Lesson #6: Basic Transformations with the Absolute Value Function
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1 Lesson #6: Basic Transformations with the Absolute Value Function Recall: Piecewise Functions Graph:,, What parent function did this piecewise function create? The Absolute Value Function Algebra II with Trigonometry: Unit 1 1
2 Basic Parent Function y The Is the absolute value function one to one? Explain. No, because the graph fails the horizontal line test. [y values repeat] Domain:, Range:, Is the absolute function even, odd, or neither? Even, because or the graph is reflected over the axis. 1. If, what happens to the graph if 3 is added to the function? The graph is shifted UP 3 units. 2. How does this transformation The range becomes or,. The equation becomes 3. Algebra II with Trigonometry: Unit 1 2
3 3. If, what happens to the graph if 3 is subtracted from the function? The graph is shifted DOWN 3 units. 4. How does this transformation change the range of the function? The range becomes 3or,. The equation becomes If, write the new equation if the graph is moved 2 units to the right How does this transformation change the range of the function? The range remains 0or,. *Note: Horizontal translations are opposite of what they appear to be. Algebra II with Trigonometry: Unit 1 3
4 7. If, write the new equation if the graph is moved 2 units to the left How does this transformation change the range of the function? The range remains 0or,. *Note: Horizontal translations are opposite of what they appear to be. Transformations Using Function Notation Vertical Translations If k is positive, the graph shifts up k units. If k is negative, the graph shifts down k units Horizontal Translations If h is positive, the graph shifts to the right h units. (Remember: if h looks negative h is positive) If h is negative, the graph shifts to the left h units. (Remember: if h looks positive h is negative) Algebra II with Trigonometry: Unit 1 4
5 9. If, write the new equation if the graph is reflected over the -axis. *Recall: 10. How does this transformation The range becomes 0or,. To reflect over axis, negate the value. 11. If, write the new equation if the graph is reflected over the -axis. 12. How does this transformation The graph remains exactly the same! *Recall: To reflect over axis, negate the value. Algebra II with Trigonometry: Unit 1 5
6 Transformations Using Function Notation Given: Reflection over the -axis If the -value is negated, is reflected over the -axis. Reflection over the -axis If the -value is negated, is reflected over the -axis. 13. If, what happens to the graph if the function is multiplied by 3? The graph is vertically stretched by a factor of How does this transformation The range remains 0or,. The equation becomes 3. Algebra II with Trigonometry: Unit 1 6
7 15. If, what happens to the graph if the function is multiplied by? The graph is vertically shrunk or compressed by a factor of. 16. How does this transformation The range remains 0or,. The equation becomes. Transformations Using Function Notation Given: Vertically Stretching or Shrinking a Graph If 1, multiply each -coordinate of by, vertically stretching the graph of by the factor of. If 0 1, multiply each -coordinate of by, vertically shrinking the graph of by the factor of. Algebra II with Trigonometry: Unit 1 7
8 17. If, what happens to the graph if the function changed to? The graph is horizontally shrunk or compressed by a factor of. 18. How does this transformation The range remains 0or,. 19. If, what happens to the graph if the function changed to? The graph is horizontally stretched by a factor of. 20. How does this transformation The range remains 0or,. Algebra II with Trigonometry: Unit 1 8
9 Transformations Using Function Notation Given: Horizontally Stretching or Shrinking a Graph If 1, divide each -coordinate of by, horizontally shrinking the graph of by the factor of. If 0 1, divide each -coordinate of by, horizontally stretching the graph of by the factor of. Transformations Using Function Notation Vertical Stretch & Shrink Horizontal Stretch & Shrink Algebra II with Trigonometry: Unit 1 9
10 Transformations Using Function Notation Combinations of Transformations: A function involving more than one transformation can be graphed by performing transformations in the following order: 1. Horizontal Shifting 2. Stretching or Shrinking 3. Reflecting 4. Vertical Shifting Example 1: Write the equation of each graph below. Algebra II with Trigonometry: Unit 1 10
11 Example 2: Use the graph of and transformations to sketch the graph of 3 2. Also, describe the transformations in words. H: Left 3 S: NONE R: NONE V: DOWN 2 Example 3: Use the graph of and transformations to sketch the graph of Also, describe the transformations in words. H: Right 1 Vertical Stretch S: by factor of 2 *multiply values by 2* R: NONE V: UP 1 Algebra II with Trigonometry: Unit 1 11
12 Example 4: Use the graph of and transformations to sketch the graph of 2. Also, describe the transformations in words. H: Left 2 Horizontal Stretch S: by factor of 2 *divide values by * R: Reflect over -axis *since is negated* V: None Using the Calculator In order to graph an absolute value equation: Go to **The quickest way to find the absolute value: Second 0 and press enter. abs( will appear or will appear. This is the absolute value function. Algebra II with Trigonometry: Unit 1 12
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