet and a vertical distance of 19.27 feet.
corresponding positive x-values. If f(–3) =
, then
f(|–3|) = .
To graph h(x) = f(|x|), replace the range for x < 0
with a reflection of the range for x > 0 with respect
to the y-axis.
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1-5 Parent Functions and Transformations
42. f(x) =
SOLUTION:
|f(x)| replaces all of the negative y-values with the
corresponding positive y-values.If f(–6) = undefined ,
then f(|–6|) = undefined.
|f(x)| replaces all of the negative y-values with the
corresponding positive y-values. If f(–3) =
, then |f(–3)| =
.
To graph g(x) = |f(x)|, reflect the range with respect
to the x-axis for all elements of the domain where
f(x) is less than zero.
f(|x|) replaces all of the negative x-values with the
corresponding positive x-values. If f(–6) = undefined
, then f(|–6|) =
.
To graph h(x) = f(|x|), replace the range for x < 0
with a reflection of the range for x > 0 with respect
to the y-axis.
To graph g(x) = |f(x)|, reflect the range with respect
to the x-axis for all elements of the domain where
f(x) is less than zero.
f(|x|) replaces all of the negative x-values with the
corresponding positive x-values. If f(–2) =
, then f(|–3|) =.
To graph h(x) = f(|x|), replace the range for x < 0
with a reflection of the range for x > 0 with respect
to the y-axis.
43. f(x) = x4 – x3 – 4x2
SOLUTION:
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1-5 Parent Functions and Transformations
44. f(x) =
x3 + 2x2 – 8x – 2
SOLUTION:
|f(x)| replaces all of the negative y-values with the
corresponding positive y-values. If f(–7) =
, then
.
|f(–3)| =
To graph g(x) = |f(x)|, reflect the range with respect
to the x-axis for all elements of the domain where
f(x) is less than zero.
f(|x|) replaces all of the negative x-values with the
corresponding positive x-values. If f(–3) =
, then
f(|–3|) =
.
To graph h(x) = f(|x|), replace the range for x < 0
with a reflection of the range for x > 0 with respect
to the y-axis.
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1-5 Parent Functions and Transformations
45. f(x) =
+5
46. f(x) =
–6
SOLUTION:
SOLUTION:
|f(x)| replaces all of the negative y-values with the
corresponding positive y-values. If f(–4) =
, then |f(–4)| =
.
|f(x)| replaces all of the negative y-values with the
corresponding positive y-values. If f(–1) =
, then |f(–1)| =
.
To graph g(x) = |f(x)|, reflect the range with respect
to the x-axis for all elements of the domain where
f(x) is less than zero.
To graph g(x) = |f(x)|, reflect the range with respect
to the x-axis for all elements of the domain where
f(x) is less than zero.
f(|x|) replaces all of the negative x-values with the
corresponding positive x-values. If f(–4) =
, then f(|–4|) =
.
f(|x|) replaces all of the negative x-values with the
corresponding positive x-values. If f(–1) =
, then f(|–1|) =
To graph h(x) = f(|x|), replace the range for x < 0
with a reflection of the range for x > 0 with respect
to the y-axis.
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.
To graph h(x) = f(|x|), replace the range for x < 0
with a reflection of the range for x > 0 with respect
to the y-axis.
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1-5 Parent Functions and Transformations
47. TRANSPORTATION In New York City, the
standard cost for taxi fare is shown. One unit is equal
to a distance of 0.2 mile or a time of 60 seconds,
when the car is not in motion.
a. Write a greatest integer function f(x) that would
represent the cost for x units of cab fare, where x >
0. Round to the nearest unit.
b. Graph the function.
c. How would the graph of f(x) change if the fare for
the first unit increased to $3.70 while the cost per
unit remained at $0.40? Graph the new function.
SOLUTION:
a. When there is only a fraction of a unit, we must
round up. For example, if 3.4 units are used, the
customer will be charged for 4 units. To accomplish
this, use [[x + 1]] when x is not a whole number.
b.
c. If the fare for one unit increased to $3.70, and the
cost per unit was still $0.40, then the cost per trip
must have increased to $3.30. This will cause a
vertical translation of the graph of $0.80. The graph
of f(x) is translated 0.8 unit up.
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1-5 Parent Functions and Transformations
48. PHYSICS The potential energy in joules of a spring
that has been stretched or compressed is given by
p(x) =
, where c is the spring constant
and x is the distance from equilibrium. When x is
negative, the spring is compressed, and when x is
positive, the spring is stretched.
Write and graph the function with the given
parent function and characteristics.
49. f(x) =
; expanded vertically by a factor of 2,
translated 7 units to the left and 5 units up
SOLUTION:
g(x) =
+5
a. Describe the transformation(s) of the parent
function f(x) = x2 used to graph p(x) = 4.5x2.
b. The graph of the potential energy for a second
spring passes through the point (3, 315). Find the
spring constant for the spring and write the function
for the potential energy.
The shift 5 units up is represented by an addition of 5
after f(x), or f(x) + 5. The shift 7 units left is
represented by the addition of 7 inside f(x), or f(x +
7). The vertical expansion by a factor of 2 is
represented by the coefficient 2 outside f(x), or 2f(x).
Therefore, g(x) = 2f(x + 7) + 5.
SOLUTION:
a. p(x) = cf(x), so the parent function is multiplied by
a constant and the transformation is a vertical
expansion.
b. Use (3, 315) to solve for c in p(x).
50. f(x) = [[x]]; expanded vertically by a factor of 3;
reflected in the x-axis; translated 4 units down
SOLUTION:
g(x) = −3[[x]] – 4
p(x) = 35x2
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The reflection in the x-axis is represented by
negative coefficient in front of f(x), or −f(x). The
vertical expansion by a factor of 3 is represented by
the coefficient of 3 in front of f(x), of 3f(x). The shift
4 units down is represented by the subtraction of 4
after f(x), or f(x) − 4. Therefore, g(x) = −3f(x) − 4.
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1-5 Parent Functions and Transformations
PHYSICS The distance an object travels as a
function of time is given by f(t) =
at2 + v0t +
53. a = 4, v0 = 8, x0 = 1
SOLUTION:
x 0, where a is the acceleration, v0 is the initial
velocity, and x 0 is the initial position of the
object. Describe the transformation(s) of the
parent function f(t) = t2 used to graph f(t) for
each of the following.
51. a = 2, v0 = 2, x0 = 0
SOLUTION:
Substitute the values then complete the square to
identify the transformations.
translated 2 units to the left; expanded vertically;
translated 7 units down
54. a = 3, v0 = 5, x0 = 3
SOLUTION:
translated one unit left; translated one unit down
52. a = 2, v0 = 0, x0 = 10
SOLUTION:
Substitute the values to identify the transformations.
translated
units left; expanded vertically;
translated
units down
translated 10 units up
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1-5 Parent Functions and Transformations
Write an equation for each g(x).
55.
56.
SOLUTION:
SOLUTION:
The parent function is f(x) =
. The graph of g(x)
appears to be f(x) shifted 4 units up and 3 units to the
right. Therefore, we have
.
Use (4, 6) to determine if there is a dilation.
The parent function is f(x) = x3. Reference the
critical point of the parent graph at (0, 0). The graph
of g(x) appears to have this point at (5, –8). The
graph also appears to be reflected. So, the graph of
g(x) appears to be f(x) shifted 5 units to the left,
shifted 8 units down, and then reflected in the y-axis.
The reflection causes the shift left to appear to be a
shift right in the graph. Therefore, we have g(x) = −
(x − 5)3 − 8. Use (3, −4) to determine if there is a
dilation.
There is a dilation of 2, so g(x) =
+ 4.
The dilation is 0.5, so g(x) = −0.5(x – 5)3 – 8.
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1-5 Parent Functions and Transformations
57.
58.
SOLUTION:
SOLUTION:
The parent function is f(x) =
. Use the initial
point of the parent graph at (0, 0) to estimate the
translation.The graph of g(x) appears to be f(x)
shifted 4 units left and 6 units down.
The parent function is f(x) = [[x]]. With a greatest
integer function, more than one transformation can
lead to the same graph of g(x). One possibility is a
dilation with no shift at all. Let a represent the
dilation and use (4, 1), a point on g(x), to determine
a.
There is also an obvious dilation. Let a represent the
dilation and use (0, 2) to solve for a.
There is no reflection, so a = 4 and g(x) = 4
– 6.
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Therefore, one possible function is g(x) =
[[x]].
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1-5 Parent Functions and Transformations
59. SHOPPING The management of a new shopping
Identify the parent function f(x) of g(x), and
describe the transformation of f(x) used to
graph g(x).
mall originally predicted that attendance in thousands
would follow f(x) =
for the first 60 days of
operation, where x is the number of days after
opening and x = 1 corresponds with opening day.
Write g(x) in terms of f(x) for each situation below.
a. Attendance was consistently 12% higher than
expected.
b. The opening was delayed 30 days due to
construction.
c. Attendance was consistently 450 less than
expected.
60.
SOLUTION:
SOLUTION:
The graph is a quadratic, so the parent function is
a. A consistent percentage change is represented by
a dilation, or a coefficient in front of f(x). Therefore,
g(x) = 1.12f(x).
f(x) = x2. The graph is reflected in the x-axis
because it resembles an upside-down version of the
parent graph.
b. There is no affect on f(x). While the opening is
delayed, the number of days after the opening, which
determines the domain of the function, is unaffected.
Use like points to gauge the translation. In the parent
graph, the minimum is located at x = 0. The
maximum of the graph is the minimum of the parent
graph translated down 3 units and left 4 units.
c. To represent a consistently less value, subtract the
difference from f(x). Therefore, g(x) = f(x) − 0.45.
So far, we have g(x) = −a(x + 4)2 − 3 where a
represents the unknown compression or expansion.
Use the given point to identify the value of a.
Substituting 0.5 for a, g(x) = −0.5(x + 4)2 − 3, which
means that the parent graph was compressed
vertically.
Therefore, the graph of g(x) is the graph of f(x)
translated 4 units to the left, compressed vertically,
reflected in the x-axis, and translated 3 units down.
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1-5 Parent Functions and Transformations
62.
61.
SOLUTION:
The parent function is f(x) = x3. The graph is
reflected in the x-axis because it resembles an
upside-down version of the parent graph.
Use like points to gauge the translation. In the parent
graph, the point of inflection, or the point where the
graph appears to curve in at the middle is located at x
= 0. The point of inflection of g(x) is 2 units up and 4
units to the right.
So far, we have g(x) = −a(x − 4)3 + 2 where a
represents the unknown compression or expansion.
Use the given point to identify the value of a.
SOLUTION:
The parent function is f(x) =
.
Use like points to gauge the translation. In the parent
graph, the vertical asymptote is located at x = 0. This
is unchanged in g(x), so there is no horizontal
translation. In the parent function, the horizontal
asymptote is located at y = 0 and in g(x) it is located
at y = −6, so there is a translation down of 6 units.
The graph is also expanded vertically by an unknown
factor.
So far, we have g(x) =
− 6 where a represents
the unknown expansion. Use the given point to
identify the value of a.
Substituting 3 for a, g(x) = −3(x − 4)3 + 2, which
means that the parent graph was expanded vertically.
Therefore, the graph of g(x) is the graph of f(x)
translated 4 units to the right, expanded vertically,
reflected in the x-axis, and translated 2 units up..
Substituting 4 for a, g(x) =
− 6 , which means that
the parent graph was expanded vertically.
Therefore, the graph of g(x) is the graph of f(x)
expanded vertically and translated 6 units down.
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1-5 Parent Functions and Transformations
Use f(x) to graph g(x).
63.
SOLUTION:
The parent function is f(x) =
. The graph is
reflected in the x-axis because it resembles an
upside-down version of the parent graph.
Use like points to gauge the translation. In the parent
graph, the point where the graph begins is located at
x = 0. In g(x), this point located 3 units to the right
and 5 units up.
64. g(x) = 0.25f(x) + 4
SOLUTION:
f(x) is dilated by a factor of 0.25 and then translated
up 4. Do this to each piece of the graph. Remember
that in the graph of f(x), (x, y) = (x, f(x)). Therefore,
in the graph of g(x), (x, y) = (x, g(x)) or (x, 0.25f(x)
+ 4).
1st line
So far, we have g(x) = −a
+ 5 where a
represents the unknown compression or expansion.
Use the given point to identify the value of a.
2nd line
Ray
(x, f(x))
(−6, 6)
to
(−2, −4)
(−2, 1)
to
(6, −1)
(6, 4)
(x, g(x))
(−6, 5.5)
to
(−2, 3)
(−2, 4.25)
to
(6, 3.75)
(6, 5)
Therefore, the graph of g(x) is the graph of f(x)
translated 3 units to the right, reflected in the x-axis,
and translated 5 units up.
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1-5 Parent Functions and Transformations
65. g(x) = 3f(x) – 6
66. g(x) = f(x – 5) + 3
SOLUTION:
SOLUTION:
f(x) is dilated by a factor of 3 and then translated
down 6. Do this to each piece of the graph.
Remember that in the graph of f(x), (x, y) = (x, f(x)).
Therefore, in the graph of g(x), (x, y) = (x, g(x)) or
(x, 3f(x) − 6).
f(x) is 5 shifted units right and 3 units up. Do this to
each piece of the graph. In this graph, the x- and ycoordinates are both shifted. For each segment, add
5 to the x-coordinate and add 3 to the y-coordinate.
1st line
2nd
line
Ray
(x, f(x))
(−6, 6)
to
(−2, −4)
(−2, 1)
to
(6, −1)
(6, 4)
(x, g(x))
(−6, 12)
to
(−2, −18)
(−2, −3)
to
(6, −9)
(6, 6)
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1st line
2nd line
Ray
(x, f(x))
(−6, 6)
to
(−2, −4)
(−2, 1)
to
(6, −1)
(6, 4)
(x, g(x))
(−1, 9)
to
(3, −1)
(3, 4)
to
(11, 2)
(11, 7)
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1-5 Parent Functions and Transformations
67. g(x) = −2f(x) + 1
Use f(x) =
SOLUTION:
68. g(x) = 2f(x) + 5
f(x) is dilated by a factor of −2 and then translated up
1. Do this to each piece of the graph. Remember that
in the graph of f(x), (x, y) = (x, f(x)). Therefore, in
the graph of g(x), (x, y) = (x, g(x)) or (x, −2f(x) + 1).
1st
2nd
Ray
(x, f(x))
(−6, 6)
to
(−2, −4)
(−2, 1)
to
(6, −1)
(6, 4)
– 4 to graph each function.
SOLUTION:
(x, g(x))
(−6,
−11)
to
(−2, 9)
(−2, −1)
to
(6, 3)
(6, −7)
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1-5 Parent Functions and Transformations
69. g(x) = −3f(x) + 6
71. g(x) = f(2x + 1) + 8
SOLUTION:
SOLUTION:
70. g(x) = f(4x) – 5
SOLUTION:
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1-5 Parent Functions and Transformations
72. MULTIPLE REPRESENTATIONS In this
problem, you will investigate operations with
functions. Consider f(x) =
x2
+ 2x + 7, g(x) = 4x + 3,
and h(x) = x2 + 6x + 10.
a. TABULAR Copy and complete the table below
for three values for a.
73. ERROR ANALYSIS Danielle and Miranda are
describing the transformation g(x) = [[x + 4]].
Danielle says that the graph is shifted 4 units to the
left, while Miranda says that the graph is shifted 4
units up. Is either of them correct? Explain.
SOLUTION:
Sample answer: Both; the greatest integer function is
the step function in which a shift of a units left also
shifts the graph a units up.
b. VERBAL How are f(x), g(x), and h(x) related?
c. ALGEBRAIC Prove the relationship from part b
algebraically.
SOLUTION:
a.
So, a shift of a units left is identical to a shift of a
units up.
b. The values for h(a) and f(a) + g(a) are equal for
each value of a, so h(x) may be the sum of f(x) and
g(x).
c. We think h(x) = f(x) + g(x), so we need to set
these equal to each other and simplify until we get
congruent expressions on each side of the equal sign.
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1-5 Parent Functions and Transformations
74. REASONING Let f(x) be an odd function. If g(x) is
75. Writing in Math Explain why order is important
a reflection of f(x) in the x-axis and h(x) is a
reflection of g(x) in the y-axis, what is the
relationship between f(x) and h(x)? Explain.
when transforming a function with reflections and
translations.
SOLUTION:
Sample answer: Order is important because different
graphs can be obtained depending on the order the
transformations are performed. For example, if (a, b)
is on the original graph and there is a translation 6
units up and then a reflection in the x-axis, the
resulting point will be (a, −b – 6).
f(x) and g(x) represent the same function; If f(x) =
x3, an odd function, then g(x) = −x3, a reflection of
f(x) in the x-axis. Likewise, h(x) = −(−x3) or x3, a
reflection of g(x) in the y-axis. Therefore, f(x) =
h(x).
SOLUTION:
f(x) = x3
However, if (a, b) is reflected in the x-axis first and
then translated 6 units up, the resulting point will be
(a, −b + 6).
g(x) = −f(x) = −x3
h(x) = g(−x) = −f(−x) = −(−x)3 = x3
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1-5 Parent Functions and Transformations
REASONING Determine whether the following
statements are sometimes, always, or never true.
Explain your reasoning.
76. If f(x) is an even function, then f(x) = |f(x)|.
SOLUTION:
x2. All of the
Consider the basic even function, f(x) =
values of f(x) are positive, so |f(x)| = f(x) for all
values of x.
77. If f(x) is an odd function, then f(−x) = −|f(x)|.
SOLUTION:
Consider the odd function f(x) = x3.
f(–x) = (–x)3 = –x3
–|f(x)| = –|x3|
Notice from the graphs of these functions that they
are not equal.
Now consider an even function in which negative
values are a part of the output. Translate the graph
down 4 units and we get g(x) = x2 − 4. This function
is still even because g(x) = g(−x).
However, there are some values where g(x) is
negative. |g(x)| will only have positive values, so g(x)
≠ |g(x)|.
However, f(x) = 0 is also an odd function and f(–x) =
–| f(x)| for all x, so the statement is sometimes true.
78. If f(x) is an even function, then f(−x) = −|f(x)|.
SOLUTION:
Consider the basic even function, f(x) = x2. All of the
values are positive, so all of the values of f(–x) will
also be positive. This is becaue the square eliminates
the negatives
f(–x) = (–x)2 = x2
–|f(x)| = –|x2| = –x2
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1-5 Parent Functions and Transformations
79. CHALLENGE Describe the transformation of f(x)
=
if (−2, −6) lies on the curve.
SOLUTION:
Sample answer: There are many different curves that
can include (–2, –6). One example would be to simply
translate the endpoint of the parent function. It is
normally at (0, 0), so the translation will be 2 units left
and 6 units down.
From the graph, –|f(x)| ≠ f(–x).
g(x) = f(x + 2) – 6 =
Now consider an even function in which some output
values are negative, like f(x) = –x2.
f(–x) = –(–x)2 = –x2
–|f(x)| = –|–x2| = –x2
From the graph, –|f(x)| = f(–x).
Thus, sometimes when f(x) is an even function, then
f(−x) = −|f(x). f(x) = –x2 is an even function and
f(–x) = –| f(x)| for all x. However, f(x) = x2 is an
even function and f(–x) ≠ –| f(x)|.
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1-5 Parent Functions and Transformations
80. REASONING Suppose (a, b) is a point on the
graph of f(x). Describe the difference between the
transformations of (a, b) when the graph of f(x) is
expanded vertically by a factor of 4 and when the
graph of f(x) is compressed horizontally by a factor
of 4.
SOLUTION:
Sample answer: A vertical expansion of f(x) by a
factor of 4 would move (a, b) to (a, 4b). A horizontal
compression by a factor of 4 would move (a, b) to
Find the average rate of change of each function
on the given interval.
82. g(x) = −2x2 + x – 3; [−1, 3]
SOLUTION:
g(x) = −2x2 + x – 3; [−1, 3]
g(3) = −2(3)2 + (3) – 3
g(3) = −18 + 3 − 3 = −18
g(−1) = −2(−1)2 + (−1) – 3
g(−1) = −2 − 1 − 3 = −6
.
For example, if f(x) = x2, then a vertical expansion by
a factor of 4 would be g(x) = 4x2. A horizontal
compression by a factor of 4 would be h(x) = (4x)2.
Let (a, b) = (2, 4). Then in the vertical expansion, (2,
4) would move to (2, 16). In the horizontal
compression, (2, 4) moves to
.
83. g(x) = x2 – 6x + 1; [4, 8]
SOLUTION:
g(x) = x2 – 6x + 1; [4, 8]
g(8) = (8)2 – 6(8) + 1
g(8) = 64 – 48 + 1
g(8) = 17
g(4) = (4)2 – 6(4) + 1
g(4) = 16 – 24 + 1
g(4) = –7
81. Writing in Math Use words, graphs, tables, and
equations to relate parent functions and
transformations. Show this relationship through a
specific example.
SOLUTION:
See students’ work.
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1-5 Parent Functions and Transformations
84. f(x) = −2x3 – x2 + x – 4; [−2, 3]
SOLUTION:
f(x) = −2x3 – x2 + x – 4; [−2, 3]
f(3) = −2(3)3 – (3)2 + (3) – 4
f(3) = –54 – 9 + 3 – 4
f(3) = –64
Use the graph of each function to describe its
end behavior. Support the conjecture
numerically.
85. q(x) = –
SOLUTION:
f(−2) = −2(−2)3 – (−2)2 + (−2) – 4
f(–2) = 16 – 4 – 2 – 4
f(–2) = 6
x
–100
–10
–5
5
10
100
q(x)
0.12
1.2
2.4
–2.4
–1.2
–0.12
As x → ∞, the denominator of the fraction will
increase and the value of the fraction will approach
0, so q(x) will approach 0.
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1-5 Parent Functions and Transformations
86. f(x) =
87. p(x) =
SOLUTION:
x
–25
–10
–5
5
10
25
SOLUTION:
f(x)
0.0008
0.005
0.02
0.02
0.005
0.0008
As x → ∞, the denominator of the fraction will
increase and the value of the fraction will approach
0, so f(x) will approach 0.
x
–1000
–100
–10
–5
5
10
100
1000
p(x)
0.99501
0.95146
0.82143
0.375
0.5
1.71429
1.05155
1.00502
As x → ∞, the fraction will get closer and closer to
, so p(x) will approach 1.
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1-5 Parent Functions and Transformations
Use the graph of each function to estimate its yintercept and zero(s). Then find these values
algebraically.
89.
SOLUTION:
88.
SOLUTION:
From the graph, it appears that f(x) will intersect the
y-axis at (0, 0). Find f(0).
From the graph, it appears that f(x) will intersect the
y-axis at about (0, 12). Find f(0).
Because f(0) = 0, there is a y-intercept at (0, 0).
Because f(0) = 13, there is a y-intercept at (0, 13).
From the graph, it appears that there is an x-intercept
near x = −1, x = 0, and x = 2. Let f(x) = 0 and solve
for x.
From the graph, it appears that there is an x-intercept
near x = 2 and x = 6. Let f(x) = 0 and solve for x.
Therefore, the zeros of f are 0, 2, and −1.
Therefore, the zeros of f are about 5.73 and 2.27.
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1-5 Parent Functions and Transformations
92. LOTTERIES In a multi-state lottery, the player
must guess which five of the white balls numbered
from 1 to 49 will be drawn. The order in which the
balls are drawn does not matter. The player must
also guess which one of the red balls numbered from
1 to 42 will be drawn. How many ways can the
player complete a lottery ticket?
90.
SOLUTION:
SOLUTION:
From the graph, it appears that there is no y-intercept
for the graph of f(x). Find f(0).
For the 5 white balls:
Order does not matter, so the number of possibilities
can be represented by a combination. There are 49
possible choices and we are choosing 5, so we use
49C5.
Because f(0) = is undefined, there is no y-intercept.
For the red ball, there are 42 options. Therefore, the
total number of possibilities is 49C5 · 42 = 80,089,128.
From the graph, it appears that there is an x-intercept
near x = 3. Let f(x) = 0 and solve for x.
no y-intercept; zero: 3;
93. SAT/ACT The figure shows the graph of y = g(x),
which has a minimum located at (1,−2). What is the
maximum value of the function h(x) = −3g(x) − 1?
Therefore, the zero of f is 3.
91. GOVERNMENT The number of times each of the
first 42 presidents vetoed bills are listed below. What
is the standard deviation of the data?
2, 0, 0, 7, 1, 0, 12, 1, 0, 10, 3, 0, 0, 9, 7, 6, 29, 93, 13, 0,
12, 414, 44, 170, 42, 82, 39, 44, 6, 50, 37, 635, 250,
181, 21, 30, 43, 66, 31, 78, 44, 25
SOLUTION:
Enter the data in your calculator and find the
standard deviation of the population of data.
A6
B5
C3
D2
E It can not be determined from the information
given
SOLUTION:
The minimum of g(x) is –2. h(x) is a transformation
of g(x). In order to obtain the critical point of h(x)
that corresponds with the minimum of g(x), multiply
this value by –3 and then subtract 1.
–2(–3) – 1 = 5
The standard deviation is about 118.60.
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1-5 Parent Functions and Transformations
96. REVIEW What is the effect on the graph of y = k x2
94. REVIEW What is the simplified form of
as k decreases from 3 to 2?
?
A The graph of y = 2x2 is a reflection of the graph of
y = 3x2 in the y-axis.
B The graph is rotated 90° about the origin.
C The graph becomes narrower.
D The graph becomes wider.
SOLUTION:
SOLUTION:
As k decreases from 3 to 2, the graph of y = k x2
becomes wider.
95. What is the range of y =
F {y | y ≠ ±2
G {y | y ≥ 4}
H {y | y ≥ 0}
J {y | y ≤ 0}
?
}
The correct choice is D.
SOLUTION:
The smallest possible value for x2 is 0. Therefore, the
smallest possible value of
is
= 4. As
x approaches positive or negative infinity, x2
approaches positive infinity. Therefore, the value of
also approaches infinity. The correct
choice is G.
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