How do I find the rate of change from a table if the x-values don't go up by 1 each time?

Pick any two rows, subtract the y-values to get the change in output, subtract the x-values to get the change in input, then divide the change in output by the change in input. This gives the rate of change per single unit of input, even if the table itself jumps by 2, 5, or any other amount between rows.

What if a graph doesn't show the y-intercept directly on the grid?

Use the slope to work backward from any visible point. Find the rate of change from two points on the line, then move backward from one point one input-unit at a time, adjusting the output by the rate of change each step, until you reach an input of 0.

Does the function that starts higher always stay ahead?

No. A function with a lower starting value but a larger rate of change can catch up and eventually pass a function that started higher. The two functions will be equal at their intersection point, and after that point the one with the larger rate of change will be ahead.

Why does the initial value matter if two functions eventually become almost equal?

The initial value determines who is ahead before the functions meet, which matters a lot in real situations with a limited time frame. For example, if you only need a plan for 3 months, the plan that starts cheaper might be the better choice even if another plan would eventually become cheaper after month 12.

M8MATH-6.3

Comparing Functions in Different Representations

Learn to compare linear functions shown as tables, graphs, equations, or word problems by finding rate of change and starting value for each.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Comparing Functions in Different Representations, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

Imagine two savings plans, one shown as a table of deposits and the other described in a sentence. How do you know which one grows faster or which one starts with more money? In this lesson you'll learn to pull the same two pieces of information, rate of change and initial value, out of any representation of a linear function, whether it's dressed up as a table, a graph, an equation, or a paragraph of words. Once those two numbers are in front of you, comparing functions becomes as simple as comparing numbers.

The Two Numbers That Matter

Every linear function is built from exactly two pieces of information: the rate of change (how fast the output changes as the input increases by 1) and the initial value (the output when the input is 0). In an equation written as

y = mx + b

, the rate of change is

m

and the initial value is

b

. Once you find these two numbers for a function, no matter how it was presented, you can compare it to any other linear function.

The skill in this lesson is not computing something new. It's extracting

m

and

b

from four different disguises: a table of values, a graph, an equation, and a written description. Students often treat each representation as a totally different type of problem, but they are really the same two numbers wearing different costumes.

A table shows the rate of change as the amount the output changes each time the input increases by 1 (or however much the input jumps, divided by that jump). The initial value is the output paired with an input of 0, or found by working backward from another row. A graph shows the rate of change as the slope, rise over run, between any two points, and the initial value as the

y

-intercept, where the line crosses the vertical axis. A verbal description usually states the rate of change as a per-unit amount ('increases by 5 each week') and the initial value as a starting amount ('starts at 20').

Extracting Information From Each Representation

The table below shows exactly what to look for in each format so you never get stuck translating.

Representation	How to find rate of change	How to find initial value
Equation $y = mx + b$	$m$ , the coefficient of $x$	$b$ , the constant term
Table	$\frac{\Delta y}{\Delta x}$ between two rows	the $y$ -value when $x = 0$ , or extend the pattern backward
Graph	slope between two clear points, $\frac{\text{rise}}{\text{run}}$	the point where the line crosses the $y$ -axis
Verbal description	the words 'per', 'each', 'every' usually attach to it	the words 'starts at', 'begins with', 'flat fee' usually attach to it

A common mistake with tables is assuming the rate of change is just the difference between consecutive

y

-values without checking the

x

-values also increase by 1. If the input jumps by 2 or 3 each row, you must divide the change in

y

by the actual change in

x

, not just read the difference in

y

.

A common mistake with graphs is misreading the

y

-intercept when the graph doesn't start at

x = 0

. Always trace the line all the way to where it crosses the vertical axis, even if you have to extend it mentally.

A common mistake with verbal descriptions is mixing up which number is the rate and which is the starting value, especially when the sentence order is reversed, like 'Starting from 40 dollars, the tank drains at 5 dollars per hour,' where 40 is still the initial value even though it's mentioned first.

Comparing Rate of Change: Who Grows Faster

Once you have the rate of change for each function, comparing which one grows faster (or shrinks faster) is just comparing two numbers. The function with the larger rate of change increases more quickly as the input grows. If one rate is negative and the other is positive, the positive one is increasing while the negative one is decreasing, so the positive-rate function will eventually be larger no matter where they start.

Be careful with negative rates: a rate of change of

-3

is smaller than

-1

, even though 3 is bigger than 1 in absolute value, because

-3

means the function is dropping faster. When comparing two negative rates, the one closer to zero is actually the slower-changing, and therefore in a sense the 'faster growing' (less negative) function.

Comparing initial values is more straightforward: whichever function has the larger

b

value (or larger starting output) begins higher on the graph, at

x = 0

. But a higher starting value does not guarantee a function stays ahead. A function that starts lower but has a much larger rate of change will eventually catch up and overtake the one that started higher. This is exactly what happens at the point where two lines intersect.

Finding Where Two Functions Meet

The point where two linear functions have the same output for the same input is called their intersection or meeting point. To find it, set the two equations equal to each other and solve for

x

, then substitute back to find

y

. For example, if

y = 3x + 2

and

y = x + 8

, setting them equal gives

3x + 2 = x + 8

, so

2x = 6

and

x = 3

; then

y = 3(3) + 2 = 11

, so they meet at

(3, 11)

.

If the functions are given in mixed representations, first convert each one into an equation

y = mx + b

using the rate of change and initial value you extracted, then solve as above. This is why extracting

m

and

b

accurately in the earlier steps matters so much: any error there will shift the intersection point.

A meeting point makes real-world sense in many problems: two competing offers, plans, or savings accounts often cross at the input value where they become equally good, and knowing whether one is better before or after that point helps answer questions like 'which plan should you choose if you only need it for 3 months.'

Key terms

rate of change.: How much the output of a function changes for each unit increase in the input; called slope on a graph and often the coefficient of $x$ in an equation.
initial value.: The output value of a function when the input is 0; the starting amount before any change has occurred.
slope.: The rate of change of a linear function read from a graph, calculated as rise divided by run, $\frac{\Delta y}{\Delta x}$ , between any two points on the line.
y-intercept.: The point where a line crosses the vertical axis, where $x = 0$ ; its $y$ -value equals the initial value of the function.
intersection point.: The input-output pair where two functions produce the same output for the same input; found by setting the two equations equal and solving.
linear function.: A function whose rate of change is constant, producing a straight line when graphed, and that can be written as $y = mx + b$ .

Worked example

A gym membership plan A is described by the equation

y = 15x + 40

, where

x

is the number of months and

y

is the total cost in dollars. Plan B is shown in this table: after 1 month the total cost is 65 dollars, after 2 months it is 85 dollars, after 3 months it is 105 dollars. Which plan starts cheaper, which plan costs more per month, and after how many months do the two plans cost the same total amount?

Start with Plan A since it's already an equation in the form

y = mx + b

. The rate of change is

m = 15

, meaning it costs 15 dollars more for each additional month. The initial value is

b = 40

, meaning there is a 40 dollar starting fee before any monthly charges.

Now extract the same two numbers from Plan B's table. The input increases by 1 each row, and the output increases from 65 to 85 to 105, a change of 20 each time, so the rate of change for Plan B is 20 dollars per month.

To find Plan B's initial value, work backward one step from month 1. Since the cost increases by 20 each month, the cost at month 0 would be

65 - 20 = 45

dollars. So Plan B's equation is

y = 20x + 45

.

Compare initial values: Plan A starts at 40 dollars, Plan B starts at 45 dollars, so Plan A starts cheaper by 5 dollars.

Compare rates of change: Plan A increases by 15 dollars per month, Plan B increases by 20 dollars per month, so Plan B costs more per month and grows faster.

Since Plan A starts cheaper and grows slower, Plan A will always cost less than Plan B for every month after the start, unless they meet somewhere. Set the equations equal to check:

15x + 40 = 20x + 45

. Subtract

15x

from both sides to get

40 = 5x + 45

. Subtract 45 from both sides to get

-5 = 5x

, so

x = -1

.

Since a negative number of months doesn't make sense in this real situation, the two plans never actually meet for any month greater than or equal to 0. Plan A stays cheaper than Plan B the entire time someone could realistically be a member.

Practice questions

Function P is graphed as a line passing through the points (0, 6) and (2, 10). Function Q is given by the equation

y = 3x + 2

. Which function has the greater rate of change?

Function P, because its rate of change is 2
Function Q, because its rate of change is 3
Function P, because it starts higher
Function Q, because it starts lower

Answer: Function Q, because its rate of change is 3

For Function P, use the two points to find slope:

\frac{10-6}{2-0}=\frac{4}{2}=2

, so its rate of change is 2. Function Q's equation already shows the rate of change as the coefficient of

x

, which is 3. Since 3 is greater than 2, Function Q has the greater rate of change, even though Function P starts higher at 6 compared to Function Q's initial value of 2.

A savings account is described as starting with 200 dollars and growing by 25 dollars every week. A second account is shown in a table: week 2 has 260 dollars, week 4 has 310 dollars. Write an equation for each account in the form

y = mx + b

, and determine which account has more money after 10 weeks.

Answer: Account 1 is $y = 25x + 200$ ; Account 2 is $y = 25x + 210$ ; after 10 weeks Account 2 has more money.

Account 1's description directly gives initial value 200 and rate of change 25, so

y = 25x + 200

. For Account 2, find the rate of change using the table: from week 2 to week 4, the input increases by 2 and the output increases by

310-260=50

, so the rate is

50\div 2=25

dollars per week. Working backward from week 2, at week 0 the value would be

260 - 2(25) = 210

, so Account 2 is

y = 25x + 210

. Both accounts grow at the same rate of 25 dollars per week, but Account 2 starts 10 dollars higher, so it stays ahead by exactly 10 dollars at every week, including week 10, where Account 1 has 450 dollars and Account 2 has 460 dollars.

Two functions never intersect when graphed. What must be true about their rates of change and initial values?

They must have the same rate of change but different initial values
They must have different rates of change but the same initial value
They must have both the same rate of change and the same initial value
They must have different rates of change and different initial values

Answer: They must have the same rate of change but different initial values

Two lines that never cross are parallel, meaning they have identical slopes (rates of change) but different y-intercepts (initial values). If the rates of change were different, the lines would eventually cross at some intersection point, no matter how far apart their starting values were. If both the rate of change and initial value were the same, the two functions would actually be the exact same line, overlapping everywhere rather than never meeting.

FAQ

How do I find the rate of change from a table if the x-values don't go up by 1 each time?: Pick any two rows, subtract the y-values to get the change in output, subtract the x-values to get the change in input, then divide the change in output by the change in input. This gives the rate of change per single unit of input, even if the table itself jumps by 2, 5, or any other amount between rows.
What if a graph doesn't show the y-intercept directly on the grid?: Use the slope to work backward from any visible point. Find the rate of change from two points on the line, then move backward from one point one input-unit at a time, adjusting the output by the rate of change each step, until you reach an input of 0.
Does the function that starts higher always stay ahead?: No. A function with a lower starting value but a larger rate of change can catch up and eventually pass a function that started higher. The two functions will be equal at their intersection point, and after that point the one with the larger rate of change will be ahead.
Why does the initial value matter if two functions eventually become almost equal?: The initial value determines who is ahead before the functions meet, which matters a lot in real situations with a limited time frame. For example, if you only need a plan for 3 months, the plan that starts cheaper might be the better choice even if another plan would eventually become cheaper after month 12.

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