TRANSFORMATIONS of the Quadratic Function: Y = X2 - Part I

Name: ______

TRANSFORMATIONS of the Quadratic function: y = x2 - Part I

A. Graph the quadratic function: 2 yx Describe the basic shape of this graph:

______

What are the coordinates of the vertex of this graph?

______

B. Keep this quadratic function in y1, and graph each of the following: 1. y = (x – 3)2 2. y = (x – 2)2 3. y = (x – 1)2

Vertex:_____ Vertex:_____ Vertex:_____

C. Compare graphs 1, 2, and 3 with the “parent-graph” ( y= (x)2. How are “the kids” different? How are they the same?

D. Investigate the behavior of some other “kids” by graphing each of the following functions:

4.y = (x + 3)2 5. y = (x + 2)2 6. y = (x + 1)2

E. Compare graphs 4, 5, 6 with the “parent-graph” ( y= x2). How are “the kids” different? How are they the same? F. Next, investigate the graphs of the following 6 functions:

7. y = x2 - 3 8. y = x2 - 2 9. y = x2 - 1

Vertex:______Vertex:______Vertex:______

10. y = x2 + 3 11. y= x2 + 2 12. y = x2 + 1

G. In a short paragraph, describe the similarities and differences between graphs 1-6 and 7-12.

H. If you did not have access to a graphing calculator, how might you explain to a new student in the class how to draw the graphs of each of the following functions: 13. y = x2 + 4 14. y = (x – 8)2

15. y = (x + 12)2 16. y = x2 - 7

I. Without use of a graphing calculator, how might you draw the graph of each of these functions? 17. y = (x – 3)2 + 2 18. y = (x + 1)2 - 4

19. y = (x + 2)2 + 3 20. y = (x – 2)2 - 1 TRANSFORMATIONS of the Quadratic Function: y = x2 - Part II

A. Graph the Quadratic function: (remember to keep this graph in y1) 2 yx Describe the basic shape of this graph:

______

What are the coordinates of the vertex of this graph?

______Vertex:_____

B. Draw the graph of the following function: y = - x2

Compare this graph with the “parent” graph in part A.

______

Why do you think this graph is called a “vertical reflection” of the quadratic function?

Vertex:______

C. Now try graphing the following quadratic function equations.

1. y = - x2 + 1 2. y = - x2 - 2

Vertex:____ Vertex:_____

3. y = - (x – 3)2 4. y = - (x + 4)2

Vertex:_____ Vertex:_____

D. Explain what happens to the graph if the quadratic is multiplied by a negative. E. Now we’re going to get tricky! - Graph the following: 5. y = 2x2 Describe the similarities and differences between the graph of y = |x| and this graph.

______

Vertex:_____

F. Now try graphing the following quadratic equations.

6. y = ½ x2 7. y = (3/2) x2 8. y = 0.75x2

G. What is the effect on the graph of multiplying the quadratic function by a positive number that is greater than 1?

H. What is the effect on the graph of multiplying the quadratic function by a positive number that is less than 1?

I. PUTTING IT ALL TOGETHER ! In the following functions, I have combined the transformations on the quadratic function that you just discovered. Try graphing the following functions without a graphing calculator, then use your calculator to check your answer.

9. y = (x – 1)2 + 2 10. y = - (x + 3)2 11. y = 3(x)2 - 6

Vertex:_____ Vertex:_____ Vertex:_____ Homework: 4. y = -2(x – 4)2 5. f(x) = ½ (x – 2)2 – 1 6. g(x) = - (x + 2)2 + 5

Show your tables here:

Make a conjecture about functions that come in the form: y = a(x – h)2 + k. Explain the effect of a, h and k on the quadratic function graph.