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All courses Algebra I · F-IF.4 64 of 76
Interpret key graph/table features of quadratic models in context.

Interpret the vertex of a quadratic model in context

Problem
Interpret the vertex of \(h(t)=-16(t-2)^{2}+70\) when \(t\) is time in seconds and \(h\) is height in feet. State the vertex with units, use the coefficient sign to classify the extremum, confirm the time occurs after launch, and give the physical meaning.
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Hint

Label h with input units and k with output units before interpreting.

The vertex(h,k) pairs contextual input h with output k; sign(a) determines whether k is a maximum or minimum.

Solution walkthrough

01

Read the vertex with units

\[\begin{aligned} h(t)=-16(t-2)^2+70 \\ \text{vertex}:~(2~\text{seconds},~70~\text{feet}) \end{aligned}\]

The model is in a(t-h)²+k form. The input coordinate h=2 is time after launch, and the output coordinate k=70 is height.

02

Classify the extremum

\[a=-16<0~->~\text{opens}~\text{downward}~->~\text{vertex}~\text{is}~a~\text{maximum}\]

A negative quadratic coefficient makes the parabola open downward, so every nearby modeled height is below the vertex height.

03

Check the time against launch

\[\begin{aligned} \text{launch}~\text{time}~t=0 \\ \text{vertex}~\text{time}~t=2 \\ 2>0 \end{aligned}\]

Time is measured after launch, and the vertex occurs two seconds later than t=0. Thus the maximum is part of the post-launch motion.

04

Interpret the vertex

\[\text{maximum}~\text{height}=70~\text{feet}~\text{at}~t=2~\text{seconds}\]

The input and output coordinates together say that the model predicts its greatest height, 70 feet, two seconds after launch.

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Another way

  1. Evaluate h(2)=70 and note that -16(t-2)² is never positive, so no modeled height can exceed 70.

!

Common mistake

Do not reverse the vertex coordinates. Time is the input and belongs first; height is the output and belongs second.

Answer graph of h(t)=−16(t−2)²+70 with time t in seconds and height h in feet; axis t=2 and downward-opening vertex (2, 70) mean a maximum height of 70 feet at 2 seconds.
vertex=(2,70); opening=down; extremum type=maximum; contextual input=2 seconds; contextual output=70 feet; domain status=vertex occurs after launch; interpretation=model predicts peak height70ft at2s

Solution walkthrough video