Warm-Up: you should be able to answer the following without the use of a calculator 2) Graph the following function and state the domain, range and axis of symmetry for the following function:

Absolute Value and Exponential Functions and Their Transformations

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

The Parent Graph of the Absolute Value Function

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.

Absolute Value Function Vertex Axis of Symmetry

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

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

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.

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?

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|.

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

Example 1:

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

Absolute Value on your calculator

Graphing example 2 on your calculator

You Try:

Example 3: Write a function for the graph shown.

You Try: Write a function for the graph shown.

Exponential Functions

Exponential Parent Graph Key Characteristics: There are no lines of symmetry These functions will always have an asymptote There is no vertex point

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.

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

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

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

You Try:

Homework Worksheet #4