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Write the equation of the parabola that passes through the points shown in the table. $(-3,12)$ $(-2 ,0)$ $(-1,0)$ 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 shown in the table.

(-3,12)

(-2 ,0)

(-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 $p$ and $q$ : Identify the values of $p$ and $q$ from the given points. The points $(-2, 0)$ and $(-1, 0)$ indicate the x-intercepts of the parabola, so $p = -2$ and $q = -1$ .
Write general form: Write the general form of the equation using the identified $p$ and $q$ . Substitute $-2$ for $p$ and $-1$ for $q$ into the equation $y = a(x - p)(x - q)$ . This gives $y = a(x + 2)(x + 1)$ .
Use point $(-3, 12)$ : Use the point $(-3, 12)$ to find the value of $a$ . Substitute $-3$ for $x$ and $12$ for $y$ into the equation $y = a(x + 2)(x + 1)$ . This results in $12 = a(-3 + 2)(-3 + 1)$ .
Simplify and solve: Simplify and solve for $a$ . The equation becomes $12 = a(-1)(-2)$ , which simplifies to $12 = 2a$ . Solving for $a$ gives $a = \frac{12}{2} = 6$ .
Write final equation: Write the final equation of the parabola using the found value of $a$ . Substitute $6$ for $a$ into $y = a(x + 2)(x + 1)$ . This results in $y = 6(x + 2)(x + 1)$ .

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