Question

Type the correct answer in each box. Use numerals instead of words.

This graph represents a quadratic function. What is the function’s equation written in factored form and in vertex form?

Answer 1

The quadratic function is represented by the factored form "f(x) = 4/3 * x * (x - 5)" and the vertex form "f(x) = 4/3 * (x - 2)^2 - 8."

The given graph of the quadratic function provides key points to determine its equation in factored and vertex forms.

Factored Form:

Since the parabola passes through the points (0, 0) and (5, 0), the factored form can be expressed as :

"f(x) = a * x * (x - 5),"

Where "a" is the leading coefficient.

To find "a," use the given vertex (2, -8) by substituting these coordinates into the equation:

-8 = a * 2 * (2 - 5).

Solving for "a,"

We get

"a = 4/3."

So, the factored form is "f(x) = 4/3 * x * (x - 5)."

Vertex Form:

The vertex form is:

"f(x) = a * (x - h)^2 + k."

Using the given vertex (2, -8), substitute these values:

f(x) = 4/3 * (x - 2)^2 - 8.

Answer 2

f(x) = 2x(x - 8)

f(x) = 2(x - 2)² - 8

Explanation:

Let the equation of the quadratic function is,

f(x) = a(x - h)² + k

Here, (h, k) is the vertex of the function.

From the graph attached,

Vertex of the parabola → (2, -8)

Therefore, equation of the function will be,

f(x) = a(x - 2)² - 8

Since, the graph passes through origin (0, 0),

f(0) = a(0 - 2)² - 8

0 = 4a - 8

a = 2

Equation of the given parabola will be,

f(x) = 2(x - 2)² - 8

= 2(x² - 4x + 4) - 8

= 2x² - 8x + 8 - 8

= 2x² - 8x

= 2x(x - 8)

Therefore, factored form of the function will be,

f(x) = 2x(x - 8)