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All courses Math II · F-IF.9 25 of 73
Compare function properties across representations, especially quadratic comparisons.

Compare two quadratic vertex records

Problem
Compare these quadratics by vertex: A(x)=(x−2)²+3; B(x)=−(x−5)²+8.
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Hint

Write A and B as vertex/type records before selecting relations.

A vertex has two coordinates and an extremum type; compare each separately.

Solution walkthrough

01

Read vertex A from vertex form

\[A(x)~=~(x-2)^2+3~\text{has}~\text{vertex}~(2,3)\]

This matches the form \(a(x-h)^2+k\), so the vertex is \((h,k)=(2,3)\).

02

Determine whether A has a minimum or maximum

\[a~=~1>0\]

Because the coefficient is positive, quadratic A opens upward and has a minimum at its vertex.

03

Read vertex B and its type

\[B(x)~=~-(x-5)^2+8~\text{has}~\text{vertex}~(5,8)~\text{and}~a~=~-1<0\]

The vertex is \((5,8)\), and the negative coefficient means B opens downward and has a maximum.

04

State the comparison

\[A~\text{vertex}=(2,3);~A~\text{type}=\text{minimum};~B~\text{vertex}=(5,8);~B~\text{type}=\text{maximum};~x-\text{location}~\text{relation}=A~\text{left}~\text{of}~B;~\text{vertex}-\text{value}~\text{relation}=A~\text{lower}~\text{than}~B;~\text{same}~x-\text{coordinate}=\text{no};~\text{same}~\text{vertex}~\text{value}=\text{no};~\text{comparable}~\text{same}-\text{type}~\text{extrema}=\text{no}\]

This compares both quadratics by the location and type of their vertices.

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Another way

  1. Sketch the opening direction from the sign of each squared term coefficient, then pair that with the vertex coordinates.

!

Common mistake

Reading \((x-5)^2\) as vertex x-coordinate \(-5\) instead of \(5\) in vertex form.

Answer comparison showing the exact vertices, zeros, axes, or rate evidence required by the variant.
A vertex=(2,3); A type=minimum; B vertex=(5,8); B type=maximum; x-location relation=A left of B; vertex-value relation=A lower than B; same x-coordinate=no; same vertex value=no; comparable same-type extrema=no