Question

1. determine the quadratic function that defines a parabola with x-intercepts ( x = -1 ) and ( x = 3 ) and y-intercept ( y = -6 ). provide a sketch to support your work.

Explanation:

Step1: Use factored form of quadratic

A quadratic with x - intercepts \( x = - 1 \) and \( x=3 \) can be written in factored form as \( y=a(x + 1)(x - 3) \), where \( a\) is a constant.

Step2: Find the value of \( a \) using y - intercept

The y - intercept is \( y=-6 \), which occurs when \( x = 0 \). Substitute \( x = 0 \) and \( y=-6 \) into the equation:

$$
LATEXBLOCK0
$$

Divide both sides by - 3: \( a=\frac{-6}{-3}=2 \)

Step3: Expand the factored form

Substitute \( a = 2 \) into the factored form \( y=a(x + 1)(x - 3) \):

$$
LATEXBLOCK1
$$

Answer:

The quadratic function is \( y = 2x^{2}-4x - 6 \)