Linear, Quadratic and Exponential Models

Linear, quadratic and exponential models are three common ways to describe data. To choose one from a table, look at how $y$ changes as $x$ goes up by the same step. Equal first differences mean linear. Equal second differences mean quadratic. Equal ratios mean exponential. Then write the equation and use it to predict.

Before this lesson: You should know slope-intercept form, graphing quadratic functions, exponential functions and linear vs. exponential functions.

How Each Model Changes

Each model has its own pattern. A linear model adds the same amount each step. An exponential model multiplies by the same amount each step.

A quadratic model is in between. The amount it adds changes, but it changes by the same amount each time. The table below shows all three.

Three tables for x = 0 to 4. Linear: y = 3, 5, 7, 9, 11, and each first difference is +2. Quadratic: y = 1, 2, 5, 10, 17; the first differences are 1, 3, 5, 7 and each second difference is +2. Exponential: y = 3, 6, 12, 24, 48, and each y is 2 times the one before. — Linear adds the same amount. Quadratic has equal second differences. Exponential multiplies by the same amount.

A first difference is the change from one $y$ to the next. A second difference is the change between first differences. A ratio is one $y$ divided by the one before it.

How to Choose a Model From a Table

Pattern in y	Model	Equation form
Same first differences	Linear	$y = mx + b$
Same second differences	Quadratic	$y = ax^2 + bx + c$
Same ratio	Exponential	$y = a \cdot b^x$

Check that the $x$ -values go up by the same step.
Find the first differences. If they are all the same, the model is linear.
If not, find the second differences. If they are all the same, the model is quadratic.
If not, divide each $y$ by the one before. If the ratios are all the same, the model is exponential.
Write the equation, then check it with a point from the table.

Tip: when $x$ goes up by 1, the $a$ in $y = ax^2 + bx + c$ is half the second difference.

Worked Examples

Example 1: A linear table

$x$	0	1	2	3	4
$y$	4	7	10	13	16

The first differences are all +3, so the model is linear. The slope is 3 and the starting value is 4, so $y = 3x + 4$ .

Example 2: An exponential table

$x$	0	1	2	3	4
$y$	2	6	18	54	162

The first differences are 4, 12, 36 and 108, so it isn’t linear. Each $y$ is 3 times the one before, so the model is exponential. It starts at 2, so $y = 2 \cdot 3^x$ .

Example 3: A quadratic table

$x$	0	1	2	3	4
$y$	0	2	8	18	32

The first differences are 2, 6, 10 and 14. The second differences are all 4, so the model is quadratic. Half of 4 is 2, so $a = 2$ . The table starts at 0, and $y = 2x^2$ fits every point.

Example 4: Find a, b and c

$x$	0	1	2	3	4
$y$	5	8	13	20	29

The first differences are 3, 5, 7 and 9. The second differences are all 2, so $a = 1$ . At $x = 0$ , $y = 5$ , so $c = 5$ . Use the point $(1, 8)$ to find $b$ .

\begin{aligned} 1^2 + b(1) + 5 &= 8 \\ b &= 2 \end{aligned}

The model is $y = x^2 + 2x + 5$ . Check $x = 4$ : $16 + 8 + 5 = 29$ .

Example 5: Choose a model from a situation

Which model fits each situation?

A phone plan costs 20 dollars a month plus 5 dollars for each gigabyte. The cost goes up by the same amount for each gigabyte, so it is linear.
A town’s population grows by 4% each year. It is multiplied by 1.04 each year, so it is exponential.
The height of a ball thrown in the air rises, then falls. Its graph is a parabola, so it is quadratic.

Example 6: Word problem

A lab counts bacteria every hour. The counts are 50, 100, 200 and 400 for hours 0 to 3. Find a model and predict the count at hour 6.

Each count is 2 times the one before, so the model is exponential: $y = 50 \cdot 2^x$ . At hour 6, $y = 50 \cdot 2^6 = 50 \cdot 64 = 3{,}200$ bacteria.

Which Model Grows Fastest?

An increasing exponential model always passes a linear or quadratic model in the long run. Here is $y = 10x$ , $y = x^2$ and $y = 2^x$ side by side.

$x$	2	4	6	8	10
$10x$	20	40	60	80	100
$x^2$	4	16	36	64	100
$2^x$	4	16	64	256	1,024

At first, $2^x$ is the smallest. By $x = 10$ , it is more than 10 times bigger than the others. To learn more, see exponential growth and decay.

Common Mistakes

Mistake	Example of the error	Fix
Stopping at first differences	Calling 1, 2, 5, 10, 17 “not a pattern”	Find the second differences too. Here they are all 2, so it is quadratic.
Uneven x-values	Using differences when $x$ = 0, 1, 3, 4	Differences only work when $x$ goes up by the same step.
Mixing up adding and multiplying	Calling 3, 6, 12, 24 linear because it “goes up by doubling”	Doubling is multiplying by 2. That is exponential, not linear.
Using the second difference as a	Second difference 4, so $a = 4$	When $x$ steps by 1, $a$ is half the second difference: $a = 2$ .
Missing exponential decay	Calling 80, 40, 20, 10 linear because it goes down	The first differences are not equal. Each $y$ is half the one before, so it is exponential.

Practice Problems

Which model fits $x$ = 0, 1, 2, 3 and $y$ = 10, 7, 4, 1? Write the equation.
Which model fits $x$ = 0, 1, 2, 3 and $y$ = 1, 4, 16, 64? Write the equation.
Which model fits $x$ = 0, 1, 2, 3, 4 and $y$ = 1, 3, 9, 19, 33? Write the equation.
Which model fits $x$ = 0, 1, 2, 3 and $y$ = 80, 40, 20, 10? Write the equation.
Which model fits $x$ = 0, 1, 2, 3, 4 and $y$ = 0, −1, 0, 3, 8? Write the equation.
A car rental costs 30 dollars a day. Which model fits the total cost?
An investment grows by 5% each year. Which model fits its value?
Which model fits the area of a circle as its radius grows?
Which model fits $x$ = 1, 2, 3, 4 and $y$ = 6, 12, 24, 48? Write the equation.
Which is larger at $x = 7$ : $2^x$ or $x^2$ ?

Show answers

The first differences are all −3, so it is linear: $y = -3x + 10$ .
Each $y$ is 4 times the one before, so it is exponential: $y = 4^x$ .
The first differences are 2, 6, 10, 14 and the second differences are all 4, so it is quadratic. $a = 2$ and $c = 1$ . From $(1, 3)$ : $2 + b + 1 = 3$ , so $b = 0$ . $y = 2x^2 + 1$ .
Each $y$ is half the one before, so it is exponential: $y = 80 \cdot \left(\frac{1}{2}\right)^x$ .
The first differences are −1, 1, 3, 5 and the second differences are all 2, so it is quadratic. $a = 1$ and $c = 0$ . From $(1, -1)$ : $1 + b = -1$ , so $b = -2$ . $y = x^2 - 2x$ .
Linear. The cost goes up by the same 30 dollars each day.
Exponential. The value is multiplied by 1.05 each year.
Quadratic. The area is $A = \pi r^2$ .
Each $y$ is 2 times the one before, so it is exponential. At $x = 1$ , $a \cdot 2 = 6$ , so $a = 3$ . $y = 3 \cdot 2^x$ .
$2^7 = 128$ and $7^2 = 49$ , so $2^x$ is larger.

FAQ

How do you know if a table is linear, quadratic or exponential?

Check the first differences, then the second differences, then the ratios. Equal first differences mean linear, equal second differences mean quadratic, and equal ratios mean exponential.

What are second differences?

They are the differences between the first differences. Equal second differences tell you the data is quadratic.

Which grows faster, exponential or quadratic?

An increasing exponential function always grows faster in the long run. A quadratic may be bigger at first, but the exponential passes it.

Summary

Linear models add the same amount each step.
Quadratic models have equal second differences.
Exponential models multiply by the same ratio each step.
The $x$ -values must go up by the same step before you compare.
Increasing exponential models grow fastest in the long run.

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