Vertex Form of a Quadratic

VERTEX FORM OF A QUADRATIC

1. Identify the stretch/compression factor and the vertex for each of the following.

a. y = 0.3(x – 1)2 +8

b. y = 30x2 – 11

2. For each of the following functions, identify the vertex and specify whether it represents a maximum or minimum and then sketch its graph by hand.

a. y = (x – 2)2

b. y = ½ (x – 2)2 + 3

3. For each quadratic function identify the vertex, axis of symmetry, and specify whether it represents a maximum or minimum. Evaluate the function at x = 0 and then sketch a graph of the function by hand.

a. f(t) = 0.25(t – 2)2 + 1

b. g(x) = 3 – (x – 5)2

4. Create a quadratic function in the vertex form y = a(x – h)2 + k, given the specified values for a and the vertex (h, k). Then rewrite the function in the standard form y = ax2 + bx + c. Use your graphing calculator to check that the graphs of the two forms are the same.

a. a = 1, (h, k) = (2, -4)

b. a = -2, (h, k) = (-3, 1)

5. Transform the function f(x) = x2 into a new function g(x) by shrinking f(x) by a factor of ¼, then shifting it down by 6 units. Find the equation of g(x) and sketch it by hand.

6. Transform the function f(x) = 3x2 into a new function h(x) by reflecting f(x) over the x axis, shifting the result horizontally to the right 4 units, and then shifting it up by 5 units.

a. What is the equation for h(x)?

b. What is the vertex of h(x)?

c. What is the vertical intercept of h(x)?

7. For each quadratic function use the method of “completing the square” to convert to the a-h-k form, and then identify the vertex. Use your graphing calculator to confirm that the two forms are the same.

a. y = x2 + 8x + 15

b. f(x)= x2 – 4x – 5

c. z = 2m2 + 6m – 5