Graph linear and quadratic functions and identify intercepts, maxima, and minima.
Construct and validate a quadratic graph from key features
Problem
Construct a quadratic with zeros \(1\) and \(5\), opening upward, and \(y\text{-}\text{intercept}\) \((0,5)\). Start with \(y=a(x-1)(x-5)\), use the \(y\text{-}\text{intercept}\) to solve for \(a\), then verify the axis, vertex, and opening direction against the graph scaffold. Identify the construction that passes every check.
Given zeros determine a factored family of quadratics, but an additional point is what fixes the vertical scale. We’ll build that family, use the supplied intercept to solve the scale, derive the axis and vertex from symmetry, and validate every required point and the opening direction before tracing the curve.
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