Question

What is the equation of the quadratic function represented by this graph

Answer 1

f(x) = - x² + 2x + 8

or f(x) = - (x - 1)² + 9

or f(x) = - (x + 2)(x - 4)

Explanation:

f(0) = 8 {y-axis intercept = (0, 8)}

Vertex = (1, 9)

f(x) = a(x - 1)² + 9

8 = a(0-1)² + 9

8 = a + 9

a = -1

f(x) = - (x - 1)² + 9

f(x) = - (x² - 2x + 1) + 9

f(x) = - x² + 2x - 1 + 9

f(x) = - x² + 2x + 8

x₁ = -2 and x₂ = 4, so also:

f(x) = -(x + 2)(x - 4)

Answer 2

g(x) = -1 (x + 2)(x − 4)

Explanation:

To write the equation of the function in factored form, g(x) = a(x − r1)(x − r2), find the values of r1, r2, and a.

The function has x-intercepts of -2 and 4, so (x − (-2)), or (x + 2), and (x − 4) are factors of the equation.

To find a, look at how the value of y changes for 1 unit to the right of the vertex. The y-value goes down 1 unit for a point 1 unit to the right of the vertex, so a = -1.

The equation of this quadratic function is g(x) = -(x + 2)(x − 4).

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