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Write the equation of the parabola that passes through the points $(-6,0$ ), $(1,0$ ), and $(-5,-42$ ). 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

(-6,0

),

(1,0

), and

(-5,-42

). Write your answer in the form

y = a(x - p)(x - q)

, where

a

,

p

, and

q

are integers, decimals, or simplified fractions.

Identify $p$ and $q$ : Identify the values of $p$ and $q$ from the given points where the parabola crosses the x-axis. $\newline$ The points $(-6, 0)$ and $(1, 0)$ are the x-intercepts of the parabola, so $p = -6$ and $q = 1$ .
Write parabola equation: Write the equation of the parabola in the form $y = a(x - p)(x - q)$ using the identified values of $p$ and $q$ . Substitute $-6$ for $p$ and $1$ for $q$ into the equation $y = a(x - p)(x - q)$ . $y = a(x + 6)(x - 1)$
Find value of $a$ : Use the third point $(-5, -42)$ to find the value of $a$ . $\newline$ Substitute $-5$ for $x$ and $-42$ for $y$ into the equation $y = a(x + 6)(x - 1)$ . $\newline$ $-42 = a(-5 + 6)(-5 - 1)$
Solve for $a$ : Solve the equation to find the value of $a$ . $\newline$ $-42 = a(1)(-6)$ $\newline$ $-42 = -6a$ $\newline$ $a = -42 / -6$ $\newline$ $a = 7$
Final parabola equation: Write the final equation of the parabola using the value of $a$ found in Step $4$ . $\newline$ Substitute $7$ for $a$ into the equation $y = a(x + 6)(x - 1)$ . $\newline$ $y = 7(x + 6)(x - 1)$

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