Problem

Build a complete sketch fingerprint for \(y=(x-1)(x-5)\). Find the zeros, use their midpoint for the axis, evaluate the vertex and \(y\text{-}\text{intercept}\), use the leading coefficient for the opening direction, and verify the symmetric inputs \(x=2\) and \(x=4\).

Big Picture

What this problem is really about

An exact sketch fingerprint should derive enough checked landmarks to determine one parabola unambiguously. We’ll combine roots and their midpoint with the vertex, opening, vertical intercept, and a reflected point pair, then infer domain, range, and end behavior from that structure.

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Zeros 1,5; axis x=3; vertex (3,−4); opens down; y-intercept (0,5); symmetric points (2,−3), (4,−3).

Zeros 1,5; axis x=1; vertex (1,0); opens up; y-intercept (0,5); symmetric points (0,5), (2,−3).

Zeros 1,5; axis x=3; vertex (3,−4); opens up; y-intercept (0,5); symmetric points (2,−3), (4,−3).

Zeros −1,−5; axis x=−3; vertex (−3,32); opens up; y-intercept (0,5); symmetric points (−4,45), (−2,21).

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Build a complete sketch fingerprint for \(y=(x-1)(x-5)\). Find the zeros, use their midpoint for the axis, evaluate the vertex and \(y\text{-}\text{intercept}\), use the leading coefficient for the opening direction, and verify the symmetric inputs \(x=2\) and \(x=4\).

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