GRAPHS OF FUNCTIONS

Contributed by:

Sharp Tutor Mon, Jan 17, 2022 09:33 PM UTC

TOPICS:
1. increasing and decreasing functions
2. even and odd functions
3. vertical and horizontal shifts
4. reflecting graphs

1. Pre-Calculus Honors
1.3: Graphs of Functions
HW: p.37 (8, 12, 14, 23-26 all,
38-42 even, 80-84 even)
Copyright © Cengage Learning. All rights reserved.

2. Increasing and Decreasing Functions
Determine the intervals on which each function is
increasing, decreasing, or constant.
(a) (b) (c)
2

3. Increasing and Decreasing
The more you know about the graph of a function, the more
you know about the function itself. Consider the graph
shown in Figure 1.20. Moving from left to right, this graph
falls from x = –2 to x = 0, is constant from x = 0 to x = 2,
and rises from x = 2 to x = 4.
Figure 1.20
3

4. Even and Odd Functions
• A function whose graph is symmetric with respect to the
y -axis is an even function.
• A function whose graph is symmetric with respect to the
origin is an odd function.
• A graph has symmetry with respect to the y-axis if
whenever (x, y) is on the graph, then so is the point (–x, y).
• A graph has symmetry with respect to the origin if whenever
(x, y) is on the graph, then so is the point (–x, –y).
• A graph has symmetry with respect to the x-axis if
whenever (x, y) is on the graph, then so is the point (x, –y).
4

5. Even and Odd Functions
A graph that is symmetric with respect to the x-axis is not
the graph of a function (except for the graph of y = 0).
Symmetric to y-axis. Symmetric to origin. Symmetric to x-axis.
Even function. Odd function. Not a function.
5

6. Even and Odd Functions
Algebraic Test for Even and Odd Functions:
•A function f is even when, for each x in the domain of f,
f(-x) = f(x).
•A function f is odd when, for each x in the domain of f,
f(-x) = -f(x).
6

7. Example 10 – Even and Odd Functions
Determine whether each function is even, odd, or neither.
a. g(x) = x3 – x
b. h(x) = x2 + 1
c. f (x) = x3 – 1
a. This function is odd because
g (–x) = (–x)3+ (–x)
= –x3 + x
= –(x3 – x)
= –g(x).
7

8. Example 10 – Solution
b. h(x) = x2 + 1
b. This function is even because
h (–x) = (–x)2 + 1
= x2 + 1
= h (x).
c. f (x) = x3 – 1
c. Substituting –x for x produces
f (–x) = (–x)3 – 1
= –x3 – 1.
So, the function is neither even nor odd.
8

9. Pre-Calculus Honors
1.4: Shifting, Reflecting, and
Stretching Graphs
Copyright © Cengage Learning. All rights reserved.
9

10. Library of Parent Functions: Commonly Used Functions
Label important characteristics of each parent function.
2
f ( x)  x f ( x)  x f ( x)  x
3 1
f ( x)  x f ( x) x f ( x) 
x
10

11. Vertical Shift
Change each function so it shifts
up 2 units from the parent function.
2
f ( x)  x f ( x)  x f ( x) x
3 1
f ( x)  x f ( x)  x f ( x) 
x
11

12. Horizontal Shift
Change each function so it shifts
right 3 units from the parent function.
3 1
f ( x)  x f ( x) x f ( x) 
x
12

13. Vertical and Horizontal Shifts
13

14. Example 1 – Shifts in the Graph of a Function
Compare the graph of each function with the graph of
f (x) = x3.
a. g (x) = x3 – 1 b. h (x) = (x – 1)3 c. k (x) = (x + 2)3 + 1
a. You obtain the graph of g by shifting the graph of f one
unit downward.
Vertical shift: one unit downward
Figure 1.37(a)
14

15. Example 1 – Solution cont’d
Compare the graph of each function with the graph of f (x) = x3.
b. h (x) = (x – 1)3 : You obtain the graph of h by shifting the
graph of f one unit to the right.
Horizontal shift: one unit right
Figure 1.37 (b)
15

16. Example 1 – Solution cont’d
Compare the graph of each function with the graph of
f (x) = x3.
c. k (x) = (x + 2)3 + 1 : You obtain the graph of k by shifting the
graph of f two units to the left and then one unit upward.
Two units left and one unit upward
Figure 1.37 (c)
16

17. Reflecting Graphs
17

18. Example 5 – Nonrigid Transformations
Compare the graph of each function with the graph of
f (x) = | x |.
a. h (x) = 3| x |
b. g (x) = | x |
a. Relative to the graph of
f (x) = | x |, the graph of
h (x) = 3| x | = 3f (x)
is a vertical stretch (each
y-value is multiplied by 3)
of the graph of f (See Figure 1.45.)
Figure 1.45 18

19. Example 5 – Solution cont’d
b. Similarly, the graph of g (x) = | x | = f (x) is a vertical
shrink (each y-value is multiplied by ) of the graph of f .
(See Figure 1.46.)
Figure 1.46
19

21. Example 8 – Sketching a Piecewise-Defined Function
Sketch the graph of
f (x) = 2x + 3, x ≤ 1
–x + 4, x > 1
by hand.
21

22. Sketch the piecewise function.
 4  x , x  0
f ( x) 
 4  x , x 0
22

23. Do Now: Sketch the piecewise function.
2 x  1, x  1
f ( x)  2
 x  2, x   1
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