Exploring the Equations of Parabolas

(a lesson plan using TI graphing calculators by Rebecca A. Zeier)

 

 

Objective:

After answering a variety of guided questions about the graphs of parabolas, students should be able to identify the different properties of parabolas solely by looking at the equations.

 

Activity: 

Have the students work independently or in pairs to answer the following questions.

 

 

Directions:

Go to the graphing part of the TI Graphing Calculator and answer the following questions.

 

I.  Graph y = x2

 

1. Describe the graph: ___________________________________________

______________________________________________________________

 

2. Now graph the following equations

y = x2 + 1

y = x2 + 3

           

3. Where does each graph cross the y-axis?

y = x2 ____________

y = x2 + 1 _________

y = x2 + 3 _________

 

4. Where do you think the graph of y = x2 + 6 will cross the y-axis? _________

 

5. Graph y = x2 + 6.  Was your answer in number 4 correct? ________________

 

6. Make a conjecture.  What does the “b” in the general equation y = x2 + b tell you about the graph of the line? ___________________________________________________________________________________

 

7. Does your conjecture hold true for the following equations?

y = x2 + 5 ________

y = x2 + 2 ________

 

 

Clear the graphs.

 

II.  Graph (x + 1) 2

1. What happens? ____________________________________________________________________

 

2. Graph (x + 3) 2.  What happens to this graph?______________________________________________

 

3. Describe what happens differently to these graphs from part I. __________________________________________________________________________________

 

4. Where do you think the graph of y = (x – 6) 2 will look like? __________________________________________________________________________________

 

5. Graph y = (x – 6) 2.  Was your guess in number 4 correct?  ___________________________________

 

6. Make a conjecture.  What does the “d” in the general equation y = (x + d) 2 do to the graph?_____________________________________________________________________________

 

 

 

III. Graph the following equations on the same screen of your calculator:

y = (x + 1) 2 + 2

y = (x + 1) 2 + 5

y = (x + 1) 2 + 7

                       

1. How are the graphs the same? _________________________________________________________

 

2. How do the graphs differ? ____________________________________________________________

 

 

IV. Graph the following equations on the same screen of your calculator:

y = (x + 3) 2 + 2

y = (x - 5) 2 + 2

y = (x + 7) 2 + 2

 

1. How are the graphs the same? __________________________________________________________

 

2.  How do the graphs differ? _____________________________________________________________

 

 

V.  Use your knowledge you have gained to predict what will be true about the following graphs:

y = (x - 3) 2 + 7 _______________________________________________________________________

 

y = (x + 7) 2 – 6 _______________________________________________________________________

 

y = x 2 + 3 ___________________________________________________________________________

 

y = (x + 0) 2 – 2 _______________________________________________________________________

 

 

VI. Graph the graphs from part V.  Were you correct? __________________________________________

 

 

VII.  Extra Credit:  What do you think will happen when you graph the following equation?

 

y = - (x + 2) 2 + 7   _____________________________________________________________________