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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.

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Write a quadratic function from its x-intercepts and another point

Full solution

Q. 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.

Identify x-intercepts: We have: $\newline$ Points: $(-4,18)$ , $(-3,0)$ , and $(-1,0)$ $\newline$ Identify x-intercepts from the given points. $\newline$ The x-intercepts are the x-values of the points where the y-values are zero. $\newline$ x-intercepts: $-3$ , $-1$
Simplify using x-intercepts: We have: $\newline$ $p = -3$ $\newline$ $q = -1$ $\newline$ Simplify $y = a(x - p)(x - q)$ using x-intercepts. $\newline$ $y = a(x - p)(x - q)$ $\newline$ $y = a(x + 3)(x + 1)$
Simplify with point: We have: $\newline$ $y = a(x + 3)(x + 1)$ $\newline$ Point: $(-4,18)$ $\newline$ Simplify $y = a(x + 3)(x + 1)$ . $\newline$ $y = a(x + 3)(x + 1)$ $\newline$ $18 = a(-4 + 3)(-4 + 1)$ $\newline$ $18 = a(-1)(-3)$ $\newline$ $18 = 3a$
Solve for $a$ : We have: $\newline$ $18 = 3a$ $\newline$ Solve for $a$ . $\newline$ $18 = 3a$ $\newline$ $\frac{18}{3} = \frac{3a}{3}$ $\newline$ $a = 6$
Write parabola equation: We have: $\newline$ $a = 6$ $\newline$ $p = -3$ $\newline$ $q = -1$ $\newline$ Write the equation of the parabola in the form $y = a(x - p)(x - q)$ . $\newline$ $y = a(x + 3)(x + 1)$ $\newline$ $y = 6(x + 3)(x + 1)$

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