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Write the equation of the parabola that passes through the points $(4,0)$ , $(6,0)$ , $(8,4)$ . 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

(4,0)

,

(6,0)

,

(8,4)

. 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 where the parabola crosses the x-axis, which are $(4,0)$ and $(6,0)$ . Thus, $p = 4$ and $q = 6$ .
Write parabola equation: Write the equation of the parabola in the form $y = a(x - p)(x - q)$ . Substituting $p = 4$ and $q = 6$ , we get $y = a(x - 4)(x - 6)$ .
Substitute point $(8, 4)$ : Substitute the point $(8, 4)$ into the equation to find the value of $a$ . Plugging in $x = 8$ and $y = 4$ , we get $4 = a(8 - 4)(8 - 6)$ .
Solve for a: Simplify and solve for a: $4 = a(4)(2)$ , $4 = 8a$ , $a = \frac{4}{8}$ , $a = 0.5$ .
Final equation of parabola: Write the final equation of the parabola using the values of $a$ , $p$ , and $q$ . Substituting $a = 0.5$ , $p = 4$ , and $q = 6$ into $y = a(x - p)(x - q)$ , we get $y = 0.5(x - 4)(x - 6)$ .

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