Question

Find the roots, y-intercept, and vertex of the quadratic function f(x) = x2 − 2x − 3.

Answer 1

see explanation

Explanation:

To find the roots let f(x) = 0, that is

x² - 2x - 3 = 0

(x - 3)(x + 1) = 0 ← in factored form

Equate each factor to zero and solve for x

x - 3 = 0 ⇒ x = 3

x + 1 = 0 ⇒ x = - 1

Roots are x = - 1, x = 3

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To find the y- intercept let x = 0, that is

f(0) = 0 - 0 - 3 = - 3 ← y- intercept

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The vertex lies on the axis of symmetry which is situated at the midpoint of the roots.

`x_(vertex)` =
`(-1+3)/(2)` = 1

substitute x = 1 into f(x) for y- coordinate of vertex

`y_(vertex)` = 1² - 2(1) - 3 = 1 - 2 - 3 = - 4

vertex = (1, - 4)