1Cademy - Finding the Equation of a Parabolic Arch (20 Feet High and 40 Feet Wide)

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Finding the Equation of a Parabolic Arch Bridge

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Finding the Equation of a Parabolic Arch (20 Feet High and 40 Feet Wide)

To find the equation of a parabolic arch that is $20$ feet high and $40$ feet wide at its base, first establish a coordinate system. Place the lower left side of the arch at the origin $(0, 0)$ , which makes the lower right side $(40, 0)$ . The vertex, representing the highest point of the arch, lies halfway between the base points due to symmetry, making its $x$ -coordinate $20$ . Because the arch is $20$ feet high, the vertex is $(20, 20)$ . Substitute the vertex $(h, k) = (20, 20)$ into the standard form equation $y = a(x - h)^2 + k$ to get $y = a(x - 20)^2 + 20$ . Next, use another point on the parabola, such as the origin $(0, 0)$ , to solve for the coefficient $a$ . Substituting $x = 0$ and $y = 0$ yields $0 = a(0 - 20)^2 + 20$ . Solving this equation gives $-20 = 400a$ , which simplifies to $a = -\frac{1}{20}$ . Substituting $a$ back into the vertex equation gives the final standard form: $y = -\frac{1}{20}(x - 20)^2 + 20$ .

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Updated 2026-05-25

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