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All courses Algebra II · F-IF.4 28 of 55
Interpret key features of rational, square-root, cube-root, and other function models in context.

Audit contextual x- and y-intercepts with units

Problem
For \(P(x)~=~(x~-~5)/(x~+~2)\), where \(x~\ge~0\) is weeks and \(P\) is a unitless profit ratio, find \(\text{both}~\text{intercepts}\), check their feasibility, and interpret each contextual intercept.
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Hint

Solve output=0 for x-intercepts and evaluate input 0 for y-intercepts, checking all restrictions.

Find algebraic intercepts first, then check the model's domain before attaching contextual meaning and units.

Solution walkthrough

01

Find the x-intercept

\[P(x)=0~->~(x-5)/(x+2)=0~->~x-5=0~->~x=5\]

A rational expression is zero when its numerator is zero and its denominator is not. At x=5, the denominator is 7, so the x-intercept exists at (5,0).

02

Find the y-intercept

\[P(0)=(0-5)/(0+2)=-5/2\]

The y-intercept uses input x=0. Substitution gives the exact unitless profit ratio -5/2, so the coordinate is (0,-5/2).

03

Check contextual feasibility

\[5\ge~0~\text{and}~0\ge~0\]

The model permits all week inputs x>=0. Both intercept inputs, week 5 and week 0, lie in that contextual domain.

04

Interpret both intercepts

\[\text{Algebra}:~x-\text{intercept}~(5,0),~y-\text{intercept}~(0,-5/2).~\text{Feasibility}:~\text{both}~\text{inputs}~5~\text{and}~0~\text{satisfy}~x\ge~0.~\text{Interpretation}:~\text{at}~\text{week}~5~\text{the}~\text{ratio}~\text{is}~0;~\text{at}~\text{week}~0~\text{it}~\text{is}~-5/2.\]

The input coordinate is measured in weeks and the output is a unitless profit ratio. This gives every requested algebraic, feasibility, and contextual field, choice A.

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Another way

  1. Evaluate P(5) and P(0) directly after identifying the two intercept inputs.

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Common mistake

For the x-intercept, setting the numerator to zero is valid only after checking that the denominator is nonzero. Also interpret the coordinates using weeks and a unitless ratio.

Solution walkthrough video