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All courses Algebra II · A-REI.11 19 of 55
Solve f(x)=g(x) approximately using intersections of polynomial, rational, radical, absolute-value, exponential, and logarithmic graphs.

Estimate solutions by reading intersections of a polynomial graph and a line

Problem
Use the graph of the parabola and line to estimate every intersection input and point.
Coordinate graph of a parabola and a line on the same grid, intersecting near (1,3) and (4,6). Open full size
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Hint

Read the x-values where the two graphs cross, then pair them with the nearby intersection coordinates.

An intersection estimate from a graph gives both the approximate x-solutions and the approximate crossing points when those coordinates are described.

Solution walkthrough

01

Use the intersection criterion

\[\text{intersection}~->~\text{same}~x~\text{and}~\text{same}~y\]

A solution of f(x)=g(x) occurs wherever the parabola and line occupy the same plotted point.

02

Scan the full displayed window

\[\text{two}~\text{crossings}~\text{visible}\]

The inspected graph shows one crossing on the parabola's left side and a second crossing on its rising right side. Checking the whole window prevents stopping after one solution.

03

Read both coordinates from the grid

\[(1,3)~\text{and}~(4,6)\]

The first shared point aligns with x=1 and y=3; the second aligns with x=4 and y=6.

04

Report inputs and points

\[x~\text{approx}~1~\text{and}~x~\text{approx}~4;~\text{intersections}~\text{near}~(1,3)~\text{and}~(4,6)\]

The requested intersection inputs are the x-coordinates 1 and 4, paired with their common outputs 3 and 6, choice A.

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Another way

  1. If equations were available, solve their equality; here the task specifically calls for reading every shared point from the graph.

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Common mistake

Do not report only the lower crossing. The line meets the parabola again on the right side of the displayed graph.

Coordinate graph showing a line and a parabola with the intersection points open parenthesis 1 comma 3 close parenthesis and open parenthesis 4 comma 6 close parenthesis marked and labeled.

Solution walkthrough video