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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 Points: Identify the values of $p$ and $q$ from the given points where the parabola crosses the x-axis, which are the points $(-6,0)$ and $(1,0)$ . Thus, $p = -6$ and $q = 1$ .
Write General Form: Write the general form of the equation using the identified $p$ and $q$ : $y = a(x - p)(x - q)$ . Substituting $p = -6$ and $q = 1$ , we get $y = a(x + 6)(x - 1)$ .
Find Value of a: Use the third point $(-5, -42)$ to find the value of a. Substitute $x = -5$ and $y = -42$ into the equation: $-42 = a(-5 + 6)(-5 - 1)$ .
Solve for $a$ : Simplify and solve for $a$ : $-42 = a(1)(-6)$ , $-42 = -6a$ , $a = 7$ .

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