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

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

Recognize Basic Functions Linear function Constant function Identity function

Recognize Basic Functions Quadratic function Cube function

Recognize Basic Functions Square root function Cube root function

Recognize Basic Functions Absolute value function Reciprocal function

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

Apply Vertical and Horizontal Translations (Shifts) 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) Consider a function defined by y = f(x). Let c and h represent positive real numbers. 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. Parent Families x 1 2 3 4

Example 1 continued:

Example 2: Graph the functions. Parent Families x 1 2 3 4

Example 2 continued:

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

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

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

Apply Vertical and Horizontal 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 1 2 3 4 x 1 2 3 4

Example 5 continued:

Apply Vertical and 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 1 2 3 4 x 1 2 3 4

Example 6 continued:

Example 7:

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

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. Recognize Basic Functions 2. Apply Vertical and Horizontal Translations (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 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 y = f(x) – c Shift upward c units Shift downward c units Replace (x, y) by (x, y + c) Replace (x, y) by (x, y – c)

Summarize of Transformations of Graphs 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 Horizontal translation (shift) y = f(x – h) y = f(x + h) Shift right h units Shift left h units Replace (x, y) by (x + h, y). Replace (x, y) by (x – h, y).

Summarize of Transformations of Graphs 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 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 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 Horizontal stretch/shrink y = f(ax) H. shrink (if a > 1) H. stretch (if 0 < a < 1) Graph is stretched/shrunk horizontally by a factor of . Replace (x, y) by .

Summarize of Transformations of Graphs 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 Reflection y = –f(x) y = f(–x) Reflection across the x-axis Reflection across the y-axis Replace (x, y) by (x, –y). Replace (x, y) by (–x, y).

Summarize of Transformations of Graphs To graph a function requiring multiple transformations, use the following order. 1. Horizontal translation (shift) 2. Horizontal and vertical stretch and shrink 3. Reflections across x- or y-axis 4. Vertical translation (shift)

. Example 9: Parent function:

Shift the graph to the left 1 unit . 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: