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Write the equation of the parabola that passes through the points $(-2,0)$ , $(2,3)$ , $(4,0)$ 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

(-2,0)

,

(2,3)

,

(4,0)

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 and vertex: Identify the x-intercepts and vertex from the given points $(-2,0$ ), $(2,3)$ , and $(4,0)$ . The x-intercepts are at $(-2,0)$ and $(4,0)$ , suggesting $p = -2$ and $q = 4$ . The vertex form is not directly given, but we can use symmetry to find the x-coordinate of the vertex, which is the midpoint between $-2$ and $4$ .
Calculate vertex x-coordinate: Calculate the x-coordinate of the vertex using the midpoint formula: $(p + q) / 2 = (-2 + 4) / 2 = 1$ . However, the given vertex point $(2,3)$ suggests the x-coordinate is $2$ , not $1$ . This indicates a need to adjust our understanding of the vertex or recheck the intercepts.

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