Question

write the equation of the parabola that passes through the points shown in the table.

x	y
-2	0
4	-24
6	0

write your answer in the form y = a(x - p)(x - q), where a, p, and q are integers, decimals, or simplified fractions.

Explanation:

Step1: Identify roots from table

The parabola has roots at \( x = -2 \) and \( x = 6 \) (since \( y = 0 \) at these \( x \)-values). So the factored form is \( y = a(x - (-2))(x - 6)=a(x + 2)(x - 6) \).

Step2: Find \( a \) using the point \( (4, -24) \)

Substitute \( x = 4 \) and \( y = -24 \) into the equation:
\( -24 = a(4 + 2)(4 - 6) \)
Simplify the right - hand side: \( (4 + 2)=6 \) and \( (4 - 6)=-2 \), so \( -24 = a\times6\times(-2) \)
\( -24 = -12a \)
Solve for \( a \): \( a=\frac{-24}{-12}=2 \)

Step3: Write the final equation

Substitute \( a = 2 \) into the factored form: \( y = 2(x + 2)(x - 6) \)

Answer:

\( y = 2(x + 2)(x - 6) \)