Linear Functions

Read slope as a rate with units

A delivery service charges $C(d)=4+1.5d$, where $d$ is distance in miles and $C$ is cost in dollars. The slope 1.5 means each additional mile adds 1.50 dollars. The intercept 4 is the fixed charge when the distance is zero.

A question about the coefficient of $d$ asks about change, not total cost. At 6 miles, the total is $C(6)=13$ dollars, but the rate is still 1.50 dollars per mile.

Build the rule from a table

Suppose a table contains $(2,11)$ and $(6,23)$. The slope is $m=\frac{23-11}{6-2}=3$. Use either point in $y=3x+b$: $11=6+b$, so $b=5$. The function is $f(x)=3x+5$.

The output difference 12 is not the slope because the input changed by 4. Divide changes in the same order: second output minus first output over second input minus first input.

Use the intercept that matches the question

The vertical intercept comes from setting the input to zero. The horizontal intercept comes from setting the output to zero. For $f(x)=3x+5$, the vertical intercept is $(0,5)$ and the horizontal intercept is $(-\frac{5}{3},0)$.

In a context, a horizontal intercept may be the time when a quantity reaches zero. Check whether that time is within the model's stated domain before treating it as a real event.

Distinguish constant change from constant percent change

Outputs 10, 15, 20 at inputs 0, 1, 2 have constant differences, so they fit a linear rule. Outputs 10, 15, 22.5 have constant ratios of 1.5, so they fit an exponential rule instead.

When the table's input gaps vary, compare rates rather than raw output differences. A longer input interval should produce a proportionally larger change in a linear model.

Check your understanding

Sample linear functions questions

Practice this skill

1. Write the units of each slope and intercept before selecting an interpretation.
2. Find a rule from two points, then use a third point to check it.
3. Practice a mix of equations, tables, and verbal descriptions.

Print the linear functions worksheet and worked answers.

Then use a mixed practice module to check whether you can choose this method among other question types.