MM2A1. Students will investigate step and piecewise functions, including greatest integer and...
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Transcript of MM2A1. Students will investigate step and piecewise functions, including greatest integer and...
MM2A1. Students will investigate step and MM2A1. Students will investigate step and piecewise functions, including greatest piecewise functions, including greatest integer and absolute value functions.integer and absolute value functions.
b. Investigate and explain characteristics of a variety of piecewise functions b. Investigate and explain characteristics of a variety of piecewise functions including domain, range, vertex, axis of symmetry, zeros, intercepts, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, points of discontinuity, intervals over which the function is extrema, points of discontinuity, intervals over which the function is constant, intervals of increase and decrease, and rates of change.constant, intervals of increase and decrease, and rates of change.
c. Solve absolute value equations and inequalities analytically, graphically, and c. Solve absolute value equations and inequalities analytically, graphically, and by using appropriate technology.by using appropriate technology.
Absolute Value FunctionsGeneral Form: y = a | x – h | + k
1. The graph is V-shaped
2. Vertex of the graph: (h, k)
note: opposite of h in general form
3. “a” acts as the slope for the right hand side (the left side is the opposite)
Absolute Value FunctionsParent Graph: y = | x |
x y ordered pair
What effect does each one have on the parent graph? y = a | x – h | + k
Determines if graph opens up (+)or down (-)
Determines if graph is
fatter 0 < a < 1
or skinnier a > 1
Moves the graph left (+)
or right (-)
Moves the graph up (+)
or down (-)
1. y = 2 |x - 2| + 3 4. y = 1/3 |x| + 5
2. y = -|x + 5| - 6 5. y = |x|
3. y = -2|x + 2|
Steps for Graphing:Steps for Graphing:1) Find and plot the vertex (opposite of h,
k)
2) Find and sketch the axis of symmetry
3) 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!
run
risea
Graphing Absolute Value Functionsexample 1
32 xy
Vertex: ( , )
Slope: ________
Graphing Absolute Value Functionsexample 2
11 xy
Vertex: ( , )
Slope: ________
Graphing Absolute Value Functionsexample 3
322
1 xy
Vertex: ( , )
Slope: ________
Steps for writing an equation when Steps for writing an equation when given an absolute value graph.given an absolute value graph.
1) Identify the vertex (opposite of h, k)2) Determine if “a” will be positive or
negative (opens up or down)3) 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!)
4) For the final answer: substitute “a” and the vertex (opposite of h, k) back into
khxay
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 SYMMETRYThis 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 + brise
run
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
_________________( )
_________________f x
Step 6:
Worksheet Side 1 Try these on your own!
EOCT Challenge!
Match the absolute value equation to its piecewise function
2y x
x 2 if x 0. y
-x 2 if x 0b
x-2 if x 2. y
-x-2 if x 2a
x - 2 if x 2. y
-x 2 if x 2c
x 2 if x 2. y
-x-2 if x 2d
-x-2 if x 0. y
x-2 if x 0e
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
( ) 2 1 3f x x
( ) 2 1 3f x x ( ) 2 1 3f x x
Step 2: Substitute “0” for x and solve for y-intercept, so that f(0)= b
( ) 2 1 3f x x
( ) 2 1 3f x x
( ) 2 1 3f x x
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
_________________( )
_________________f x
Homework
Complete Writing Absolute Value as Piecewise worksheet.