College Algebra Chapter 2 Functions and Graphs Section 2.6 Transformations of Graphs

Concepts 1. Recognize Basic Functions 2. Apply Vertical and Horizontal Shifts 3. Apply Vertical and Horizontal Shrinking and Stretching 4. Apply Reflections Across the x- and y-Axes 5. Summarize Transformations of Graphs

Basic Functions and Their Graphs Linear functionConstant functionsIdentity Function

Basic Functions and Their Graphs, continued Quadratic functionCube Function

Basic Functions and Their Graphs, continued Square Root functionCube Root Function

Basic Functions and Their Graphs, continued Absolute Value functionReciprocal Function

Concepts 1. Recognize Basic Functions 2. Apply Vertical and Horizontal Shifts 3. Apply Vertical and Horizontal Shrinking and Stretching 4. Apply Reflections Across the x- and y-Axes 5. Summarize Transformations of Graphs

Vertical and Horizontal Translations 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. 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: Graph the functions. ParentFamilies x

Example 1 continued:

Example 2: Graph the functions. ParentFamilies x

Example 2 continued:

Example 3: Graph the function. Horizontal shift: ___________ Vertical shift: _____________

Example 4: Graph the function. Horizontal shift: ___________ Vertical shift: _____________

Concepts 1. Recognize Basic Functions 2. Apply Vertical and Horizontal Shifts 3. Apply Vertical and Horizontal Shrinking and Stretching 4. Apply Reflections Across the x- and y-Axes 5. Summarize Transformations of Graphs

Vertical Shrinking and Stretching 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.

Example 5: Graph the functions. x x

Example 5 continued:

Horizontal Shrinking and Stretching 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 horizontally 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 horizontally by a factor of a.

Example 6: Graph the functions. x x

Example 6 continued:

Example 7:

Concepts 1. Recognize Basic Functions 2. Apply Vertical and Horizontal Shifts 3. Apply Vertical and Horizontal Shrinking and Stretching 4. Apply Reflections Across the x- and y-Axes 5. Summarize Transformations of Graphs

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:

Concepts 1. Recognize Basic Functions 2. Apply Vertical and Horizontal Shifts 3. Apply Vertical and Horizontal Shrinking and Stretching 4. Apply Reflections Across the x- and y-Axes 5. Summarize Transformations of Graphs

Summarize of Transformations of Graphs Guidelines: Perform horizontal transformations first. These are operations on x. Perform vertical transformations next. These are operations on f(x). Horizontal transformations Vertical transformations

Summarize of Transformations of Graphs Order of transformations: 1.Horizontal shrink/stretch/reflection “b.” 2.Horizontal shift “c.” 3.Vertical shrink/stretch/reflection “a.” 4.Vertical shift “d.” Horizontal transformations Vertical transformations

Example 9:. Parent function:

Example 9 continued:. Shift the graph to the left 1 unit Apply a vertical stretch (multiply the y-values by 2) Shift the graph downward 3 units

Example 10:. Parent function:

Example 10 continued:.