ACTIVITY 6
Parent Functions (Linear)

Before doing this activity. . .



ACTIVITY:
This activity will begin the investigation of linear functions.  If I were to ask you what the graph of function  would look like before you graphed it, could you tell me?  If not, then by the end of this activity you will be able to.
Note: Before beginning, be sure that all of the Stat Plots are off.  You can do this by pressing  [PLOT] and then choosing 4:PlotsOff.
Step 1. 
Enter the function  into the  editor of your calculator by pressing  and using the  key.
Graph the function by pressing  and selecting 6:ZStandard.  Sketch the graph on your recording sheet.
What type of curve did you sketch?
Is the graph that of an increasing or decreasing function?
We will call the function  our parent function.  We will be changing the coefficients of our parent function and comparing these new graphs to the original graph you just sketched.
Notice that the parent function  produced the graph of a line that goes up from left to right, thus the parent function is an increasing function.
Step 2. Enter the following functions into the  editor.
Graph the functions by pressing  and selecting 6:ZStandard.  Sketch the graphs on your recording sheet.
As the coefficient of x increased, what happened to the graph of the parent function ?
Step 3. Enter the following functions into the  editor.
Graph the functions by pressing  and selecting 6:ZStandard.  Sketch the graphs on your recording sheet.
As the coefficient of x decreased, what happened to the graph of the parent function ?
Step 4.
Enter the following functions into the  editor.
(Note:  Be sure to use the  key instead of the  key when entering a negative value.)
Graph the functions by pressing  and selecting 6:ZStandard.  Sketch the graphs on your recording sheet.
What happened to the graph of the parent function  when the coefficient of x became negative?
Notice that the graph of the parent function  changed from that of an increasing function to a decreasing function when the coefficient of x became negative.
Step 5. Enter the following functions into the  editor.
Graph the functions by pressing  and selecting 6:ZStandard.  Sketch the graphs on your recording sheet.
As the absolute value of the coefficient of x increased, what happened?
Does this graph look like the reflection of any previous graphs you have sketched?
Step 6. Enter the following functions into the  editor.
Graph the functions by pressing  and selecting 6:ZStandard.  Sketch the graphs on your recording sheet.
As the absolute value of the coefficient of x decreased, what happened?
Does this graph look like the reflection of any previous graphs you have sketched?

Let's summarize what we have "discovered" in Steps 1 - 6.

All of the functions you entered in these steps were of the form .  We investigated what happened when |a| increased and decreased.  Fill in the following blanks on your recording sheet.
When a is positive, the graph is that of a(n) _______________ function.
When a is negative, the graph is that of a(n) _______________ function.
 
As |a| increases, the graph becomes more _______________.
As |a| decreases, the graph becomes more _______________.

 
Step 7. Enter the following functions into the  editor.
Graph the functions by pressing  and selecting 6:ZStandard.  Sketch the graphs on your recording sheet.
What happened to the graph of the parent function  when you increased the value being added to x?  The value being added on is called the constant term.
Did the graphs cross above or below the origin along the vertical axis?
Step 8. Enter the following functions into the  editor.
Graph the functions by pressing  and selecting 6:ZStandard.  Sketch the graphs on your recording sheet.
What happened to the graph of the parent function  when you decreased the value of the constant term?
Did the graphs cross above or below the origin along the vertical axis?

Let's summarize what we have "discovered" in Steps 7 - 8.

All of the functions you entered in these steps were of the form .  We investigated what happened when c increased and decreased.  Fill in the following blanks on your recording sheet.
As c increases, the graph shifts _______________.
As c decreases, the graph shifts _______________.
 
When c is positive, the graph will cross _______________ the origin along the vertical axis.
When c is negative, the graph will cross _______________ the origin along the vertical axis.
Let's put this all together now by answering the following practice problems.  (Note: You will be asked to answer questions regarding these practice problems at the end of the activity.)
 
Practice 1. Given , answer the following without graphing.
 
a) 
Will the graph be that of an increasing or a decreasing function?  What value helped you determine this?
b)
Will the graph of the function cross the vertical axis above or below the origin?  What value helped you determine this?
c)
Will the graph of the function be more vertical or horizontal when compared to the graph of the parent function ?  What value helped you determine this?
 
Practice 2.
Without graphing, which line will be more vertical, that of  or ?  Explain how you determined this.
Practice 3.
Without graphing, which line will be more vertical, that of  or ?  Explain how you determined this.
Practice 4. Without the use of a graphing calculator, match the most appropriate graph for the function .
 
a)
b)
c)
d)
Practice 5. Susie graphed the function  in her calculator.  This is what her screen looked like.
She likes the incline of the line but would like to shift it down 5 units so that her screen looks like this.
What function should Susie enter into the  editor of her calculator to accomplish this?

 
To verify your answers to all of the above practice, enter the functions in the  editor and press the  key.  Change the values so that your screen looks exactly like what is shown below.  After changing your window settings, press .

(Note: will change automatically.  There is no need to alter this value.)


LOOKING BACK:
This activity investigated linear functions.  All of the linear functions that were investigated were of the form , called the slope-intercept form.  The coefficient of x, a, is also called the slope of the line.  The slope is the measurement of the steepness or incline of the line.  As you discovered, the larger the absolute value of the slope, the steeper or the more inclined the line.  Whether the slope is positive or negative determines if the line will go up or down from left to right.  The constant term, c, is also called the vertical axis intercept (or y-intercept) because it determines where the line will cross the vertical axis.  If c is positive, then the line will cross the vertical axis above the origin. and if c is negative, then the line will cross the vertical axis below the origin.

Upon completing this activity, you can now predict what the graph of a linear function will look like before you ever graph it.  Why is this important?  One reason is so that you catch any errors.  If you know roughly what the graph should look like before graphing it, then when a calculator or a computer gives you the "wrong answer" you will be able to catch the error.  Another reason is so that you can view your graph.  Not all window settings in a graphing calculator will allow you to "see" all the graphs.  If you know something about the graph, you can set the appropriate window so that you can view it.  The third reason, and probably the most important for a teacher, is so that you can manipulate your graph.  If your line is inclined just the way you want it but, you would like to shift it up two units, what do you do?  Right, add a 2 to the constant term.  If your line is crossing the vertical axis at the appropriate location, but you would like to lessen its incline, what do you do?  You reduce the value of the slope.  As you will see later, this idea of manipulating the coefficients to control a graph, can be extended to other functions besides linear.


QUESTIONS:
After completing this activity, prepare your responses to the following questions.

  1. In Practice 1, will the graph of the function cross the vertical axis above or below the origin?  What value helped you determine this?
  2. In Practice 3, without graphing, which line will be more vertical, that of  or ?  Explain how you determined this.
  3. In Practice 4, without the use of a graphing calculator, which is the most appropriate graph for the function ?
  4. In Practice 5, what function should Susie enter into the  editor of her calculator to shift her line down five units?
When you have prepared your responses to all of the above questions, click on this link:  Unit05 Activity05 Questions.  Remember, once you submit the form, you cannot go back and change your answers.  This form must be submitted by Monday, April 8th before 5:00pm (CST).


NEXT:
This is the end of the unit learning activities.  The next step is to complete the assessment for this unit (go to the top of the page and click on Assessment).


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