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All courses Algebra I · F-IF.9 71 of 76
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 A from vertex form

\[\begin{aligned} A(x)=(x-2)^2+3 \\ \text{vertex}~(2,3);~\text{coefficient}~1>0~->~\text{minimum} \end{aligned}\]

The form a(x-h)²+k gives vertex (h,k). A's positive square coefficient makes the parabola open upward.

02

Read B from vertex form

\[\begin{aligned} B(x)=-(x-5)^2+8 \\ \text{vertex}~(5,8);~\text{coefficient}~-1<0~->~\text{maximum} \end{aligned}\]

B has h=5 and k=8. Its negative square coefficient reflects the parabola downward, so its vertex is a maximum.

03

Compare vertex coordinates

\[\begin{aligned} 2<5~->~A~\text{is}~\text{left}~\text{of}~B \\ 3<8~->~A~\text{vertex}~\text{is}~\text{lower}~\text{than}~B \end{aligned}\]

The x-coordinates differ by 3 and the vertex values differ by 5; they share neither coordinate nor value.

04

Compare the extrema precisely

\[\begin{aligned} A~\text{minimum};~B~\text{maximum} \\ \text{same}~x-\text{coordinate}=\text{no};~\text{same}~\text{value}=\text{no};~\text{comparable}~\text{same}-\text{type}~\text{extrema}=\text{no} \end{aligned}\]

A minimum and B maximum are different extremum types, so they should not be ranked as two minima or two maxima. The inspected paired graph confirms the locations and openings.

+

Another way

  1. Use square bounds: (x-2)²>=0 gives A>=3, while -(x-5)²<=0 gives B<=8.

!

Common mistake

Do not say B has a minimum just because 8 is its vertex value. The negative coefficient makes the vertex the highest point, so it is a maximum.

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

Solution walkthrough video