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

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

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.

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

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

Constant Function

Identity Function

Absolute Value Function

Square Root Function

Quadratic Function

Cubic Function

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.

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.

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) ______________________

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

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

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

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) _______________________

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

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

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.

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.

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.

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.

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.

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