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 _____________________________________________________________________