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Write the equation of the parabola that passes through the points shown in the table. $x$ $y$ $- 7$ $0$ $- 6$ $20$ $- 1$ $0$ Write your answer in the form $y=a(x-p)(x-q)$ , where $a$ , $y$ $0$ , and $y$ $1$ 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.

x

y

- 7

0

- 6

20

- 1

0

Write your answer in the form

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

, where

a

,

y

0

, and

y

1

are integers, decimals, or simplified fractions.

Identify x-intercepts: Identify the x-intercepts from the given points. Points $(7, 0)$ and $(1, 0)$ suggest x-intercepts, so $p = 7$ and $q = 1$ .
Write equation in form: Write the equation in the form $y = a(x - p)(x - q)$ . Substituting $p = 7$ and $q = 1$ , we get $y = a(x - 7)(x - 1)$ .
Use point to find $a$ : Use the point $(-6, 20)$ to find the value of $a$ . Substitute $x = -6$ and $y = 20$ into the equation: $20 = a(-6 - 7)(-6 - 1)$ .
Simplify and solve for $a$ : Simplify and solve for $a$ : $20 = a(-13)(-7) = 91a$ . Divide both sides by $91$ to find $a$ : $a = \frac{20}{91}$ .
Write final equation: Write the final equation using the values found: $y = \left(\frac{20}{91}\right)(x - 7)(x - 1)$ .

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