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All courses Math I · F-LE.2 31 of 59
Construct linear and exponential functions, including arithmetic/geometric sequences, from graphs, descriptions, or input-output pairs.

Construct a linear function from graph points

Problem
Use the displayed plotted points to construct the linear function.
Coordinate graph with plotted points (0, 3) and (2, 7). Open full size
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Hint

Use the two points \((0, 3)\) and \((2, 7)\) to find the slope: rise over run.

Once you know the slope, use the point with \(x = 0\) to read the y-intercept directly. Then write the equation in the form \(f(x)=mx+b\).

Solution walkthrough

01

Read the plotted points and bounds

\[(0,3)~\text{and}~(2,7)\]

The inspected coordinate graph labels exactly these points; both lie inside its displayed x- and y-bounds.

02

Compute the slope

\[m=(7-3)/(2-0)=4/2=2\]

Output rises 4 while input runs 2, so the signed constant rate is positive 2.

03

Identify the intercept and build the rule

\[(0,3)~->~b=3;~f(x)=\text{mx}+b=2x+3\]

The point with x=0 directly supplies the y-intercept 3. Combining it with slope 2 gives the linear function.

04

State and check the function

\[f(x)=2x+3\]

Substitution gives f(2)=2(2)+3=7, reproducing the second plotted point and checking the model.

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Another way

  1. Starting at y=3 when x=0, the graph rises 2 per input unit, giving values 3,5,7 and the same rule.

!

Common mistake

Do not use the y-change 4 as slope. Slope is rise divided by run, so 4/2=2.

Worked linear model: slope (7-3)/(2-0)=2 and intercept 3, so f(x)=2x+3.
model: f(x)=2x+3