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**Chapter 1 Functions and Their Graphs**

Pre-Calculus Chapter 1 Functions and Their Graphs

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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

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**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.

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**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

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**Common Functions Sketch graphs of the following functions:**

Constant Function Identity Function Absolute Value Function Square Root Function Quadratic Function Cubic Function

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Constant Function

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Identity Function

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**Absolute Value Function**

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Square Root Function

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Quadratic Function

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Cubic Function

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**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.

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**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.

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**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) ______________________

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**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

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

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**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

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**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) _______________________

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

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

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**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.

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**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.

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**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.

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**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.

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**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.

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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

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