What is the easiest way to tell if a function in a table is linear?

Calculate the rate of change y/ x for at least two different pairs of consecutive points. If both ratios are the same, the function is linear. If they are different, it is nonlinear. You only need to check two pairs to be sure, but checking more pairs is even safer.

Can a function be linear if its equation has addition or subtraction in it?

Yes. Linear functions can have addition or subtraction. For example, y = 2x + 7 and y = -3x - 4 are both linear. What matters is that x appears only to the first power (not squared, not cubed, not in a square root, and not in a denominator). If x is only multiplied or divided by numbers and then added or subtracted, the function is linear.

Why is y = x^2 nonlinear but y = 2x is linear?

The difference is the exponent on x. In y = x^2, the variable is squared, which means the output grows much faster and in a curved pattern. In y = 2x, the variable is only to the first power, so the output grows at a constant rate of 2 units per unit increase in x. Only equations where x appears to the first power (like y = mx + b) are linear. Any other power, root, or special form makes the function nonlinear.

If a nonlinear function has a constant rate of change somewhere, does that make it linear?

No. A nonlinear function might have sections where the rate of change stays the same for a few points, but it will not stay constant everywhere. What defines a function as linear is that the rate of change is constant across the entire domain. For a nonlinear function like y = x^2, the rate of change keeps increasing (or decreasing, depending on where you look), so it is never constant overall.

M8MATH-6.2

Linear vs Nonlinear Functions

Learn to distinguish linear functions (straight-line graphs with constant rate of change) from nonlinear functions using tables, equations, and graphs.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Linear vs Nonlinear Functions, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

You've already worked with functions and how to represent them in different ways. Now comes a crucial skill: telling the difference between two broad families of functions. A linear function always produces a straight-line graph and has a special property—the rate of change stays exactly the same no matter where you measure it. A nonlinear function breaks that rule: its graph curves or bends, and the rate of change varies. In this lesson, you'll learn how to spot both types using tables of values, equations, and graphs so you can make quick, confident identifications in real situations.

What Makes a Function Linear?

A linear function follows the equation

y = mx + b

, where

m

is the slope (the constant rate of change) and

b

is the y-intercept. When you graph a linear function, you always get a straight line—never curved, never jagged. The key feature that defines a linear function is the constant rate of change: for every fixed increase in the input (

x

), the output (

y

) increases by the same amount every single time.

For example, if

y = 2x + 3

, then when

x

increases by 1,

y

always increases by 2. When

x

goes from 0 to 1,

y

goes from 3 to 5 (a change of 2). When

x

goes from 5 to 6,

y

goes from 13 to 15 (again, a change of 2). This consistency is what makes the graph a straight line. The slope

m = 2

tells you exactly how steep that line is, and the

b = 3

tells you where it crosses the y-axis.

Testing a Table for a Constant Rate of Change

The fastest way to spot a linear function in a table is to check whether the rate of change is constant. Calculate the change in

y

divided by the change in

x

(which is

\frac{\Delta y}{\Delta x}

) for consecutive rows. If you get the same ratio every time, the function is linear.

Consider this table:

|

x

| 1 | 2 | 3 | 4 | |

y

| 5 | 9 | 13 | 17 |

From

x = 1

x = 2

\frac{\Delta y}{\Delta x} = \frac{9 - 5}{2 - 1} = \frac{4}{1} = 4

From

x = 2

x = 3

\frac{\Delta y}{\Delta x} = \frac{13 - 9}{3 - 2} = \frac{4}{1} = 4

From

x = 3

x = 4

\frac{\Delta y}{\Delta x} = \frac{17 - 13}{4 - 3} = \frac{4}{1} = 4

The rate of change is always 4, so this is a linear function. You can even write the equation:

y = 4x + 1

(since when

x = 1

y = 5

, and

4(1) + 1 = 5

Nonlinear Functions and Their Characteristics

A nonlinear function does not follow

y = mx + b

and does not have a constant rate of change. Its graph is not a straight line—it may curve, bend, or have other irregular shapes. Two common examples are quadratic and cubic functions.

The equation

y = x^2

(a quadratic function) shows what happens when you square the input. When you plot the points from a table—

(1, 1)

(2, 4)

(3, 9)

(4, 16)

—you see a U-shaped curve called a parabola. Notice the rate of change: from

x = 1

x = 2

y

increases by 3; from

x = 2

x = 3

y

increases by 5; from

x = 3

x = 4

y

increases by 7. The change keeps getting bigger—not constant.

Similarly,

y = x^3

(a cubic function) produces a different kind of curve. When plotted, it passes through

(1, 1)

(2, 8)

(3, 27)

. The rate of change here is even more dramatic: from

x = 1

x = 2

y

increases by 7; from

x = 2

x = 3

y

increases by 19. These varying rates of change are what make the graph curve instead of staying straight.

How to Distinguish Linear from Nonlinear

You have three tools to tell linear and nonlinear functions apart:

In a Table: Calculate

\frac{\Delta y}{\Delta x}

for consecutive pairs of points. If the ratio is always the same, it's linear. If the ratios vary, it's nonlinear.

In an Equation: Look at the form. If it matches

y = mx + b

(with

x

to the first power only), it's linear. If

x

is squared, cubed, under a square root, or exponents vary, it's nonlinear.

In a Graph: A straight line means the function is linear. Any curve, bend, or non-straight shape means the function is nonlinear.

A common mistake is assuming that because a function uses addition or subtraction, it must be linear. For example,

y = x^2 + 3

looks like it has the form of

y = mx + b

, but the

x^2

term makes it nonlinear. Similarly,

y = \frac{1}{x}

has

x

in the denominator, so it's definitely nonlinear even though it uses division. Always focus on whether the rate of change is constant and whether

x

appears only to the first power.

Key terms

Linear function.: A function of the form $y = mx + b$ whose graph is a straight line and whose rate of change is constant.
Nonlinear function.: A function whose graph is not a straight line and whose rate of change is not constant.
Rate of change.: The ratio $\frac{\Delta y}{\Delta x}$ that measures how much the output changes for a given change in input; in linear functions, this ratio is constant.
Constant rate of change.: The property of a linear function where every equal increase in input produces an equal increase in output; this is the slope $m$ in $y = mx + b$ .
Slope (m).: The constant rate of change in a linear function; it tells you how steep the line is and which direction it goes.
y-intercept (b).: The value of $y$ when $x = 0$ ; the point where the line crosses the y-axis in the equation $y = mx + b$ .
Parabola.: The U-shaped graph of a quadratic function like $y = x^2$ .
Quadratic function.: A nonlinear function in which the variable is raised to the second power, such as $y = x^2$ or $y = ax^2 + bx + c$ .

Worked example

The table below shows the relationship between the side length of a square and its area. Determine whether this function is linear or nonlinear, and explain your reasoning.

| Side length (

s

) | 1 | 2 | 3 | 4 | 5 | | Area (

A

) | 1 | 4 | 9 | 16 | 25 |

Step 1: Calculate the rate of change between consecutive points.

From

s = 1

s = 2

\frac{\Delta A}{\Delta s} = \frac{4 - 1}{2 - 1} = \frac{3}{1} = 3

From

s = 2

s = 3

\frac{\Delta A}{\Delta s} = \frac{9 - 4}{3 - 2} = \frac{5}{1} = 5

From

s = 3

s = 4

\frac{\Delta A}{\Delta s} = \frac{16 - 9}{4 - 3} = \frac{7}{1} = 7

From

s = 4

s = 5

\frac{\Delta A}{\Delta s} = \frac{25 - 16}{5 - 4} = \frac{9}{1} = 9

Step 2: Look for a pattern.

The rate of change is not constant. It goes 3, then 5, then 7, then 9. Each time the rate of change itself increases.

Step 3: Write the equation and confirm.

From the table, we can recognize the pattern:

A = s^2

. This is not of the form

y = mx + b

, so it cannot be linear.

Step 4: State your conclusion.

This function is nonlinear because the rate of change is not constant. The equation

A = s^2

is a quadratic function, and its graph would be a parabola opening upward, not a straight line.

Practice questions

Which table represents a linear function?

| $x$ | 1 | 2 | 3 | 4 | \n | $y$ | 2 | 4 | 8 | 16 |
| $x$ | 1 | 2 | 3 | 4 | \n | $y$ | 3 | 7 | 11 | 15 |
| $x$ | 1 | 2 | 3 | 4 | \n | $y$ | 1 | 8 | 27 | 64 |
| $x$ | 0 | 1 | 2 | 3 | \n | $y$ | 0 | 1 | 4 | 9 |

Answer: | $x$ | 1 | 2 | 3 | 4 | \n | $y$ | 3 | 7 | 11 | 15 |

In this table, the rate of change is constant: from each

x

to the next,

y

increases by exactly 4. (

\frac{7-3}{2-1} = 4

\frac{11-7}{3-2} = 4

\frac{15-11}{4-3} = 4

). This constant rate of change means the function is linear with slope

m = 4

, and it matches the form

y = mx + b

. The other tables show nonlinear functions: the first doubles each time (exponential), the third cubes the input (cubic), and the fourth squares the input (quadratic).

For the equation

y = 3x + 5

, explain why this is a linear function and what the slope and y-intercept represent.

Answer: This is a linear function because it is written in the form $y = mx + b$ . The slope is $m = 3$ , which means for every 1 unit increase in $x$ , $y$ increases by exactly 3 units. The y-intercept is $b = 5$ , which means the graph crosses the y-axis at the point $(0, 5)$ . Because the rate of change is constant (always 3), the graph is a straight line.

Students sometimes confuse the structure of the equation with whether a function is linear. The key is that

x

appears only to the first power (no exponents, no roots, no

x

in a denominator). Any equation matching

y = mx + b

will always produce a straight-line graph with a constant rate of change equal to the slope

m

The cubic function

y = x^3

passes through the points

(0, 0)

(1, 1)

(2, 8)

, and

(3, 27)

. Why is this function nonlinear, and how can you see this in the table?

Answer: This function is nonlinear because the rate of change is not constant. From $(0, 0)$ to $(1, 1)$ : $\frac{\Delta y}{\Delta x} = \frac{1 - 0}{1 - 0} = 1$ . From $(1, 1)$ to $(2, 8)$ : $\frac{\Delta y}{\Delta x} = \frac{8 - 1}{2 - 1} = 7$ . From $(2, 8)$ to $(3, 27)$ : $\frac{\Delta y}{\Delta x} = \frac{27 - 8}{3 - 2} = 19$ . The rate of change goes 1, then 7, then 19—it keeps increasing. Additionally, the equation $y = x^3$ has $x$ raised to the third power, so it does not match $y = mx + b$ . The graph of a cubic function is a smooth curve, not a straight line.

Some students see that the rate of change is growing and wonder if there is a pattern (there is—it follows a pattern related to cubic numbers—but that does not make it linear). What matters is whether every step produces the same change in

y

per unit change in

x

. Here it does not, so the function is nonlinear. The cube root and higher powers always produce nonlinear functions.

FAQ

What is the easiest way to tell if a function in a table is linear?: Calculate the rate of change $\frac{\Delta y}{\Delta x}$ for at least two different pairs of consecutive points. If both ratios are the same, the function is linear. If they are different, it is nonlinear. You only need to check two pairs to be sure, but checking more pairs is even safer.
Can a function be linear if its equation has addition or subtraction in it?: Yes. Linear functions can have addition or subtraction. For example, $y = 2x + 7$ and $y = -3x - 4$ are both linear. What matters is that $x$ appears only to the first power (not squared, not cubed, not in a square root, and not in a denominator). If $x$ is only multiplied or divided by numbers and then added or subtracted, the function is linear.
Why is $y = x^2$ nonlinear but $y = 2x$ is linear?: The difference is the exponent on $x$ . In $y = x^2$ , the variable is squared, which means the output grows much faster and in a curved pattern. In $y = 2x$ , the variable is only to the first power, so the output grows at a constant rate of 2 units per unit increase in $x$ . Only equations where $x$ appears to the first power (like $y = mx + b$ ) are linear. Any other power, root, or special form makes the function nonlinear.
If a nonlinear function has a constant rate of change somewhere, does that make it linear?: No. A nonlinear function might have sections where the rate of change stays the same for a few points, but it will not stay constant everywhere. What defines a function as linear is that the rate of change is constant across the entire domain. For a nonlinear function like $y = x^2$ , the rate of change keeps increasing (or decreasing, depending on where you look), so it is never constant overall.

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