Standard MM2A1. Students will investigate step and piecewise functions, including greatest integer and absolute value functions. b. Investigate and explain characteristics of a variety of piecewise functions including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, points of discontinuity, intervals over which the function is constant, intervals of increase and decrease, and rates of change. c. Solve absolute value equations and inequalities analytically, graphically, and by using appropriate technology.

Absolute Value Functions General Form: y = a | x – h | + k Characteristics: 1. The graph is V-shaped Vertex of the graph: (h, k) note: opposite of h in general form “a” acts as the slope for the right hand side (the left side is the opposite)

Absolute Value Functions Parent Graph: y = | x | x y ordered pair Graph Transformations What effect does each one have on the parent graph? y = a | x – h | + k Determines if graph is fatter 0 < a < 1 or skinnier a > 1 Moves the graph up (+) or down (-) Moves the graph left (+) or right (-) Determines if graph opens up (+) or down (-)

Determine the vertex of the following functions. State whether the graph will open up or down. y = 2 |x - 2| + 3 4. y = 1/3 |x| + 5 y = -|x + 5| - 6 5. y = |x| y = -2|x + 2|

Steps for Graphing: Find and plot the vertex (opposite of h, k) Find and sketch the axis of symmetry Use “a” to find the slope and the next 2 points. 4) Using symmetry, plot 2 additional points and connect them to your vertex to create a “V” shaped graph!

Graphing Absolute Value Functions example 1 Vertex: ( , ) Slope: ________

Graphing Absolute Value Functions example 2 Vertex: ( , ) Slope: ________

Graphing Absolute Value Functions example 3 Vertex: ( , ) Slope: ________

Steps for writing an equation when given an absolute value graph. Identify the vertex (opposite of h, k) Determine if “a” will be positive or negative (opens up or down) Find a point to the right of the vertex that the graph passes through exactly and count the slope from the vertex to the point. This is “a” (the slope!) For the final answer: substitute “a” and the vertex (opposite of h, k) back into

Example 1 Vertex: ( , ) A is: positive / negative Slope: ________ Equation: y =

Example 2 Vertex: ( , ) A is positive / negative Slope: ________ Equation: y =

Writing Absolute Value Equations as Piecewise Functions from a Graph

Step 1: Locate the vertex and draw a vertical dashed line to represent the breaking point of the graph This is the AXIS OF SYMMETRY This x-value will be your DOMAIN RESTRICTION for the inequality

Steps 2-3: Write the equation of the RIGHT piece first Find your slope by counting on the graph Find your y-intercept by looking at the graph (Remember: you may have to see where it WOULD cross the y-axis if the graph does not) y = mx + b

Step 4: To write the equation of the LEFT piece, make the slope from the RIGHT piece NEGATIVE Find your new y-intercept by looking at the graph (this is where the line does or WOULD cross the y-axis.)

Step 5: Just like a piecewise function is organized, write the equation of the RIGHT piece first including the DOMAIN RESTRICTION

Step 6:

Worksheet Side 1 Try these on your own!

EOCT Challenge! Match the absolute value equation to its piecewise function

Convert Absolute Value Equations to Piecewise Functions

Step 1: Split absolute value into two separate equations One with the same slope as original One with the opposite slope as original

Step 2: Substitute “0” for x and solve for y-intercept, so that f(0)= b

Step 3: Change into y=mx+b form Identify slope a = m (slope) Identify y-intercept f(0)= b

Step 4: Identify the Domain Restriction Look at the original absolute value equation, the OPPOSITE of H is your axis of symmetry. Just like graphing, this is your domain restriction Put into piecewise form

Complete Writing Absolute Value as Piecewise worksheet. Homework Complete Writing Absolute Value as Piecewise worksheet.