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Write the equation of the parabola that passes through the points $(-1,0)$ , $(4,0)$ , $(5,-18)$ 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

(-1,0)

,

(4,0)

,

(5,-18)

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 $q$ : Step $1$ : Identify the values of $p$ and $q$ from the given points where the parabola crosses the x-axis, which are $(-1,0)$ and $(4,0)$ . Thus, $p = -1$ and $q = 4$ .
Use standard form: Step $2$ : Use the standard form of the equation for a parabola with these intercepts: $y = a(x - p)(x - q)$ . Substituting $p = -1$ and $q = 4$ , we get $y = a(x + 1)(x - 4)$ .
Substitute third point: Step $3$ : Substitute the third point $(5, -18)$ into the equation to find 'a'. Plugging in $x = 5$ and $y = -18$ , we get $-18 = a(5 + 1)(5 - 4)$ .
Simplify equation: Step $4$ : Simplify the equation from Step $3$ : $-18 = a(6)(1)$ , which simplifies to $-18 = 6a$ . Solving for $a$ gives $a = -18 / 6 = -3$ .
Substitute value of a: Step $5$ : Substitute the value of $a$ back into the equation from Step $2$ . We get $y = -3(x + 1)(x - 4)$ .

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