Question

write the equation of the parabola that passes through the points (-4, 18), (-3, 0), and (-1, 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 points

The parabola has roots at \( x = -3 \) and \( x = -1 \) (since \( y = 0 \) at these points). So the factored form is \( y = a(x - (-3))(x - (-1)) = a(x + 3)(x + 1) \).

Step2: Substitute the point \((-4, 18)\)

Substitute \( x = -4 \) and \( y = 18 \) into \( y = a(x + 3)(x + 1) \):
\( 18 = a(-4 + 3)(-4 + 1) \)
Simplify inside the parentheses: \( 18 = a(-1)(-3) \)
Simplify the right - hand side: \( 18 = 3a \)

Step3: Solve for \( a \)

Divide both sides by 3: \( a=\frac{18}{3}=6 \)

Step4: Write the final equation

Substitute \( a = 6 \) back into the factored form: \( y = 6(x + 3)(x + 1) \)

Answer:

\( y = 6(x + 3)(x + 1) \)