Comparing Two Linear Functions

The two numbers that matter

The rate of change is the slope: how much $y$ changes for each increase of $1$ in $x$. In $y = mx + b$ it is the coefficient $m$ — the number attached to $x$.

The starting value is the $y$-intercept: the value of $y$ when $x = 0$. In $y = mx + b$ it is the constant $b$. On a graph, it is where the line crosses the $y$-axis.

Pulling them out of each form

From an equation: read them straight off $y = mx + b$. In $y = 3x + 2$, the rate of change is $3$ and the starting value is $2$. Do not swap them — the rate is always the number multiplying $x$.

From a table: pick two rows and divide the change in $y$ by the change in $x$ — the rate is $\displaystyle \frac{y_2 - y_1}{x_2 - x_1}$. This works even when the table skips $x$-values. For the starting value, find the row where $x = 0$; if there is no such row, step the pattern backward until you reach $x = 0$.

From a graph: the starting value is where the line crosses the $y$-axis, and the rate of change is the rise over the run between two grid points. In the graph below, both lines start at $1$, but the blue line rises $3$ for every step right while the black line rises $1$ — the steeper line has the greater rate of change.

Rates are not totals

A table that reaches a large $y$-value does not automatically have the larger rate — it may just show more rows. Always divide the change in $y$ by the change in $x$ before comparing.

Also, the function with the greater rate of change is not automatically greater at every $x$-value. A function with a smaller rate but a bigger starting value can stay ahead for a while. If a question asks which function is greater at a specific $x$, evaluate both functions there.

Worked examples

Try one yourself

Common questions