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Write the equation of the parabola that passes through the points $(-9,-16$ ), $(-5,0$ ), and $(-4,0$ ). $\newline$ 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

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Q. Write the equation of the parabola that passes through the points

(-9,-16

),

(-5,0

), and

(-4,0

).

\newline

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: $-5$ , $-4$
Write equation in form: Use the $x$ -intercepts to write the equation in the form $y = a(x - p)(x - q)$ . $p = -5$ $q = -4$ $y = a(x - p)(x - q)$ $y = a(x + 5)(x + 4)$
Find value of 'a': Use the third point $(-9,-16)$ to find the value of 'a'. $\newline$ Substitute $x = -9$ and $y = -16$ into the equation $y = a(x + 5)(x + 4)$ . $\newline$ $-16 = a(-9 + 5)(-9 + 4)$ $\newline$ $-16 = a(-4)(-5)$ $\newline$ $-16 = 20a$
Solve for 'a': Solve for 'a'. $\newline$ $-16 = 20a$ $\newline$ $a = -\frac{16}{20}$ $\newline$ $a = -\frac{4}{5}$
Write final equation: Write the final equation of the parabola using the value of $a$ and the x-intercepts. $a = \frac{-4}{5}$ $p = -5$ $q = -4$ $y = a(x - p)(x - q)$ $y = \left(\frac{-4}{5}\right)(x + 5)(x + 4)$

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