Find the equation of quadratic function determined from the graph below?

f(x) =  1 / 3 x² - 4 / 3 x - 5 / 3

using the form:

f(x) = a(x - h)² + k

The vertex coordinates are 2 and -3.

h = 2

k = - 3

therefore,

f(x) = a(x - 2)²  - 3

f(x) = a(x - 2)² - 3

let's use the coordinates (-1, 0) to find a. Therefore,

0 = a(-1 - 2)² - 3

0 = 9a - 3

3 = 9a

a = 3 / 9

a = 1 / 3

let's insert the value of a in the equation.

f(x) = a(x - 2)² - 3

f(x) = 1 / 3 ( x - 2)² - 3

f(x) = 1 / 3 (x - 2)(x -2) - 3

f(x) = 1 / 3 (x² - 4x + 4) - 3

f(x) = x² / 3 - 4x / 3 + 4 / 3 - 3

f(x) =  1 / 3 x² - 4 / 3 x - 5 / 3

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Step-by-step explanation:

The x-intercepts are x = -1 and x = 5, so:

y = k (x + 1) (x − 5)

The vertex is (2, -3), so:

-3 = k (2 + 1) (2 − 5)

-3 = -9k

k = 1/3

y = 1/3 (x + 1) (x − 5)

Simplifying:

y = 1/3 (x² − 4x − 5)

y = 1/3 x² − 4/3 x − 5/3

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step-by-step explanation: