Graphing Absolute Value Functions

Instructor: Russell Frith
The purpose of this lesson is to define what is meant by the absolute value of a number and then to extend this idea to define the parent function of absolute value.

Experiencing Absolute Value

Suppose you and your friend are driving down a long, straight country road that runs in an east-west direction. Suddenly you have a flat tire. You pull over and see that you don't have a spare tire so the two of you decide to set out for a tow truck. You decide to head west for help, and your friend decides to head east. If you both walk at the same pace, say three miles per hour, you will both be three miles from your car after an hour of walking, but in opposite directions.

Now suppose the road is the x-axis and your car is the origin on the x-axis. Your friend who has walked three miles east is '+3' miles from the origin. You have walked three miles west and you are '-3' miles from the origin. In terms of physical distances, you are both three miles in magnitude from the car (origin). The magnitude of your distance from the origin, or absolute value, is three miles, irrespective of the direction. No negative sign is ever used when stating the distance from the origin.

The Absolute Value of a Number

The absolute value of a number is the distance without regard to sign a number is from zero. For example, '7' is seven units to the right of zero, but '-7' is also seven units from zero, only in the opposite direction. Subsequently, the absolute value of 7 is 7 and the absolute value of -7 is also 7. One uses absolute value bars '|a|' to write out the absolute value for a number a. For example, |-7| = 7.

The Parent Function of Absolute Value

The parent absolute value function is the function that takes a number as input and returns the that number without a sign. So, if an input number is positive, then the output of the function is the same as the input. If an input number is negative, then the output is the input number without its sign.


The parent function is sometimes formulated as a piecewise function.


The Graph of the Absolute Value Function

Every absolute value function has either a maximal point or a minimal point which is known as the vertex. A point is maximal if no other point on the graph is positioned above it. A point is minimal if no other point on the graph is below it.

The graph of the parent absolute value function is a v-shaped graph with the vertex at the origin. This vertex is also the lowest point on the graph.


Scaling the Graph of the Absolute Value Function

Often times you need to take the parent absolute value function and multiply it by a number. The modified parent function is written as a constant a times the parent absolute value function.


Multiplying the parent absolute value function by a number is known as scaling. This action has the effect of changing the appearance of the graph.


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