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

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

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

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).

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. Even function. Symmetric to origin. Odd function. Symmetric to x-axis. Not a function.

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).

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

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

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

10 Library of Parent Functions: Commonly Used Functions Label important characteristics of each parent function.

11 Vertical Shift Change each function so it shifts up 2 units from the parent function.

12 Horizontal Shift Change each function so it shifts right 3 units from the parent function.

13 Vertical and Horizontal Shifts

14 Example 1 – Shifts in the Graph of a Function Compare the graph of each function with the graph of f (x) = x 3. a. g (x) = x 3 – 1 b. h (x) = (x – 1) 3 c. k (x) = (x + 2) Solution: 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)

15 Example 1 – Solution Compare the graph of each function with the graph of f (x) = x 3. 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) cont’d

16 Example 1 – Solution Compare the graph of each function with the graph of f (x) = x 3. c. k (x) = (x + 2) : 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) cont’d

17 Reflecting Graphs

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 | Solution: 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

19 Example 5 – Solution 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.) cont’d Figure 1.46

20 Copyright © Cengage Learning. All rights reserved. Pre-Calculus Honors 1.3: Step Functions and Piecewise-Defined Functions HW: p.38 (56-62 even)

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

22 Sketch the piecewise function.

23 Do Now: Sketch the piecewise function.