The graph approaches zero as x approaches positive
infinity so
= 0.
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.
The graph is decreasing on (−
, 0) and (0,
).
5. f (x) = c
SOLUTION:
2. f (x) =
SOLUTION:
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
}.
If c = 0, all real numbers are x-intercepts. If c ≠ 0,
there are no x-intercepts.
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 intersects the y-axis at (0, c), so there is
a y-intercept at (0, c).
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 has mirror images over the origin, thus he
graph is symmetric with respect to the origin.
The graph approaches c as x approaches negative
infinity so
.
The graph has an infinite discontinuity at x = 0.
The graph approaches zero as x approaches
negative infinity so
= 0.
The graph approaches c as x approaches infinity so
.
The graph approaches zero as x approaches positive
infinity so
= 0.
The graph is constant on (−
The graph is decreasing on (−
5. f (x) = c
SOLUTION:
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, 0) and (0,
).
Use the graph of f (x) =
function.
8. g(x) =
,
).
to graph each
SOLUTION:
g(x) = f (x + 3).
Therefore, g(x) is the graph of f (x) =
3 units to the left.
translated
Page 1
The graph approaches c as x approaches infinity so
.
1-5 Parent Functions and Transformations
The graph is constant on (−
,
Use the graph of f (x) =
function.
8. g(x) =
).
to graph each
14. g(x) =
SOLUTION:
SOLUTION:
g(x) = f (x + 7) − 4.
g(x) = f (x + 3).
Therefore, g(x) is the graph of f (x) =
3 units to the left.
Use the graph of f (x) =
–4
Therefore, g(x) is the graph of f (x) =
translated
translated 7
units left and 4 units down.
Describe how the graphs of f (x) = [[x]] and g(x)
are related. Then write an equation for g(x).
to graph each
function.
11. g(x) =
+4
SOLUTION:
g(x) = f (x) + 4.
Therefore, g(x) is the graph of f (x) =
translated 4
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.
units up.
The graph of g(x) is decreasing from left to right
which is the opposite or a reflection of f (x) = [[x]].
14. g(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]].
–4
SOLUTION:
g(x) = f (x + 7) − 4.
Therefore, g(x) is the graph of f (x) =
translated 7
Each horizontal line is 1 unit which is the same as f
(x) = [[x]].
units left and 4 units down.
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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
Page 2
when g(x) = [[5 – x]], or reflected in the y -axis and
translated 5 units up when g(x) = [[−x]] + 5.
(x) reflected in the y -axis and translated 5 units right
when g(x) = [[5 – x]], or reflected in the y -axis and
translated 5 units up when g(x) = [[−x]] + 5.
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).
Describe how the graphs of f (x) = [[x]] and g(x)
are related. Then write an equation for g(x).
20.
17.
SOLUTION:
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 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 graph of g(x) is decreasing from left to right
which is the opposite or a reflection of f (x) = [[x]].
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.
The graph of g(x) has closed dots on the left and
open on the right which is the opposite as f (x) =
[[x]].
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.
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 y -axis and
translated 5 units up when g(x) = [[−x]] + 5.
Thus, the graph of g(x) is the graph of f (x) translated
8 units to the right; g(x) = | x – 8 |.
Describe how the graphs of f (x) = |x| and g(x)
are related. Then write an equation for g(x).
23.
SOLUTION:
20.
SOLUTION:
The central characteristic of f (x) = x is the point
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
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where
the -two
linesbymeet.
For
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
Page 3
the graph.
The vertex is at (1, −2), so the vertex is translated 1
linear equation. If the coordinate is 8, the expression
inside the absolute value should be x − 8.
the absolute value symbols. It describes the vertical
shift from the origin. The y-coordinate is −2, so we
need to subtract 2.
Thus, the graph of g(x) is the graph of f (x) translated
1-5 8Parent
Functions
units to the
right; g(x) =and
| x – Transformations
8 |.
g(x) = | x − 1 | − 2
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.
26. g(x) =
23.
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).
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 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).
29. g(x) = −2| 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 1, the expression
inside the absolute value should be x − 1.
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).
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) = | x − 1 | − 2
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.
26. g(x) =
Graph each function.
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).
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32.
SOLUTION:
Page 4
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.
1-5 Parent Functions and Transformations
Graph each function.
35.
32.
SOLUTION:
SOLUTION:
2
On the interval [−∞, −2), graph y = −x .
On the interval [−2, 7), graph y = 3.
2
On the interval [7, ∞), graph y = (x − 5) .
Multiple points must be found for x = −2 and x = 7
because of the domain intervals.
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 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.
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.
35.
SOLUTION:
SOLUTION:
Let x = 0 represent 1988. The price changes at the
beginning of the year, so the dots will appear on the
left end of each segment.
Plot each of the following segments.
On the interval (−∞, −3), graph y = |x − 5|.
On the interval [−1, 3), graph y = 4x − 3.
On the
interval
[4, ∞),
graph y
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=
.
Since f (−1) = −7 and f (4) = 2, draw dots at (−1, −7)
x-coordinates
dot
circle
0
3
3
7
7
11
y-coordinates
0.25
0.29
0.32
Page 5
and (4, 2).
1-5
f(−3) ≠ −8 and f (3) ≠ 9, so the points (−3, −8) and (3,
9)
are not Functions
included in theand
graph.
Draw circles at
Parent
Transformations
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.
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
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
SOLUTION:
Let x = 0 represent 1988. The price changes at the
beginning of the year, so the dots will appear on the
left end of each segment.
Plot each of the following segments.
x-coordinates
dot
circle
0
3
3
7
7
11
11
13
13
14
14
18
18
19
19
20
20
y-coordinates
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.
0.25
0.29
0.32
0.33
0.34
0.37
0.39
0.41
0.42
44. f (x) =
3
2
x + 2x – 8x – 2
SOLUTION:
Use the graph of f (x) to graph g(x) = |f (x)| and h
(x) = f (|x|).
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41.
f (x) =
SOLUTION:
Page 6
|f (x)| replaces all of the negative y-values with the
corresponding positive y-values. If f (–7) =
1-5 Parent Functions and Transformations
|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.
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.
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.
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.
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.
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