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Write the equation of the parabola that passes through the points $(1,0)$ , $(6,0)$ , and $(7,30)$ . 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)

,

(6,0)

, and

(7,30)

. 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: We have: $\newline$ Points: $(1,0)$ , $(6,0)$ and $(7,30)$ $\newline$ Identify 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: $1$ , $6$
Simplify using x-intercepts: We have: $\newline$ $p = 1$ $\newline$ $q = 6$ $\newline$ Simplify $y = a(x - p)(x - q)$ using x-intercepts. $\newline$ $y = a(x - p)(x - q)$ $\newline$ $y = a(x - 1)(x - 6)$
Simplify with given point: We have: $\newline$ $y = a(x - 1)(x - 6)$ $\newline$ Point: $(7,30)$ $\newline$ Simplify $y = a(x - 1)(x - 6)$ . $\newline$ $y = a(x - 1)(x - 6)$ $\newline$ $30 = a(7 - 1)(7 - 6)$ $\newline$ $30 = a(6)(1)$ $\newline$ $30 = 6a$
Solve for $a$ : We have: $\newline$ $30 = 6a$ $\newline$ Solve for $a$ . $\newline$ $30 = 6a$ $\newline$ $\frac{30}{6} = \frac{6a}{6}$ $\newline$ $a = 5$
Write parabola equation: We have: $\newline$ $a = 5$ $\newline$ $p = 1$ $\newline$ $q = 6$ $\newline$ Write the equation of the parabola in the form $y = a(x - p)(x - q)$ . $\newline$ $y = a(x - 1)(x - 6)$ $\newline$ $y = 5(x - 1)(x - 6)$

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