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All courses Algebra I · F-IF.9 27 of 76
Compare two functions represented in different ways: equations, graphs, tables, or verbal descriptions.

Compare linear functions by slope and y-intercept

Problem
Compare the linear functions \(f(x)=2x+5\) and \(g(x)=4x+1\) by slope and intercept.
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Hint

Both functions are in slope-intercept form \(y=mx+b\). Read the slope from the coefficient of \(x\) and the y-intercept from the constant term.

For \(f(x)=2x+5\), the slope is \(2\) and the y-intercept is \(5\). For \(g(x)=4x+1\), the slope is \(4\) and the y-intercept is \(1\).

Solution walkthrough

01

Decode both slope-intercept forms

\[f(x)=2x+5:~\text{slope}~2,~y-\text{intercept}~5;~g(x)=4x+1:~\text{slope}~4,~y-\text{intercept}~1\]

In y=mx+b, the coefficient of x is slope and the constant is the y-intercept. Read those roles separately for each function.

02

Compare the slopes

\[4>2,~\text{so}~g~\text{has}~\text{the}~\text{greater}~\text{slope}\]

Function g rises 4 output units per input unit while f rises 2, so g is steeper.

03

Compare the y-intercepts

\[5>1,~\text{so}~f~\text{has}~\text{the}~\text{greater}~y-\text{intercept}\]

At x=0, f(0)=5 and g(0)=1, so f crosses the y-axis higher.

04

State both comparisons

\[\text{Since}~\text{the}~\text{coefficient}~\text{of}~x~\text{is}~\text{the}~\text{slope}~\text{and}~\text{the}~\text{constant}~\text{is}~\text{the}~y-\text{intercept},~g~\text{has}~\text{the}~\text{greater}~\text{slope}~\text{and}~f~\text{has}~\text{the}~\text{greater}~y-\text{intercept}.\]

This answers both requested comparisons while explicitly connecting each conclusion to the correct equation component.

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Another way

  1. Evaluate each function at x=0 for intercepts, then compare one-unit output changes for slopes.

!

Common mistake

Do not compare constants for slope or coefficients for intercept. The x-coefficients are 2 and 4; the intercept constants are 5 and 1.

Solution walkthrough video