Math 1113 Notes - Functions Revisited
Philippe B. Laval
Kennesaw State University
February 14, 2005
Abstract
This handout contains more material on functions. It continues the
material which was presented on the previous handout on functions. It
include the following topics:
• Transformation of functions
• Extreme values
• Piecewise functions
1
Transformation of Functions
We study how the graph of a function y = f (x), is modified when we perform
one or more of the transformations below:
• Replace x by x − h denoted x → x − h
• Replace x by −x denoted x → −x
• Replace x by ax, a > 0 denoted x → ax
• Replace y by y − k denoted y → y − k
• Replace y by −y denoted y → −y
• Replace y by cy, c > 0 denoted y → cy
To better understand and remember how the graph of y = f (x) will be
affected by these transformation, the student must ask two questions:
1. Which variable is the transformation being applied to? Is it x or y?
2. What kind of transformation is it? It can either be adding a quantity,
multiplying a positive quantity, or switching the sign.
The proposition below summarizes what can happen. We will then look at
examples to illustrate all the possibilities.
1
Proposition 1 If a transformation is applied to x, the graph of y = f (x) will
change horizontally. If it is applied to y, the graph of y = f (x) will change
vertically. How the graph will change depends on the transformation.
• If the transformation is adding a quantity, then the change will be a shift
(also called a translation). More precisely,
— The transformation x → x − h produces a horizontal shift of |h| units
to the right if h > 0 and |h| units to the left if h < 0.
— The transformation y → y − k produces a vertical shift of |k| units
up if k > 0 and |k| units down if k < 0.
• If the transformation is changing the sign, then the change will be a reflection of the graph. More precisely,
— The transformation x → −x produces a reflection about the y − axis.
— The transformation y → −y produces a reflection about the x − axis.
• If the transformation is multiplying by a positive quantity, the change will
be a stretching or a shrinking. More precisely,
— If a > 0, the transformation x → ax produces a horizontal shrinking
1
if a > 1 and a horizontal stretching if 0 < a < 1 by a factor of in
a
both cases.
— If c > 0, the transformation y → cy produces a vertical shrinking if
1
c > 1 and a vertical stretching if 0 < c < 1 by a factor of in both
c
cases.
We now consider each case separately. We will then see what happens when
we combine several of these transformations.
1.1
Horizontal Shift or Translation
This happens when the transformation x → x −h is applied. If h > 0, the graph
is shifted |h| units to the right. If h < 0, the graph is shifted |h| units to the
left.
Example 2 Sketch the graphs of y = (x − 3)2 and y = (x + 2)2 , using the
graph of y = x2 .
2
2
• First we do y = (x − 3) . We see that we can obtain y = (x − 3) from
y = x2 with the transformation x → x − 3. This transformation is of
2
the form x → x − h with h = 3. Therefore, the graph of y = (x − 3) is
obtained by translating the graph of y = x2 3 units to the right.
2
• Next, we do y = (x + 2)2 . We see that we can obtain y = (x + 2)2 from
y = x2 with the transformation x → x + 2 in other words x → x − (−2).
This transformation is of the form x → x − h with h = −2. Therefore,
2
the graph of y = (x + 2) is obtained by translating the graph of y = x2 2
units to the left.
The graph of y = x2 appears in black in figure 1, y = (x − 3)2 appears in
blue and y = (x + 2)2 appears in red. Looking at these graphs, we can verify
2
that the graph of y = (x − 3) is a translation of the graph of y = x2 3units to
2
the right. Similarly, we can see that the graph of y = (x + 2) is a translation
2
of the graph of y = x 2 units to the left.
Figure 1: Horizontal translation
1.2
Vertical Shift or Translation
This happens when the transformation y → y − k is applied. If k > 0, the graph
is shifted |k| units up. If k < 0, the graph is shifted |k| units down.
Example 3 Sketch the graph of y = x2 + 3 and y = x2 − 2 using the graph of
y = x2 .
3
• First, we do y = x2 + 3 . If we write the equation as y − 3 = x2 , then we
see that it can be obtained from y = x2 with the transformation y → y − 3.
This is a transformation of the form y → y − k with k = 3. Thus the
graph of y = x2 + 3 can be obtained by translating the graph of y = x2 3
units up. We can verify this works by looking at figure 2.
• Next, we do y = x2 − 2. If we write the equation as y + 2 = x2 , then we
see that it can be obtained from y = x2 with the transformation y → y + 2
or y → y − (−2). This is a transformation of the form y → y − k with
k = −2. Thus the graph of y = x2 − 2 can be obtained by translating the
graph of y = x2 2 units down. We can verify this works by looking at
figure 2.
The graph of y = x2 appears in black in figure 2, y = x2 + 3 appears in red
and y = x2 − 2 appears in blue.
Figure 2: Vertical translation
1.3
Reflection About the y-axis
This happens with the transformation x → −x.
3
Example 4 The graph of f (x) = (x − 1) + 3 is shown in black in figure 3.
Sketch the graph of y = f (−x).
4
y = f (−x) can be obtained from y = f (x) with the transformation x → −x.
Therefore, the graph of y = f (−x) is a reflection of the graph of y = f (x) about
the y − axis. This can be verified looking at 3, in which the graph of y = f (−x)
appears in blue.
Figure 3: Reflection about the y-axis
1.4
Reflection About the x-axis
This happens with the transformation y → −y.
Example 5 Sketch the graph of y = −x2 using the graph of y = x2 .
If we rewrite y = −x2 as −y = x2 , then we see that it can be obtained from
y = x2 using the transformation y → −y. Therefore, the graph of y = −x2 is a
reflection about the x − axis of the graph of y = x2 as figure 4 shows. In figure
4, y = x2 is in black, y = −x2 is in blue.
1.5
Horizontal Stretching or Shrinking
This happens when the transformation x → ax, a > 0 is applied. The transformation will produce a horizontal shrinking if a > 1 and a horizontal stretching
5
Figure 4: Reflection about the x − axis
1
if 0 < a < 1 by a factor of
in both cases. This transformation is, in gena
eral, more difficult to visualize. It can be made easier if one remembers a few
pointers.
• First, everything along the y − axis will remain unchanged. Along the
y − axis, x = 0, multiplying x by some number a will still give 0.
• The result of the transformation is a horizontal shrinking or stretching.
So, distance in the horizontal direction will be shrunk or stretched by a
factor of a1 . For example, if the transformation is x → 3x, then it is a
horizontal shrinking by a factor of 13 . So, every point on the new graph
will be three times as close to the y − axis as they were on the original
graph.
Example 6 The graph of y = |x|
is shown in black on figure 5. Use it to sketch
the graphs of y = |2x| and y = 12 x.
• First, we do y = |2x|. It can be obtained from y = |x| with the transformation x → 2x. Therefore, the graph of y = |2x| is obtained from the graph
of y = |x| by horizontal shrinking by a factor of 12 . The points on the graph
also on the y − axis will not change. Every other point will be twice as
6
close to the y-axis. The result is the blue graph on figure 5. To help you
draw the new graph, you can draw (or picture) several horizontal lines.
Along each horizontal lines, measure the distance between the y − axis
and the graph, then divide that distance by 2 (or a in general), call this
distance d. Then, on each horizontal line, plot a point d units from the
y − axis. The new graph will go through the points you just plotted.
• Next, we do y = 12 x. It can be obtained from y = |x| with the transformation x → 12 x. Therefore, the graph of y = 12 x is obtained from the
1
graph of y = |x| by horizontal stretching by a factor of 1 = 2. This is
2
similar to the previous case. This time, points will be twice as far from
the y − axis. The result is the red graph on figure 5.
Figure 5: Horizontal shrinking and stretching
1.6
Vertical Stretching or Shrinking
This happens when the transformation y → cy, c > 0 is applied. The transformation will produce a vertical shrinking if c > 1 and a vertical stretching
1
if 0 < c < 1 by a factor of
in both cases. This is similar to the horizontal
c
7
case, but it is in the vertical direction. This time, points on the x − axis do not
change. Every other point on the graph either moves away from or closer to the
1
x − axis by a factor of . For example, if the transformation is y → 13 y, then
c
the transformation will produce a vertical stretching. Every point on the graph
not on the x − axis will be three times as far from the x − axis.
Example 7 The graph of the function y = sin x is shown in black on figure 6.
Use it to sketch y = 3 sin x and y = 12 sin x.
• First, we do y = 3 sin x. If we write it as 13 y = sin x, we see that it
can be obtained from y = sin x with the transformation y → 13 y. This
1
transformation will produce a vertical stretching by a factor of 1 = 3.
3
Every point of the graph also on the x − axis will not move. Every other
point of the graph will move three times further away from the x − axis.
The result is the blue graph on figure 6.
• Next, we do y = 12 sin x. If we write it as 2y = sin x, we see that it
can be obtained from y = sin x with the transformation y → 2y. This
transformation will produce a vertical shrinking by a factor of 12 . Every
point of the graph also on the x − axis will not move. Every other point
of the graph will move twice as close to the x − axis. The result is the red
graph on figure 6.
1.7
Combining Several Transformations
Often, more than one transformation is involved. In this case, the student has
to answer several questions:
1. Which function do we start with?
2. What transformations do we apply to it?
3. In which order do we apply the transformations?
We look at an examples to illustrate this.
Example 8 Sketch the graph of y =
function.
1
4
|3x − 6| by transforming the appropriate
• First, it is easy to see that we will be transforming y = |x|. Its graph
is the black graph on figure 7.The question is which transformations will
transform |x| into
1
4 |3x − 6|, and in which order?
• Applying x → x − 6 to y = |x| produces y = |x − 6|. The graph is obtained
by translating the graph of |x| 6 units to the right. Its graph is the blue
graph on figure 7.
8
Figure 6: Vertical shrinking and stretching
• Next, we apply x → 3x to y = |x − 6|. We obtain y = |3x − 6| . Its graph
is a horizontal shrinking of the graph of y = |x − 6| . Every point on the
graph will be three times closer to the y − axis. Its graph is the red graph
on figure 7.
• Finally, we apply y → 4y to y = |3x − 6|. We obtain 4y = |3x − 6| or
y = 14 |3x − 6|. It is obtained from the previous graph by doing a vertical
shrinking by a factor of 14 . Every point on the graph is four times closer
to the x − axis. The result is the green graph on figure 7.
1.8
Even and Odd Functions
Definition 9 (even function) A function f is an even function if it satisfies
f (−x) = f (x) for every x in its domain.
Example 10 If f (x) = x2 even?
To answer this, we compute f (−x).
2
f (−x) = (−x)
= x2
= f (x)
9
Figure 7: Transformations of y = |x|
Thus, f is even.
Example 11 Is f (x) = x3 + 5 even?
Again, we compute f (−x).
3
f (−x) = (−x) + 5
= −x3 + 5
= x3 + 5
Thus, f is not even.
Every function of the form f (x) = xn , when n is an even integer will be
even functions. One of the properties of even functions is that their graph is
symmetric with respect to the y − axis. Thus, to graph an even function, we
only need to graph half of it. We obtain the other half by reflecting it about
the y − axis.
Definition 12 (odd function) A function f is an odd function if it satisfies
f (−x) = −f (x) for every x in its domain.
10
Example 13 If f (x) = x3 an odd function?
We compute f (−x).
f (−x) = (−x)3
= −x3
= −f (x)
Thus f is odd.
Example 14 Is f (x) = x3 + 5 odd?Again, we compute f (−x).
f (−x) = (−x)3 + 5
= −x3 + 5
= x3 + 5
Thus, f is not odd.
This last example shows us that some functions can be neither odd nor
even. Every function of the form f (x) = xn , where n is an odd integer are
odd functions. One of the properties of odd functions is that their graph is
symmetric with respect to the origin.
The blue and black graphs on figure 8 are graphs of even functions. The red
graph is the graph of an odd function.
1.9
Problems
In your book, do # 1, 2, 3, 4, 6, 7, 9, 11, 13, 19, 21, 31, 32, 33, 41, 43, 45, 47,
53, 95 - 102, 107 - 114 on pages 153 - 155.
2
2.1
Extreme Values of Functions
Definitions
An extreme value of a function is the largest or smallest value of the function
in some interval. It can either be a maximum value, or a minimum value. We
usually distinguish between local and global (or absolute) extreme values. Before
we give a formal definition, the graph shown in Figure 9 will help understand
these notions. We can see that the function represented on this graph seems
to have a minimum at x = −2 and x = 3. At these two points, the value
of the function is the smallest in an interval. Such minima are called local
minima. However, at x = −3, the value of the function is the smallest. Nowhere
else does the function get any smaller. Such a minimum is called a global (or
absolute) minimum. The function has a maximum at x = 2. It does not have
a global maximum at x = 2, the function gets larger at higher places. In fact,
this function has no global maximum. The local extrema of a function can be
thought of as the peaks and valleys of a mountain range. The global maximum
would then be the highest peak.
11
Figure 8: Even and odd functions
Definition 15 Let f denote a function.
1. We say that f has a local (relative) maximum at x = c if f (c) ≥ f (x) in
a region around c. The local maximum is f (c)
2. If f (c) ≥ f (x) for any x, then f has a global (absolute) maximum at
x = c. The maximum is f (c).
3. We say that f has a local (relative) minimum at x = c if f (c) ≤ f (x) in
a region around c. The local minimum is f (c)
4. If f (c) ≤ f (x) for any x, then f has a global (absolute) minimum at
x = c. The minimum is f (c).
It is very important to notice that there are two important quantities involved. The first one is the value of x at which the function achieves an extreme
value. The second is the extreme value itself. In general, finding extreme values
is difficult. It requires tools studied in a differential calculus course. For some
functions, these tools are not necessary. For other functions, even if we can’t
find exactly the extreme values, we can approximate them. Below, we illustrate
these two ideas.
12
Figure 9: Extreme Values of a Function
2.2
Extreme Values for Quadratic Functions
For more details on this, see the handout on quadratic functions. Here, we
simply summarize the results. Consider the quadratic function f (x) = ax2 +
bx + c.
1. If a > 0, then f has a global minimum, and no global maximum. Its global
−b
minimum happens at the vertex, that is when x =
. The value of the
2a
−b
.
minimum is f
2a
2. If a < 0, then f has a global maximum, and no global minimum. Its global
−b
maximum happens at the vertex, that is when x =
. The value of the
2a
−b
.
maximum is f
2a
2.3
Extreme Values for Other Functions
Until we study Calculus, the only method we have to find extreme values is
graphing the function, and approximating it extreme values looking at the
graph. If the graph is obtained with a graphing device such as a calculator,
or computer software, one can obtain a better estimate by first graphing the
function over a fairly large domain, then zooming in around the extreme value.
13
Example 16 Find the extreme values of the function f whose graph is shown
in Figure 9
Looking at the graph, we see that the function seems to have a local minima
when x = −2 and x = 3. These local minima are f (−2) ≈ −20 and f (3) ≈ 10.
−20 is also a global minimum. The function has a local maximum at x = 2. The
local maximum is f (2) ≈ 12. However, this function has no global maximum.
Example 17 Same question as above, but restrict the x-interval to [−2, 4].
The minima (local or global) will not change. Also, f still has a local maximum
at x = 2. It also has a global maximum at x = 4. The maximum is f (4) ≈ 15.
Example 18 Same question as above, restrict the x-interval to [−2, 4).
The answer is the same as above, except that the is no global maximum at x = 4,
since 4 ∈
/ [−2, 4).
3
3.1
Piecewise Functions
Definitions and Examples
Definition 19 (Piecewise function) A piecewise function is a function whose
definition changes over different intervals of its domain. The value(s) of x where
the definition changes is called the ”breaking value”.
This means that instead of having one definition, a piecewise function will
have several definitions. Only one of these definitions will be used at the same
time. The value of the independent variable will determine which definition to
use. Here are some examples. Note the notation being used.
2x + 10 if x < 0
This function consists of two
Example 20 f (x) =
x2 + 1 if x ≥ 0
pieces, hence the name ”piecewise” function. When x < 0, f (x) = 2x + 10;
when x ≥ 0, f (x) = x2 + 1. So, there is no ambiguity as to which definition to
use. 0 is the breaking value. To evaluate f (−5), we use f (x) = 2x + 10. So,
f (−5) = 2 (−5) + 10
= 0
Similarly, to find f (2), we use f (x) = x2 + 1. So,
2
f (2) = (2) + 1
= 5
Graphing this function amounts
to graphing each function for the specified value
2x + 10 if x < 0
of x. The graph of f (x) =
is shown in Figure 10
x2 + 1 if x ≥ 0
14
Figure 10: graph of f (x)
⎧
x < −3
⎨ x + 2 if
x2 − 10 if −3 ≤ x ≤ 2 This function consists of
Example 21 g (x) =
⎩
x + 3 if
x>2
three pieces. −3 and 2 are the breaking values. The first and the last piece are
linear functions. The middle piece is a quadratic function. The graph of this
piecewise function is shown on Figure 11
Remark 22 The two pieces which make f (x) do not connect. There is a gap
in the graph. When this happens, we say that the function is not continuous.
We could not draw its graph without lifting the pen from the paper.
Remark 23 The first two pieces of g (x) connect. However, the last two do
not.
Remark 24 A piecewise function can be made of two, three or more pieces.
Remark 25 Each piece making a piecewise function can be a different type of
function. For example, if the piecewise function has two pieces, they do not have
to be two linear functions or two quadratics. One piece could be linear, and the
other quadratic.
Remark 26 The TI82/83 cannot graph piecewise functions. The way around
it is to graph all the pieces making up the piecewise function. Then, remember
which portion of each function is actually being used. Figure 12 shows the graph
15
Figure 11: graph of g (x)
Figure 12: graph of f (x) and of the parts of f (x)
16
of the two functions making up f (x). Only the ”thick” portion of each function
is used for f (x).
3.2
3.2.1
Working with Piecewise Functions
Solving Equations
We will proceed by example.
Example 27 Consider
f (x) =
2x + 10 if x < 0
x2 + 1 if x ≥ 0
Solve the equation f (x) = 17
Since f (x) is made of two functions, we solve for each function, then only keep
the values of x which are acceptable.
• Using the first function, we solve
2x + 10 = 17
2x = 7
x = 3.5
However, x = 3.5 is not acceptable. 2x + 10 is only used when x < 0.
• Using the second equation, we solve
x2 + 1 = 17
x2 = 16
x = ±4
Since x2 + 1 is only used when x > 0, we keep x = 4.
• The solution for f (x) = 17 is x = 4.
3.2.2
Finding the Maximum and Minimum
To find the maximum and minimum of a piecewise function, we find the maximum and minimum of each piece. The smallest will be the minimum for the
piecewise function. The largest will be the maximum.
2
x − 6x + 11 if 0 ≤ x ≤ 5
Example 28 Find the maximum and minimum of h (x) =
x>5
x2 − 12x + 41 if
Since each piece is a parabola which opens up, each piece has a minimum, no
maximum. So, h (x) will have no maximum. To find the minimum of h (x), we
find the minimum of each piece.
17
• The minimum of x2 − 6x + 11 is the y-coordinate of its vertex. So, we find
the vertex.
−b
x =
2a
6
=
2
= 3
y = (3)2 − 6 (3) + 11
y = 2
So, the minimum is 2, it happens when x = 3
• The minimum of x2 − 12x + 41 is the y-coordinate of its vertex. So, we
find the vertex.
x =
=
y
=
=
=
−b
2a
−12
2
6
62 − 12 (6) + 41
5
So, the minimum is 5, it happens when x = 6.
• In conclusion, h (x) has no maximum. It has a minimum of 2, it happens
when x = 3.
3.3
Problems
1. In your book, do # 35, 36, 37, 39, 41, 43, on page 122.
.0018t2 − .0283t + 9.2 if t ≤ 30
2. Given M (t) =
, answer the ques.0112t2 − .8948t + 26.77 if t ≥ 30
tions below:
(a)
(b)
(c)
(d)
Find M (15). (Answer: M (15) = 9.180)
Find M (60). (Answer: M (60) = 13.402)
Solve M (t) = 11. (Answer: M (t) = 11 when t = 53.646)
Determine if M (t) has a minimum, a maximum. If it does not,
explain why. If it does, find it and find the value of t for which it
occurs. (Answer: M (t) has a minimum when t = 39.946)
3. The absolute value function, f (x) = |x|, can be defined as a piecewise
function as follows:
x if x ≥ 0
|x| =
−x if x < 0
Using the technique of piecewise functions, answer the questions below:
18
(a) Write |x − 5| as a piecewise function.
(b) Solve |x − 5| = 4 (Answer: x = 1 and x = 9)
(c) Solve |2x − 10| = 2 (Answer: x = 4 and x = 6)
(d) Solve |x + 3| = −1 (Answer: no solutions)
19