Compare Properties of Two Functions

Students compare key properties—rate of change, initial value, maximum/minimum—of two functions that may be represented differently (one as a graph, one as a table, one as an equation, or one as a verbal description). They determine which function has a greater rate of change or a higher starting value.

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Grade 8 · Functions · question 1 of 3

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One phone plan's total cost is graphed, and another plan's is the equation below, with x in months and y in dollars. Which plan's cost grows faster each month, if either?

the same rate

How it is taught

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Rate of change is how much the output grows for each unit of input.

Initial value is the output when the input is 0.

Read it: Comparing two functions on a graph

Line P starts at 2 and climbs 3 for each step across. Line Q starts at 8 and climbs 1. Q starts higher, but P grows faster: its rate of change is 3, and Q's is 1.

To read a value, go up from x to the line, then across to the vertical axis.
A line's start is where it meets the vertical axis, at x = 0.
A line's rate is how much it rises for one step across; the larger rate grows faster.

Each line pairs every x with one y, so it is a function you can read off the plane.

P rises 3 each step and Q rises 1, so P closes the gap of 6 by 2 each step.

The lines cross where they show the same value: after 3 steps both read 11.

Compare two functions by their starts, their rates and where they meet.

Picking the line that starts higher as the one that grows faster.

The start is where a line begins; the rate is how fast it climbs. P starts at 2 but climbs 3 a step, faster than Q.

Line A starts at 4 and goes up 2 for each step across. What does it read at x = 3? — 10. Start at 4 and add 2 for each of the 3 steps: 4 + 2 × 3 = 10.

y = 3x + 2 vs table of values

We have two functions, written in different ways. Which one grows faster — and which one starts higher?

Function A is given as an equation: y equals three x plus two.

In y equals m x plus b, the m is the rate of change. For A, that's three.

And b — the initial value, where x is zero — is two.

Function B is given as a table of x and y values.

Each time x grows by one, y grows by two. So B's rate of change is two.

When x is zero, y is five. That's B's initial value.

Now compare the rates. Three is greater than two, so Function A grows faster.

Compare the starts. Five is greater than two, so Function B starts higher.

Function A grows faster, but Function B starts ahead. Two functions, two stories!

Grade 8 · Comparing rate of change and initial value

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics

Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal: 1 star — one skill mastered, every rung of its ladder climbed
Climbing now: Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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