# COLLEGE ALGEBRA LAB 3

OBJECTIVES
• Understand function as an object
• Transformations and families of functions

FUNCTION AS AN OBJECT (Chapter 1: Sections 1-3 through 1-5)
The concept of function as an object that we can do things to is central to the study of algebra.
We treat a function as an object when we
• Classify functions as having properties, such as: symmetry about the y-axis or symmetry
• Transform functions by adding or multiplying constants within a function to move or
change the shape of the graph of the function
• Perform algebraic operations on functions, such as: adding, subtracting , multiplying,
dividing , and taking the composition of functions
• Find the inverse of a function

In this lab we will focus on the study of functions as objects by exploring the transformation of
functions. There are three types of transformations of function graphs that al low us to graph a
myriad of related functions for known basic functions. To do this we must first recognize the
shape of the known basic function graphs. We can then use symmetry to aid in sketching the
graph. In addition, we need to understand the related function concepts of domain and range.

SEVEN BASIC FUNCTIONS

In order to apply a transformation to a function, we must know what the original function looks
like . So we will begin by graphing seven basic functions and determining their domain, range,
and symmetries.

 Constant Function: where c is any real number (i.e. f(x) = 9) Linear Function : Square Function: Cube Function: Square Root Function : Absolute Value Function: Reciprocal Function :

The software tool we use for this lab is a dynamic graphing utility called Grapher.

We will do the basic linear function f(x) = x as an example.

 EXAMPLE 1: Determine the shape, domain, range, and any symmetry for the basic linear function f(x) = x. SOLUTION : A. Enter the function f(x) = x in Grapher on the entry line as y=x (Grapher does not recognize function notation, so we will write the functions in equation notation). Press the ENTER KEY to plot the function. The result is the graph in Figure 1 below. B. We determine the domain by examining the x-axis and asking for what values of x does the function as sign a y value ? For this linear function every x is assigned a y-value so the domain is all real numbers. In interval notation we write this as Domain of f(x) = (-∞, +∞). C. We determine the range by examining the y-axis and asking for what values of y does the function have a related x value? Again for this linear function the range is all real numbers. So in interval notation the Range of f(x) = (-∞, +∞). D. Finally we ask if the function is symmetric to the y-axis or the origin. If a function is symmetric to the y-axis then it reflects about the y-axis. Think of the y-axis as being a mirror that reflects the right side onto the left. For f(x) = x the line is not mirrored in the y-axis, so the function is not symmetric with respect to the y-axis. If a function is symmetric to the origin then a half-turn centered at the origin will place the graph back on top of itself. A half-turn of the basic linear function does place the graph back on top of itself, so the function is symmetric to the origin.

Figure 1: Plot of f(x) = x

Can you see why we would never need to check a function for symmetry about the x-axis? Now
let’s explore the properties of the six remaining basic functions.

 Exercise 1: Use Grapher to graph the remaining six basic functions in Table 1 and complete Table 1 by determining the domain, range, and symmetry for each function. Write the domains and ranges in interval notation (see Section 1.4 for help on interval notation).

Table 1: Basic Function Properties

 Basic Function Domain Range Symmetry Constant Function: Grapher enter y=5 as an example Linear Function: Grapher enter y=x (-∞, +∞) (-∞, +∞) Origin Square function: Grapher enter y=x^2 Cube function: Grapher enter y=x^3 Square root function: Grapher enter y=sqrt(x) Absolute value function: Grapher enter y=abs(x) Reciprocal function: Grapher enter y=1/x

TRANSFORMING BASIC FUNCTIONS
Now that you know the properties of some basic functions and the shape of their graphs, we can
examine transforming the basic functions. A transformation of a graph occurs when we add or
multiple a real number within the function. For example, the basic linear function f(x)=x can be
transformed into f(x)=ax+b by multiplying x by a real number a and/or adding a real number b.
The result of such transformations is a change in the graph of the function. We are going to
explore the effects of different transformations.

 Exercise 2: Let the base function be . We will use Grapher to plot the transformation of this function when we multiply by a real number a to get . The variable a that we will change is called a parameter, which means it is fixed for different cases while x and y are varied. A. On the Entry Line at the bottom of the Grapher type y=ax^2 and press the ENTER Key. The Grapher will assign a value of a = 1 to start, thus you will see the plot of the basic quadratic graph . B. To change the value of the parameter a select Options Customize Change Constants. You will be presented with a slider that allows you to change the value of a by dragging the slider. Drag the slider so that a takes on the values 2, 3, 5, 8, and 10. Describe how the graph of a function is transformed when we multiple the function by a real number a > 1. C. Now drag the slider to change a from 0 to 1. How does multiplying by a real number between 0 and 1 transform the function? D. Now drag the slider to change a from –1 to –10. Compare a for –5 to 5. What is the effect on of multiplying by a negative number a?

 Exercise 3: Clear the graph from Exercise 2 by clicking on the X button. Then reset the graphing window to 10 x 10 by clicking on the reset button. A. Let the base function be . In the Grapher type y = a*sqrt(x) and press the ENTER Key. Change the value of the parameter a from –10 to 10 using the Grapher. How does the transformation of compare to what you discovered in Exercise 2? B. How does compare to the base function ? Describe the effect of the transformation that occurs when is changed to .

 Exercise 4: Clear the graph from Exercise 3 and reset the window to 10 x 10. A. Let the base function be . In the Grapher type y = abs(x+b) and press the ENTER Key. Examine for for b= -10 to 10. Describe the transformation that results from adding  inside the base function (in essence, we are adding b directly to the x variable). B. Does it work the same for other base functions? Try some.

 Exercise 5: Clear the graph from Exercise 4 and reset the window to 10 x 10. A. Let the base function be . In the Grapher type y = x^3+c and press the ENTER Key. Examine for c = -10 to 10. Describe the transformation that results from adding c outside the base function (in essence, we are adding c to the entire base function, or y variable). B. Does it work the same for other base functions? Try some.

COMBINING TRANSFORMATIONS

If we perform more than one transformation at a time, the order they are performed in is
important. In y = a· f(x+b)+c, if the c transformation is performed before the a transformation
the graph will not be the one intended. Indeed, the graph would be of y = a· [f(x+b)+c], which
is not equivalent to the original function. This situation is avoided by following the simple rule:
given a function written in the form y = a· f(x+b)+c, always perform the “a”
transformation before the “c” transformation. Any other order except this may result in a
non-equivalent or incorrect transformation.

 Exercise 6: Let’s combine all the above work to sketch the graph of a function that is the result of multiple transformations of a base function. A. For what is the basic function being transformed? B. Before you use Grapher to graph the function , determine the effect of the three transformations performed on this base function. C. Use Grapher to graph the function . Does the graph match your predictions from Part 6B? If not, make corrections.
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