Warm-Up: you should be able to answer the following without the use of a calculator

1. Warm-Up: you should be able to answer the following without the use of a calculator 1) State the domain, range and axis of symmetry for the following parent graph 2) Graph the following function and state the domain, range and axis of symmetry for the following function: • How does the domain, range and axis of symmetry relate to the general vertex form?

2. Absolute Value and Exponential Functions and Their Transformations

3. Transformations Parabolas Revisited: Vertex Form: y = -a (x – h)2 + k *Remember that (h, k) is your vertex* Reflection across the x-axis Vertical Translation Vertical Stretcha > 1(makes it narrower)ORVertical Compression 0 < a < 1 (makes it wider) Horizontal Translation (opposite of h)

4. The Parent Graph of the Absolute Value Function

5. Vocabulary • The function f(x) = |x| is an absolute value function. • The highest or lowest point on the graph of an absolute value function is called the vertex. • An axis of symmetry of the graph of a function is a vertical line that divides the graph into mirror images. • An absolute value graph has one axis of symmetry that passes through the vertex.

6. Absolute Value Function • Vertex • Axis of Symmetry

7. Quadratic and Absolute Value Functions Quadratic and Absolute Value functions share some common characteristics: Vertex Line of Symmetry Minimum/ Maximum point

9. Vocabulary • The zeros of a function f(x) are the values of x that make the value of f(x) zero. • On this graph where x = -3 and x = 3 are where the function would equal 0. f(x) = |x| - 3

10. Other Shared Characteristics • Review the vertex form of a parabola. • Review how the changes in a , h and k transform, reflect or translate the parent graph of a parabola.

11. Parent and general equation: • Given y=|x| how do you think the general equation of a an absolute value function looks like? • How do you think each component transforms, reflects or translates the parent graph?

12. Vocabulary • A transformation changes a graph’s size, shape, position, or orientation. • A translation is a transformation that shifts a graph horizontally and/or vertically, but does not change its size, shape, or orientation. • When a = -1, the graph y = a|x| is a reflection in the x-axis of the graph of y = |x|.

13. Transformations y = -a |x – h| + k *Remember that (h, k) is your vertex* Reflection across the x-axis Vertical Translation Vertical Stretcha > 1(makes it narrower)ORVertical Compression 0 < a < 1 (makes it wider) Horizontal Translation (opposite of h)

14. Example 1:

15. Example 2: • Graph y = -2 |x + 3| + 4 • What is your vertex? • What are the intercepts?

16. Absolute Value on your calculator Where to locate the abs value

17. Graphing example 2 on your calculator

18. You Try: • Graph • Compare the graph with the graph of y = |x| (what are the transformations)

19. Example 3: • Write a function for the graph shown.

20. You Try: • Write a function for the graph shown.

21. Exponential Functions • The next family of functions we are going to look at are Exponential Functions • Our parent function being used for comparing graphs will be:

22. Exponential Parent Graph • Key Characteristics: • There are no lines of symmetry • These functions will alwayshave an asymptote • There is no vertex point

23. Exponential Parent Graph The ‘locater point’ for this function is the asymptote. Using this as our point allows for quick comparisons between the parent and transformed graphs.

24. Exponential Transformation Example #1: Comparing the asymptotes will give the vertical shift. 2

25. Exponential Transformation Example #2: Horizontal translations shift the point where the graph would have crossed the x-axis.

26. Exponential General Form Reflection across the x-axis Vertical Translation (also the asymptote) Vertical Stretcha > 1(makes it narrower)ORVertical Compression 0 < a < 1 (makes it wider) Horizontal Translation (opposite of h)

27. You Try: Identify the transformations:

28. Homework • Worksheet #4