SECTION 1.5 ABSOLUTE VALUEAND INVERSE OF QUADRATICFUNCTIONS
i) Concept of Absolute and Inverse functions
ii) Domain and Range of the inverse of a QF
iii) Graphing the absolute and inverse of a QF
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WHAT IS AN ABSOLUTE VALUE FUNCTION
The ABS function measures the distance any value isfrom zero
Since distance is always positive, the ABS of any valuewill also be positive
When graphing the absolute value of a function, any pointwith a negative y-coordinate will be positive
If the point already had a positive y-coordinate, it willstay positive
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The ABS function will reflect any part of the functionunder the x-axis to above the x-axis
The ABS of the straight line is a V-shaped graph
GRAPHING AN ABSOLUTE VALUE FUNCTION
Suppose we have the line:
Now graph the ABS ofthis line
Take the points with anegative Y-coordinates &
reflect it above the X-axis
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EX: GIVEN THE FOLLOWING PARABOLA, GRAPH THEABSOLUTE VALUE OF THIS FUNCTION:
Take any part of the graph below
the X-axis and vertically reflect it
x
y
-5
0
5
-5
5
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Note: Any number outside of the absolute value sign will beperformed afterwards
First take the abs of the parabola
Then shift the graph 4 units down
First shift the graph 4 units down
Then take the abs of the parabola
PRACTICE: GIVEN THE FOLLOWING GRAPHS, DRAWTHE ABSOLUTE VALUE OF EACH FUNCTION:
x
y
-5
0
5
-5
5
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x
y
-5
0
5
-5
5
INVERSE FUNCTIONS
The inverse of a function will switch the X and Yvariables and then isolate the Y variable afterwards
ie: Given the function, find its inverse:
Switch the two variables
The inverse function is:
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EX: GIVEN THE GRAPH OF , GRAPH THEINVERSE FUNCTION
Suppose we have a TOV for the original function
New coordinates can be found by switching the X & Ycoordinates
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PRACTICE: FIND THE INVERSE OF THE FOLLOWING FUNCTIONAND GRAPH IT.
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THE INVERSE OF A QUADRATIC FUNCTION
The inverse of a QF is not a function because it does notpass the vertical line test
So we must split the QF to two separate domains withthe axis of symmetry. When the QF is broken into 2parts, the inverse of each part will be a function
The inverse of the right sideis on top
The inverse of the left sideis on bottom
Each graph is split along theaxis of symmetry
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FIND THE INVERSE OF THE FOLLOWING FUNCTION:
Switch “x” and “y”
Isolate “y”
When you square root a value, youget both a positive and negativeanswer
represents thetop of the inverse function
represents thebottom of the inverse function
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DOMAIN & RANGE OF THE INVERSE FUNCTION:
Since we switch the x and y variables for the inverse,the domain of f(x) becomes the range of the inverse
Make sure you know which side of the domain thatyou are taking the inverse of
Likewise, the range of f(x) becomes the domain of theinverse
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EX: FIND THE DOMAIN AND RANGE OF THE INVERSE FUNCTION
We’re taking the inverse of the right side of the parabola
The graph opens up with a minimum value
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PRACTICE: FIND THE INVERSE FUNCTION AND ITSDOMAIN AND RANGE:
Switch “x” and “y”
Isolate the “y” variable
Take the negative square root becausethe domain is on the left side
Use the domain and range of f(x)to find the domain and range ofthe inverse
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GIVEN THE FUNCTION, FIND EACH OF THE FOLLOWINGVALUES
x
y
-5
0
5
-5
5
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