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Write the equation of the parabola that passes through the points $(3,0)$ , $(2,-27)$ , 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

(3,0)

,

(2,-27)

, 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: Identify the 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: $3$ , $-1$
Write equation in form: Use the $x$ -intercepts to write the equation in the form $y = a(x - p)(x - q)$ . $p = 3$ $q = -1$ $y = a(x - 3)(x - (-1))$ $y = a(x - 3)(x + 1)$
Use point to find 'a': Use the point $(2,-27)$ to find the value of 'a'. $\newline$ $y = a(x - 3)(x + 1)$ $\newline$ $-27 = a(2 - 3)(2 + 1)$ $\newline$ $-27 = a(-1)(3)$ $\newline$ $-27 = -3a$
Solve for 'a': Solve for 'a'. $\newline$ $-27 = -3a$ $\newline$ $a = \frac{-27}{-3}$ $\newline$ $a = 9$
Write final equation: Write the final equation of the parabola using the value of $a$ and the x-intercepts. $a = 9$ $p = 3$ $q = -1$ $y = a(x - p)(x - q)$ $y = 9(x - 3)(x + 1)$

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