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Write the equation of the parabola that passes through the points $(1,0)$ , $(3,0)$ , and $(5,40)$ . 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)

,

(3,0)

, and

(5,40)

. Write your answer in the form

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

, where

a

,

p

, and

q

are integers, decimals, or simplified fractions.

Identify x-intercepts: Identify the x-intercepts from the given points. $\newline$ The x-intercepts are the x-values of the points where the y-values are zero. $\newline$ x-intercepts: $1$ , $3$
Write equation in form: Use the $x$ -intercepts to write the equation in the form $y = a(x - p)(x - q)$ .
$p = 1$
$q = 3$
$y = a(x - 1)(x - 3)$
Find value of 'a': Use the third point $(5,40)$ to find the value of 'a'. $\newline$ Substitute $x = 5$ and $y = 40$ into the equation $y = a(x - 1)(x - 3)$ . $\newline$ $40 = a(5 - 1)(5 - 3)$ $\newline$ $40 = a(4)(2)$ $\newline$ $40 = 8a$
Solve for 'a': Solve for 'a'. $\newline$ $40 = 8a$ $\newline$ $\frac{40}{8} = a$ $\newline$ $a = 5$
Write final equation: Write the final equation of the parabola using the value of $a$ and the $x$ -intercepts. $\newline$ $a = 5$ $\newline$ $p = 1$ $\newline$ $q = 3$ $\newline$ $y = a(x - p)(x - q)$ $\newline$ $y = 5(x - 1)(x - 3)$

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