Do I always have to use the first and second rows of a table to find slope?

No. You can use any two rows. The slope will be the same no matter which rows you pick, as long as the relationship is linear. Pick rows that are easy to subtract, like rows where the numbers are clear and far enough apart.

What if x = 0 is not shown on the graph?

If x = 0 is off the edge of the graph, use the slope formula to find any two clear points on the line you can see, calculate m, then use b = y - mx with either of those points. Rearrange to find b without needing to see the y-intercept itself.

Can the rate of change be negative?

Yes. A negative slope means y decreases as x increases. For example, if a tank is draining, the amount of water goes down over time, so the rate of change is negative. If the equation is y = -5x + 20, then m = -5, meaning y decreases by 5 units for every 1-unit increase in x.

Why is it called 'initial value'?

Because in many real situations, x = 0 represents the starting point or the beginning of time, and the y-value at that point is what you start with or what happens before anything else occurs. For example, in a phone bill, the initial value is the base fee charged before you use any minutes. In a tank of water, it's how much water is there at the beginning.

M8MATH-7.1

Rate of Change & Initial Value from Tables & Graphs

Learn how to find the slope (m) and y-intercept (b) from tables and graphs, then write the equation y = mx + b for any linear relationship.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Rate of Change & Initial Value from Tables & Graphs, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

Every straight-line relationship has two key numbers that describe it: how steep it is, and where it crosses the y-axis. In this lesson you'll learn to spot those numbers in data tables and on graphs, and use them to write the equation of any linear function. These skills let you turn a real-world pattern—like distance traveled over time, or cost per item—into a mathematical model you can use to predict values and solve problems.

What Are Rate of Change and Initial Value?

Rate of change is how much the output (y) changes every time the input (x) increases by 1 unit. Mathematically, that's slope:

m = \frac{\Delta y}{\Delta x}

, which means "the change in y divided by the change in x." If you travel 120 miles every 2 hours, your rate of change is

\frac{120 \text{ miles}}{2 \text{ hours}} = 60

miles per hour.

Initial value is the y-value when x equals 0. It's where the line crosses the y-axis, called the y-intercept or b. If a phone plan charges a 15 dollar base fee before you use any minutes, that 15 dollars is the initial value. Initial value is called "initial" because it's often what you start with or what you pay before anything happens.

Together, rate of change (m) and initial value (b) fully describe any linear relationship. Once you know both, you can write

y = mx + b

and predict any output for any input.

Finding Rate of Change from a Table

To find slope from a table, pick any two rows and calculate

m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}

.

Let's try an example. The table shows how much water (in gallons) is in a tank over time (in minutes):

Time (x)	Water (y)
2	50
5	110
8	170

Using rows 1 and 2:

m = \frac{110 - 50}{5 - 2} = \frac{60}{3} = 20

gallons per minute.

Using rows 2 and 3:

m = \frac{170 - 110}{8 - 5} = \frac{60}{3} = 20

gallons per minute.

You get the same answer either way—that's how you know you're reading the table correctly. A common mistake: students reverse the subtraction or mix up which number goes on top. Always subtract the first row from the second row, and remember that change in y goes on top.

Finding Initial Value from a Table or by Using the Equation

If x = 0 is in the table: just read the y-value. If the table shows x = 0 in the first row and y = 12, then b = 12.

If x = 0 is NOT in the table: use the rate of change you found and pick any row. Substitute into

b = y - mx

.

Using the water tank example above with

m = 20

, pick the first row:

x = 2, y = 50

b = 50 - 20(2) = 50 - 40 = 10

So the tank started with 10 gallons. You can check using any other row:

b = 170 - 20(8) = 170 - 160 = 10

. ✓

The equation is

y = 20x + 10

. This means at time 0 (before any water flows in), there are 10 gallons. Every minute, 20 more gallons enter.

Why does this work? The equation

y = mx + b

is always true on the line. Rearrange it to

b = y - mx

, and you can find b using any point (x, y) that you know.

Finding Rate of Change and Initial Value from a Graph

On a graph, slope is the steepness. Start at any point and move right 1 unit. How far up (or down) do you go? That's the slope. If you go up 3 units when you move right 1 unit, then

m = 3

. If you go down 2 units when you move right 1 unit, then

m = -2

.

More formally, pick two clear points on the line (preferably where grid lines intersect) and use

m = \frac{\Delta y}{\Delta x}

.

Example: A line passes through

(1, 3)

and

(4, 9)

m = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2

Initial value is easy: look at the graph and find where the line crosses the y-axis. That point is

(0, b)

. If the line crosses at

(0, 5)

, then

b = 5

.

If the line doesn't cross the y-axis on your visible graph (maybe x = 0 is off the page), use the rate of change and any visible point:

b = y - mx

.

Common mistake: reading the slope backwards (rise over run instead of rise over run, but getting the fraction upside down) or misidentifying where the y-intercept is by misreading the scale on the axes.

Writing the Equation y = mx + b

Once you have m and b, you're done. Just write

y = mx + b

with your numbers plugged in.

From the water tank:

y = 20x + 10

.

From the graph example:

y = 2x + 5

.

If slope is negative, include the negative sign: if

m = -3

and

b = 8

, write

y = -3x + 8

.

If the y-intercept is 0, you can write

y = mx

y = mx + 0

; either is correct. For example, if a bike travels 12 miles per hour and you measure distance from the starting point, the equation is

y = 12x

(no initial distance to account for).

Double-check your equation by substituting a point you know. Using

y = 20x + 10

and the point

(5, 110)

y = 20(5) + 10 = 100 + 10 = 110

. ✓ This confirms your equation is correct.

Key terms

Rate of change (slope).: The ratio $\frac{\Delta y}{\Delta x}$ , which describes how much y increases (or decreases) for every 1-unit increase in x. In real life, it's the speed, cost per item, or any unit rate.
Initial value (y-intercept).: The output value when the input is 0, written as b. On a graph, it's the point where the line crosses the y-axis.
Slope (m).: A number that measures the steepness and direction of a line. Positive slope goes up to the right; negative slope goes down to the right.
Linear equation (standard form).: An equation of the form $y = mx + b$ where m is the slope and b is the y-intercept. It describes a straight-line relationship between two variables.
Change (Δ).: The difference between a final value and an initial value. $\Delta y = y_2 - y_1$ and $\Delta x = x_2 - x_1$ .
Y-intercept.: The point where a line crosses the y-axis, always at $x = 0$ . Its value is b in the equation $y = mx + b$ .

Worked example

A baker tracks the cost to make bread. After using 0 pounds of flour, there is a 5 dollar equipment cost. After 3 pounds of flour are used, the total cost is 14 dollars. After 5 pounds, the total cost is 20 dollars. Find the rate of change and initial value, then write the equation for cost in terms of pounds of flour.

First, organize the data:

(0, 5)

(3, 14)

(5, 20)

.

Since

x = 0

is given, we can read the initial value directly:

b = 5

dollars.

Now find the rate of change using any two points. Using

(0, 5)

and

(3, 14)

m = \frac{14 - 5}{3 - 0} = \frac{9}{3} = 3 \text{ dollars per pound}

Check using

(3, 14)

and

(5, 20)

m = \frac{20 - 14}{5 - 3} = \frac{6}{2} = 3 \text{ dollars per pound}

Both give the same slope, which confirms our answer. The rate of change is 3 dollars per pound of flour.

Now write the equation:

y = 3x + 5

where y is total cost (in dollars) and x is pounds of flour.

Check: At

x = 5

y = 3(5) + 5 = 15 + 5 = 20

. ✓ This matches our data point

(5, 20)

Practice questions

A linear relationship shows that when x = 2, y = 7, and when x = 6, y = 19. What is the rate of change?

Answer: 3

Use the slope formula:

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{19 - 7}{6 - 2} = \frac{12}{4} = 3

. The rate of change is 3 units of y per 1 unit of x.

A table shows time in hours (x) and distance in miles (y). When x = 0, y = 10; when x = 2, y = 50. Write the linear equation.

Answer: y = 20x + 10

The initial value is b = 10 (where x = 0). The rate of change is

m = \frac{50 - 10}{2 - 0} = \frac{40}{2} = 20

. So the equation is

y = 20x + 10

. This means you start 10 miles away and travel at 20 miles per hour.

A graph shows a line passing through the points (1, 4) and (3, 8). Where does the line cross the y-axis? Show your work.

(0, 0)
(0, 2)
(0, 1)
(0, 4)

Answer: (0, 2)

First, find the slope:

m = \frac{8 - 4}{3 - 1} = \frac{4}{2} = 2

. Then use any point to find b. Using (1, 4):

b = y - mx = 4 - 2(1) = 4 - 2 = 2

. The line crosses the y-axis at (0, 2).

FAQ

Do I always have to use the first and second rows of a table to find slope?: No. You can use any two rows. The slope will be the same no matter which rows you pick, as long as the relationship is linear. Pick rows that are easy to subtract, like rows where the numbers are clear and far enough apart.
What if x = 0 is not shown on the graph?: If x = 0 is off the edge of the graph, use the slope formula to find any two clear points on the line you can see, calculate m, then use $b = y - mx$ with either of those points. Rearrange to find b without needing to see the y-intercept itself.
Can the rate of change be negative?: Yes. A negative slope means y decreases as x increases. For example, if a tank is draining, the amount of water goes down over time, so the rate of change is negative. If the equation is $y = -5x + 20$ , then m = -5, meaning y decreases by 5 units for every 1-unit increase in x.
Why is it called 'initial value'?: Because in many real situations, x = 0 represents the starting point or the beginning of time, and the y-value at that point is what you start with or what happens before anything else occurs. For example, in a phone bill, the initial value is the base fee charged before you use any minutes. In a tank of water, it's how much water is there at the beginning.

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