Functions

Comparing Functions

Comparing properties of two functions given in different forms (table, graph, equation).

Intermediate15 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

What Is a Function?

Comparing Functions in Different Disguises

One function is given as a table of values. A friend's function is given as an equation. Before reading on, think about how you'd fairly compare which one grows faster — since a table and an equation don't look anything alike at first glance.

The trick is to translate both functions into the same kind of information — usually their rate of change, found the same way regardless of how the function was originally presented. A rate of change from a table comes from dividing the change in output by the change in input between two rows; from an equation like y = mx + b, it's simply the slope, m.

Worked Example — Comparing a Table to an Equation

Function A is given by the table: x = 0, 2, 4 with y = 3, 11, 19. Function B is given by y = 5x + 1. Find each function's rate of change: for A, (11 − 3)/(2 − 0) = 4. For B, the slope is directly 5. Since 5 is greater than 4, Function B increases faster.

Worked Example — Comparing Two Graphs

Two linear graphs are shown, one passing through (0, 2) and (3, 8), the other passing through (0, 1) and (4, 9). Find each slope: the first is (8−2)/(3−0) = 2. The second is (9−1)/(4−0) = 2. Both functions have the exact same rate of change, even though they start at different y-values.

Function Explorer

f(x) =

sin(x)

Transform: g(x) = a·f(b(x − h)) + k

a — vertical stretch/reflect1

b — horizontal stretch/reflect1

h — horizontal shift0

k — vertical shift0

Composition

Compose with another function

Trace mode

Show inverse view

Analysis (of the transformed function, in view)

y-intercept: (0, 0)
x-intercepts: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Extrema: local min at (-7.85, -1); local max at (-4.71, 1); local min at (-1.57, -1); local max at (1.57, 1); local min at (4.71, -1); local max at (7.85, 1)
Inflection points: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-1, 1]

Tip

Whatever form a function is given in — table, graph, or equation — reduce it to its rate of change (and, if needed, its starting value) before comparing, rather than trying to compare the raw table, graph, and equation directly.

Common Mistakes

Comparing two functions' starting values (y-intercepts) when the question is actually asking about their rate of change, or vice versa.
Check exactly what the question is comparing — 'which increases faster' asks about rate of change (slope), while 'which starts higher' asks about the initial value.
Computing a rate of change from a table using non-adjacent or mismatched rows inconsistently.
Pick any two rows and consistently divide the difference in y-values by the difference in x-values — for a genuinely linear function, any two rows give the same result.

Key Takeaways

Functions given in different forms — table, graph, or equation — can be compared by translating each into its rate of change.
A rate of change from a table is found by dividing the change in output by the change in input.
Two functions can share the same rate of change while having different starting values, or vice versa.

Summary

Reducing any function's representation to its rate of change makes fair comparison possible. The final lesson in this unit uses that same rate-of-change thinking to build a linear function model from a real situation.

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