College Algebra Chapter 2 Functions and Graphs Section 2.6 Transformations of Graphs Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Concepts Recognize Basic Functions Apply Vertical and Horizontal Translations (Shifts) Apply Vertical and Horizontal Shrinking and Stretching Apply Reflections Across the x- and y-Axes Summarize Transformations of Graphs

Concept 1 Recognize Basic Functions

Recognize Basic Functions (1 of 4) Linear function f(x) = mx + b Constant function f(x) = b Identity function f(x) = x

Recognize Basic Functions (2 of 4) Quadratic function Cube function

Recognize Basic Functions (3 of 4) Square root function Cube root function

Recognize Basic Functions (4 of 4) Absolute value function F(x) = |x| Reciprocal function

Concept 2 Apply Vertical and Horizontal Translations (Shifts)

Apply Vertical and Horizontal Translations (Shifts) (1 of 2) Consider a function defined by y = f(x). Let c and h represent positive real numbers. Vertical shift: The graph of y = f(x) + c is the graph of y = f(x) shifted c units upward. The graph of y = f(x) – c is the graph of y = f(x) shifted c units downward.

Apply Vertical and Horizontal Translations (Shifts) (2 of 2) Horizontal shift: The graph of y = f(x – h) is the graph of y = f(x) shifted h units to the right. The graph of y = f(x + h) is the graph of y = f(x) shifted h units to the left.

Example 1 (1 of 2) Graph the functions.

Example 1 (2 of 2) g(x) 2 up

Example 2 (1 of 2) Graph the functions.

Example 2 (2 of 2) h(x) 2 down K(x) 2 right

Skill Practice 1 Use translations to graph the given function.

Skill Practice 2 Graph the function defined by g(x) = |x+2|.

Example 3 Graph the function. Horizontal shift: 1 left Vertical shift: 4 down

Example 4 Graph the function. P(x) = |x - 2| + 3 Horizontal shift: 2 right Vertical shift: 3 up

Skill Practice 3 Use translations to graph the function defined by

Concept 3 Apply Vertical and Horizontal Shrinking and Stretching

Apply Vertical and Horizontal Shrinking and Stretching (1 of 2) Consider a function defined by y = f(x). Let a represent a positive real number. Vertical shrink/stretch: If a>1 , then the graph of y=a||f(x) is the graph of y = f(x) stretched vertically by a factor of a. If 0<a<1 , then the graph of y=a||f(x) is the graph of y = f(x) shrunk vertically by a factor of a.

Apply Vertical and Horizontal Shrinking and Stretching (2 of 2) Consider a function defined by y = f(x). Let a represent a positive real number. Horizontal shrink/stretch: If a>1 , then the graph of y=f(a||x) is the graph of y = f(x) shrunk horizontal by a factor of a. If 0<a<1 , then the graph of y=f(a||x) is the graph of y = f(x) stretched horizontal by a factor of a.

Example 5 (1 of 2) Graph the functions.

Example 5 (2 of 2)

Skill Practice 4 Graph the functions.

Example 6 (1 of 2) Graph the functions.

Example 6 (2 of 2)

Example 7 Use the graph of f(x) to graph y=f(4x)

Skill Practice 5 The graph of y=f(x) is shown. Graph.

Concept 4 Apply Reflections Across the x- and y-Axes

Apply Reflections Across the x- and y-Axes Consider a function defined by y = f(x). Reflection across the x-axis: The graph of y = – f(x) is the graph of y = f(x) reflected across the x-axis. Reflection across the y-axis: The graph of y = f(– x) is the graph of y = f(x) reflected across the y-axis.

Example 8 (1 of 2) The graph of y=f(x) is given Graph y=-f(x) and y=f(-x)

Example 8 (2 of 2)

Skill Practice 6 The graph of y=f(x) is given. Graph y=-f(x).

Concept 5 Summarize Transformations of Graphs

Summarize of Transformations of Graphs (1 of 6) Transformations of Functions Consider a function defined by y = f(x). If c, h, and a represent positive real numbers, then the graphs of the following functions are related to y = f(x) as follows. Transformation Effect on the Graph of f Changes to Points on f Vertical translation (shift) y = f(x) + c Shift upward c units Replace (x, y) by (x, y + c) y = f(x) – c Shift downward c units Replace (x, y) by (x, y – c)

Summarize of Transformations of Graphs (2 of 6) Effect on the Graph of f Changes to Points on f Horizontal translation (shift) y = f(x-h) Shift right h units Replace (x, y) by (x + h, y). y = f(x + h) Shift left h units Replace (x, y) by (x – h, y).

Summarize of Transformations of Graphs (3 of 6) Effect on the Graph of f Changes to Points on f Vertical stretch/shrink y = a[f(x)] Vertical stretch (if a > 1) Vertical shrink (if 0 < a < 1) Graph is stretched/shrunk vertically by a factor of a. Replace (x, y) by (x, ay).

Summarize of Transformations of Graphs (4 of 6)

Summarize of Transformations of Graphs (5 of 6) Effect on the Graph of f Changes to Points on f Reflection y = -f(x) Reflection across the x-axis Replace (x, y) by (x, -y). y = f(-x) Reflection across the y-axis Replace (x, y) by (-x, y).

Summarize of Transformations of Graphs (6 of 6) To graph a function requiring multiple transformations, use the following order. Horizontal translation (shift) Horizontal and vertical stretch and shrink Reflections across x- or y-axis Vertical translation (shift)

Example 9 (1 of 3) Graph the function defined by f(x) =2|x + 1|-3 Parent function: f(x)=|x|

Example 9 (2 of 3) f(x) = 2|x + 1|-3 Shift the graph to the left 1 unit f(x) = 2|x + 1|-3 Apply a vertical stretch (multiply the y-values by 2) f(x) = 2|x + 1|-3 Shift the graph downward 3 units

Example 9 (3 of 3)

Skill Practice 7 Use transformation to graph the function defined by m(x) = -3|x - 2| -4.

Example 10 (1 of 3) Graph the function defined by Parent function:

Example 10 (2 of 3) - → vertical reflect. ½ → vertical shrink -x → horizontal reflect 3 → Horizontal shift 3 right

Example 10 (3 of 3)

Skill Practice 8 Use transformations to graph the function defined by