1. Chapter 1 Functions and Their Graphs
Pre-Calculus
Chapter 1
Functions and Their Graphs

2. Warm Up 1.4
A Norman window has the shape of a square with a semicircle mounted on it. Find the width of the window if the total area of the square and the semicircle is to be 200 ft2. x

3. 1.4 Transformation of Functions
Objectives:
Recognize graphs of common functions.
Use vertical and horizontal shifts and reflections to graph functions.
Use nonrigid transformations to graph functions.

4. Vocabulary Constant Function Identity Function Absolute Value Function
Square Root Function
Quadratic Function
Cubic Function
Transformations of Graphs
Vertical and Horizontal Shifts
Reflection
Vertical and Horizontal Stretches & Shrinks

5. Common Functions Sketch graphs of the following functions:
Constant Function
Identity Function
Absolute Value Function
Square Root Function
Quadratic Function
Cubic Function

6. Constant Function

7. Identity Function

8. Absolute Value Function

9. Square Root Function

10. Quadratic Function

11. Cubic Function

12. Exploration 1 y = x2 + c, where c = –2, 0, 2, and 4.
Graph the following functions in the same viewing window:
y = x2 + c, where c = –2, 0, 2, and 4.
Describe the effect that c has on the graph.

13. Exploration 2 y = (x + c)2, where c = –2, 0, 2, and 4.
Graph the following functions in the same viewing window:
y = (x + c)2, where c = –2, 0, 2, and 4.
Describe the effect that c has on the graph.

14. Vertical and Horizontal Shifts
Let c be a positive real number.
Shifts in the graph of y = f (x) are as follows:
h(x) = f (x) + c ______________________
h(x) = f (x) – c ______________________
h(x) = f (x – c) ______________________
h(x) = f (x + c) ______________________

15. Example 1 g(x) = x3 – 1 h(x) = (x – 1)3 k(x) = (x + 2)3 + 1
Compare the graph of each function with the graph of f (x) = x3 without using your graphing calculator.
g(x) = x3 – 1
h(x) = (x – 1)3
k(x) = (x + 2)3 + 1

16. Example 2
Use the graph of f (x) = x2 to find an equation for g(x) and h(x).

17. Exploration 3 g(x) = –x2 h(x) = (–x)2
Compare the graph of each function with the graph of f (x) = x2 by using your graphing calculator to graph the function and f in the same viewing window. Describe the transformation.
g(x) = –x2
h(x) = (–x)2

18. Reflections in the Coordinate Axes
Reflections in the coordinate axes of the graph of y = f (x) are represented as follows:
h(x) = –f (x) _______________________
h(x) = f (–x) _______________________

19. Example 3
Use the graph of f (x) = x4 to find an equation for g(x) and h(x).

20. Example 4
Compare the graph of each function with the graph of

21. Exploration 4 y = cx3, where c = 1, 4 and ¼.
Graph the following functions in the same viewing window:
y = cx3, where c = 1, 4 and ¼.
Describe the effect that c has on the graph.

22. Exploration 5 y = (cx)3, where c = 1, 4 and ¼.
Graphing the following functions in the same viewing window:
y = (cx)3, where c = 1, 4 and ¼.
Describe the effect that c has on the graph.

23. Nonrigid Transformations
Changes position of the graph but maintains the shape of the original function.
Horizontal or vertical shifts and reflections.
Nonrigid Transformation
Causes a distortion in the graph Changes the shape of the original graph.
Vertical or horizontal stretches and shrinks.

24. Vertical Stretch or Shrink
Original function y = f (x).
Transformation y = c f (x).
Each y-value is multiplied by c.
Vertical stretch if c > 1.
Vertical shrink if 0 < c < 1.

25. Horizontal Stretch or Shrink
Original function y = f (x).
Transformation y = f (cx).
Each x-value is multiplied by 1/c.
Horizontal shrink if c > 1.
Horizontal stretch if 0 < c < 1.

26. Homework 1.4
Worksheet 1.4
#5, 7, 11, 13, 16, 20, 24, 26, 27, 33, 37, 39, 42, 45, 47, 51, 53, 57, 61, 63, 67