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Write the equation of the parabola that passes through the points $(-6,-15)$ , $(-1,0)$ , and $(-7,0)$ . 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

(-6,-15)

,

(-1,0)

, and

(-7,0)

. 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: $-1$ , $-7$
Write equation in form: Use the x-intercepts to write the equation in the form $y = a(x - p)(x - q)$ . $p = -1$ $q = -7$ $y = a(x - (-1))(x - (-7))$ $y = a(x + 1)(x + 7)$
Find value of 'a': Use the third point $(-6,-15)$ to find the value of $a$ . Substitute $x = -6$ and $y = -15$ into the equation $y = a(x + 1)(x + 7)$ . $-15 = a(-6 + 1)(-6 + 7)$ $-15 = a(-5)(1)$ $-15 = -5a$ $a = \frac{-15}{-5}$ $a = 3$
Write final parabola equation: Write the final equation of the parabola using the value of $a$ and the x-intercepts. $a = 3$ $p = -1$ $q = -7$ $y = a(x - p)(x - q)$ $y = 3(x + 1)(x + 7)$

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