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Try It 9.126: Determining a Quadratic Function from its Graph

When finding a quadratic function from a given graph, use the vertex form f(x)=a(x−h)2+kf(x) = a(x-h)^2 + k by identifying the vertex and a secondary point. For instance, given an upward-facing parabola with a vertex at (−3,−1)(-3, -1) and a yy-intercept at (0,8)(0, 8), the vertex provides h=−3h = -3 and k=−1k = -1. Substituting these into the vertex form gives f(x)=a(x−(−3))2−1f(x) = a(x - (-3))^2 - 1, which simplifies to f(x)=a(x+3)2−1f(x) = a(x + 3)^2 - 1. Next, use the yy-intercept (0,8)(0, 8) to solve for aa by substituting x=0x = 0 and f(x)=8f(x) = 8: 8=a(0+3)2−18 = a(0 + 3)^2 - 1. This becomes 8=9a−18 = 9a - 1, which simplifies to 9=9a9 = 9a, resulting in a=1a = 1. Thus, the quadratic function for this graph is f(x)=(x+3)2−1f(x) = (x + 3)^2 - 1.

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Updated 2026-05-25

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References

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OpenStax

Intermediate Algebra @ OpenStax

Ch.9 Quadratic Equations and Functions - Intermediate Algebra @ OpenStax

Algebra

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