Sketching & Comparing Functions
Real questions often give two functions in different forms — one as an equation, one as a table — and ask which grows faster or starts higher. The tool is rate of change plus a few key features.
For a line, the rate of change is just the slope. Comparing two lines is comparing their slopes; the steeper slope wins the growth race.
Rate of change as the comparison tool
For a linear function the rate of change is the slope — the coefficient of . In the rate is ; in it is .
The larger rate means the function climbs faster, even if it starts lower. From a table, compute rate of change as the change in over the change in .
Reading across forms
To compare an equation with a table, pull the same features from each: the rate of change and the starting value (-intercept). Put them side by side.
A steeper line eventually overtakes a shallower one, no matter the head start — so the greater rate of change tells you who wins in the long run. In the graph below, the steeper line starts lower but crosses above the shallower one.
Worked examples
Example 1: which grows faster
Which has the greater rate of change: or ?
Answer:
Example 2: rate from a table
A table shows y going 3, 7, 11 as x goes 0, 1, 2. Find the rate of change.
Answer:
Try one yourself
Common questions
Does a bigger starting value mean a function is bigger everywhere?
No. A function with a smaller -intercept but a larger rate of change will eventually overtake one that started higher.
How do I find rate of change from a table?
Divide the change in y by the change in x between two rows. If it is constant, the function is linear and that value is the slope.
Is rate of change the same as slope?
For linear functions, yes. For curves the rate of change varies, but over any interval it is still (change in y) / (change in x).
Want the video version?
Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.