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SPECIAL FUNCTIONS AND TRANSFORMATIONS

Unit Overview
Some real-world relationships can be modeled with special functions.  For example, the dosage of medicine given to a child can be modeled with a step-function.  This unit will look at “special” functions − piecewise, step and absolute-value functions.  The unit concludes with an overview of basic transformations of the graphs of functions.

Special Functions

Piecewise Functions

To graph:

1.)  make a table of values for each domain
2.)  substitute values into each given equation
3.)  plot points
4.) decide whether to use open points (< or >) or closed points (≤, ≥); and whether or not to use a line segment or a continuous ray, and then connect the points.

Graph each function separately.

Step Functions

A piecewise function where the graph looks like a set of steps is called a step function.

A step function pairs every number in an interval with a single value. *Recall that that the set of integers are:  {...  −3, −2, −1, 0, 1, 2, 3 ...}. The greatest-integer function  f (x) = [x] is a step function where [x] represents the greatest integer less than or equal to x.  This does not mean to round on truncate the number.  For example, [7.8] = 7 because 7 is the greatest integer less than 7.8.  Similarly, [−2.9] = −3 because −3 is the greatest integer less than  −2.9. *Note the answer to [−2.9] ≠ −2 since −2 is not less than −2.9.  Also, [4] = 4, since 4 is the integer equal to 4. **The greatest integer value will be the first integer to the left of the given decimal value on the number line.

What is the value of [12.6]?

Example #3:  A downtown parking garage charges $4 for the first hour.  The charge is $2 for each additional hour or part of an hour.  The maximum charge to park in the garage is $10.  The maximum time to leave the car in the garage is 12 hours.  (a) Write a function and graph the step function that shows the fee ( f ) in terms of the number of hours (t).  A table can be useful to organize the given information. *During the hours between 1 and 3, the fee is based on usage of an hour or part of an hour; thus, we use the integer function. The graph is a step function because each hour is paired in intervals with a single fee. *Notice on this graph the open circle is on the left of the line segment and the closed circle is on the right.  (b) Use the graph to find the cost of parking in the garage for 2  1 ⁄ 2 hours. What is the cost of parking for 2  1 ⁄ 2 hours? The cost for 2  1 ⁄ 2 hours is $8. "Click here" to check the answer.  (c)  State the domain and the range of the function.  What is the domain of the function? The domain is 0 < t ≤ 12 hours. "Click here" to check the answer. What is the range of the function? The range is {4, 6, 8, 10}. "Click here" to check the answer. The domain is 0 < t ≤ 12 since the limit for parking in the garage is no more than 12 hours. The range is {4, 6, 8, 10} as these are the only different fees that can be charged.

Absolute Value Functions

The absolute value function is another piecewise function. 

The simplest example of an absolute value function is  f (x) = | x |. 

     *Recall that absolute value is the positive value of  the number or expression within the bars.    

To graph the absolute value function, make a table of values, and then graph.

The graph of an absolute value function is V-shaped and symmetric about a vertical line called the axis of symmetry.  The graph has either a maximum point (highest) or a minimum (lowest) point called the vertex. 

For f (x) = | x | the vertex is the point (0, 0).  It is a minimum point since it is the lowest point on the graph.  The axis of symmetry is the line x = 0, the y-axis.  The axis of symmetry divides the graph into two mirror images. 

The domain of  f (x) = | x | is all real numbers.

The range of  f (x) = | x | is y ≥ 0 (in other words, zero and all positive numbers).

Stop!  Go to Questions #1-5 about this section, then return to continue on to the next section.


Transformations of Absolute Value Functions

How will  f (x) = | x − 7 | be transformed from the parent function  f (x) = | x |?

How will  f (x) = | x + 5 | be transformed from the parent function  f (x) = | x |?

*Note:  In both previous questions, the constant is contained with the variable; thus,  each one transforms horizontally in the opposite direction of the sign.

Vertical Transformations

How will  f (x) = | x | + 2 be transformed from the parent function  f (x) = | x |?

How will  f (x) = | x | − 10 be transformed from the parent function  f (x) = | x |?

Vertical and Horizontal Transformations

How will  f (x) = | x − 7 | + 12 be transformed from the parent function  f (x) = | x | ?

What is the axis of symmetry of  f (x) = | x − 7 | + 12 ?

*The axis of symmetry (x = 7) is the vertical line that divides the graph into two mirror images.  The vertex (7, 12) is the lowest point of the graph.

In the graph of  f (x) = | x |, the right branch has a slope of +1 and the left branch has a slope of −1. 

The graph of  f (x) = a | x | is a stretch or compression of the graph of  f (x) = | x |.  The right branch has a slope of a and the left branch has a slope of −a. 

Summary of Absolute Value Transformations

The general form of an absolute value function is:

The transformations that occur on the parent function  f (x) = | x | are:

Stop!  Go to Questions #6-30 to complete this unit.