• As you investigate a new function today, you will completely describe graphs and explain how modifying the equation of the function changes its graph. As you work with your team, keep the multiple representations of functions in mind.
  • 1-29.Your team will investigate a function of the form.As a team, choose a value forhbetween –10 and10.For example, ifh=7, then.
  • Your Task:On a piece of graph paper, write down the function you get when you use your value forh. Then make anx→ytable and draw a complete graph of your function. Is there any more information you need to gather to be sure that you can see the entire shape of your graph? Discuss this question with your team and add any new information you think is necessary. You may use a scientific calculator, but not a graphing calculator, for this investigation.
  • Make sure you fully describe the graph (using the attributes from problem 113).
  • How can we be sure that our graph is complete?
  • How can we get output values that are greater than 1 or less than –1?

1-30.This function is different from others you have seen in the past. To get a complete graph, you will need to make sure your table includes enough information.

  1. Make a table with integerxvalues from 5 less than your value ofhto 5 more than your value ofh.For example, if you are working withh= 7, you would begin your table atx=2 and end it atx= 12. What do you notice about all of youryvalues?
  1. Is there anyxvalue that has noyvalue for your function? Why does your answer make sense?
  1. Plot all of the points that you have in your table so far.
  1. Now you will need to add more values to your table to see what happens to your function as your input values get close to your value ofh. Choose eight input values that are very close to your value ofhand on either side ofh. For example, if you are working withh=7, you might choose input values such as 6.5, 6.7, 6.9, 6.99, 7.01, 7.1, 7.3, and 7.5. For each new input value, calculate the corresponding output and add the new point to your graph.
  1. When you have enough points to be sure that you know the shape of your graph, sketch the curve.
  1. Fully describe the graph (using the attributes from problem113).
  • 1-31.Now you will continue your investigation of .
  1. The graph of has a vertical asymptote and a horizontal asymptote. To learn more about asymptotes, read the Math Notes box in this lesson. If you have not done so already, add the asymptotes as dashed lines to your graph, and write the equations of the asymptotes in your description of your function.
  1. Now each team member should choose a different value ofhbetween –10 and10, and make a table and a complete graph for the new function. You may use a scientific calculator but not a graphing calculator for this investigation.
  2. Examine all of your team’s functions. Together, consider how theh-value affects the graph of each function and add this to your description. Do you need to add any new attributes to your descriptions? Be as thorough as possible and be prepared to share your ideas with the class.

1-32.What will the graph oflook like?

  1. Discuss this question with your team and make a sketch of what you predict the graph will look like. Give as many reasons for your prediction as you can.
  1. Use your graphing calculator to graph.Do you see what you expected to see? Why or why not?
  2. Adjust the viewing window if needed. When you see the full picture of your graph, make a sketch of the graph on your paper. Label any important points.
  1. How close was your prediction?
  • 1-33.FAMILIES OF FUNCTIONS
  1. You investigated several functions of the formy=. What do all graphs ofy=have in common? What differences do they have?
  2. Since each function of the formy=has the same basic relationship betweenxandy, this set of functions can be called afamilyof functions. The functiony=(withh= 0), is called theparent function.
    In the equationy=,the letterhis called aparameter, whilexandyare calledvariables. What is the difference between a parameter and a variable? What role do parameters and variables have in a family of functions? Use examples from this lesson to illustrate your understanding of parameters and variables.
  3. What other families of functions have you studied in this course or previous courses? What is the parent function for each family?
  • 1-34.LEARNING LOG
  • Throughout this course, you are asked to reflect on your understanding of mathematical concepts in a Learning Log. Your Learning Log should contain explanations and examples to help you remember what you have learned throughout the course, and also contains questions you are trying to understand and answer. It is important to write each entry of the Learning Log in your own words so that later you can use your Learning Log as a resource to refresh your memory. Your teacher will tell you where to write your Learning Log entries and how to structure them. Remember to label each entry with a title and a date so you can refer to it or add to it later.
  • In your Learning Log, explain how families of functions, parameters, and the descriptions of graphs are related. What do graphs within a family of functions have in common? What is different within a family? Title this entry “Families of Functions” and include today’s date.

Graphs with Asymptotes

A mathematically clear and complete definition of an asymptote requires some ideas from calculus, but some examples of graphs withasymptotesshould help you recognize them when they occur. In the following examples, the dotted lines are the asymptotes, and the equations of the asymptotes are given. In the two lower graphs, theyaxis,x=0, is also an asymptote.

As you can see in the examples, asymptotes can be diagonal lines or even curves. However, in this course, asymptotes will almost always be horizontal or vertical lines. The graph of a function has ahorizontal asymptoteif, as you trace along the graph out to the left or right (that is, as you choosexvalues farther and farther away from zero, either toward negative infinity or toward positive infinity), the distance between the graph of the function and the asymptote gets smaller and smaller.

A graph has avertical asymptoteif, as you choosexcoordinates closer and closer to a certain value, from either the left or right (or both), theycoordinate gets farther away from zero, either toward infinity or toward negative infinity.