Chapter 1: Functions and Graphs

1. Chapter 1:Functions and Graphs Section 1-6: Graphical Transformations

2. Objectives • You will learn about: • Transformations • Horizontal and Vertical Translations • Reflections Across Axes • Vertical and Horizontal Stretches and Shrinks • Combining Transformations • Why: • Studying transformations will help you to understand the relationships between graphs that have similarities but are not the same.

3. Vocabulary • Transformations • Rigid transformations • Non-rigid transformations • Vertical translation • Horizontal translation • Reflections of each other across the x-axis • Reflections of each other across the y-axis • Reflections of each other across that line • proof

4. Translation • Let c be a positive real number. Then the following transformations result in translations of the graph of y = f(x): • Horizontal Translations: • y=f(x-c): a translation right by c units. • y=f(x+c) :a translation left by c units. • Vertical Translations: • y=f(x)+c: a translation up c units. • y=f(x) - c: a translation down c units.

5. Example 1Vertical Translations • Describe how the graphs of y = |x| can be transformed to the graph of the given equation: • y = |x| - 4 • y = |x + 2|

6. Example 2Finding equations for translations • Write an equation for the translation. • The red function below is y = x3 • Find the equation of the blue function.

7. Reflections • The following transformations result in reflections of the graph of y = f(x). • Across the x-axis: y = - f(x) • Across the y-axis: y = f(-x)

8. Example 3Finding Equations for Reflections • Find an equation for the reflection across each axis:

9. Example 4Reflecting Even Functions • Prove that the graph of an even function remains unchanged when it is reflected across the y-axis.

10. Graphing Absolute Value Compositions • Given the graph of y=f(x): • The graph of y =|f(x)| can be obtained by reflecting the portion of the graph below the x-axis across the x-axis, leaving the portion above the x-axis unchanged. • The graph of y=f(|x|) can be obtained by replacing the portion of the graph to the left side of the y-axis by a reflection of the portion to the right of the y-axis across the y-axis, leaving the portion to the right of the y-axis unchanged. (The result will show even symmetry)

11. Stretches and Shrinks • Let c be a positive real number. Then the following transformations result in stretches or shrinks of the graph y=f(x): • Horizontal Stretches or Shrinks: y=f(x/c) • A stretch by a factor of c if c > 1 • A shrink by a factor of c if c < 1 • Vertical Stretches or Shrinks: y=c∙f(x) • A stretch by a factor of c if c >1 • A shrink by a factor of c if c < 1

12. Example 5Finding Equations for Stretches and Shrinks • Let C1 be the curve defined by y = f(x) = x3 – 16x. Find the equations for the non-rigid transformations of C1: • C2: a vertical stretch of C1 by a factor of 3. • C3: a horizontal shrink of C1 by a factor of ½.

13. Combining Transformations • Transformations may be performed in succession—one after another. • If these include stretches, shrinks, or reflections, order may make a difference.

14. Example 6Combining Transformations • The graph of y=x2 undergoes the following transformations: • A horizontal shift of 2 units to the right. • A vertical stretch by a factor of 3 • A vertical translation up 5 units. • Apply the transformations in the opposite order and find the equation that results.

15. Example 7Transforming a Graph Geometrically • Consider the graph of y = f(x) (Shown below). Determine the graph of y = 2 f(x+1) -3. • The following transformations were performed on f(x): • Vertical stretch by a factor of 2. • Horizontal translation one unit to the left. • Vertical shift down three units.