Applied Math- functions

Question

A parabola has a vertex of (4, 8) and points down.

Write two possible equations of quadratic functions in vertex form such that the parabola has exactly two zeroes.

Find the zeroes for both functions.

1. 1 mark for setting up the first function with the vertex substituted appropriately
2. 1 mark for choosing an appropriate a value for the first function
3. 1 mark for finding both the zeroes for the first function
4. 1 mark for setting up the second function with the vertex substituted appropriately
5. 1 mark for choosing an appropriate a value for the second function
6. 1 mark for finding both the zeroes for the second function

Answer 1

The vertex form of a quadratic function is

y = a(x - h)² + k, where (h, k) is the vertex.

Given: Vertex is (4, 8), so h = 4, k = 8.

Given: The parabola points down, so a must be negative.

Given: Parabola has exactly two zeros, so k must be positive (because a < 0), which is the case: k = 8.

1. 1 mark for setting up the first function with the vertex substituted appropriately

y = a(x - 4)² + 8

1. 1 mark for choosing an appropriate a value for the first function (-2)

y = (-2)(x - 4)² + 8

1. 1 mark for finding both the zeroes for the first function

(-2)(x - 4)² + 8 = 0

(x - 4)² = 4

x - 4 = ±2, x = 4 - 2 or x = 4 + 2, x = 2 or x = 6

Also called x-intercepts, zeros of the function are x = 2 and x = 6

In the point form, points (2, 0) and (6, 0).

1. 1 mark for setting up the second function with the vertex substituted appropriately

y = a(x - 4)² + 8

1. 1 mark for choosing an appropriate a value for the second function (-1/2)

y = (-1/2)(x - 4)² + 8

1. 1 mark for finding both the zeroes for the second function

(-1/2)(x - 4)² + 8 = 0

(x - 4)² = 16

x - 4 = ±4, x = 4 - 4 or x = 4 + 4, x = 0 or x = 8

Also called x-intercepts, zeros of the function are x = 0 and x = 8

In point form, points (0, 0) and (8, 0)

Answer 2

f(x) = -(x - 4)² + 8

f(x) = -3(x - 4)² + 8