The graph of a function can be transformed in a number of ways. We will consider three:

The function we will use is f(x) = x². The graph of this function is shown at the right.

1. Below is a table of values for the function f(x) = x². Complete the remaining columns

2. Use the table to graph f(x)=2x², g(x)=x² -3, and h(x) = (x - 3)² on the coordinate axes below. Plot only the points that have a y-coordinate small enough to fit.

3. Compare each graph you drew to the graph of f(x) = x².

a. Which function has a graph that is a vertical shift of the graph of f(x) = x² ?
b. Is the vertical shift upward or downward?
c. Which function has a graph that is a horizontal shift of the graph of f(x) = x² ?
d. Is the horizontal shift right or left?
e. Which function has a graph that is a vertical stretch of the graph of f(x) = x² ?

4. Instead of being stretched vertically, a graph may be shrunk. What function would have a graph that is a vertical shrink of the graph of f(x) = x² by a factor of ½?____________

5. Describe how the graph of each of the following functions could be obtained from the graph of f(x) = x².

a. g(x) = (x+2)²
b. g(x) = x² + 1
c. g(x) = 3x²
d. g(x) = (x-2)² - 4
e. g(x) = 2(x-2)² - 4

  6. Using what you discovered you can write a function to match a graph. In the case of quadratic functions, the vertex of the parabola is the place to look to determine the vertical and the horizontal shifts. Estimate each shift for the graph shown below:

Vertical shift ___________

Horiaontal shift _________

To determine the stretch or the shrink, place your pencil at the vertex of the parabola and move one unit to the right. Then move upward until you are on the parabola. How far up did you move? This is your estimate of the vertical stretch or shrink. Use it and the two shifts to write the function.

f(x) = ____________________

7. Write a function for the graph shown at the right:

g(x) = ____________________