Comparing Linear, Quadratic, and Exponential Functions: Understanding Growth Differences

Linear Function	$y = 2 x + 1$
Quadratic Function	$y = x^{2} + 3 x + 1$
Exponential Function	$y = 3 \cdot 2^{x}$

Linear Function

y = 2 x + 1

Quadratic Function

y = x^{2} + 3 x + 1

Exponential Function

y = 3 \cdot 2^{x}

Fish Population (Thousands)
Time (Months)	Lake Verdastro	Lake Rumoroso	Lake Mezzo
$0$	$1356$	$721$	$1207$
$1$	$6676$	$2163$	$3007$
$2$	$11996$	$6489$	$7807$
$3$	$17316$	$19467$	$15607$
$4$	$22636$	$58401$	$26407$

Fish Population (Thousands)

Time (Months)

Lake Verdastro

Lake Rumoroso

Lake Mezzo

0

1356

721

1207

1

6676

2163

3007

2

11996

6489

7807

3

17316

19467

15607

4

22636

58401

26407

Lake	Model
Verdastro	Linear Function
Rumoroso	Exponential Function
Mezzo	Quadratic Function

Lake

Model

Verdastro

Linear Function

Rumoroso

Exponential Function

Mezzo

Quadratic Function

Three Distinctive Characteristics of Quadratic Functions
Linear and exponential functions either only increase or only decrease. Conversely, quadratic functions always decrease and increase. The order depends on the direction in which the parabola opens.
Because a parabola either opens upward or downward, there is always one point that is the absolute minimum or absolute maximum of the function. This point is called the vertex.
While non-horizontal linear functions and exponential functions have at most one $x -$ intercept, quadratic functions can have up to two $x -$ intercepts.

Three Distinctive Characteristics of Quadratic Functions

Linear and exponential functions either only increase or only decrease. Conversely, quadratic functions always decrease and increase. The order depends on the direction in which the parabola opens.

Because a parabola either opens upward or downward, there is always one point that is the absolute minimum or absolute maximum of the function. This point is called the vertex.

While non-horizontal linear functions and exponential functions have at most one

x -

intercept, quadratic functions can have up to two

x -

intercepts.

Do the points appear to represent a linear function, an exponential function, or a quadratic function?

To determine the most appropriate model, we will start by recalling how linear, quadratic, and exponential graphs look like.

With this information in mind, let's now connect the points in the given diagram. By doing this, we will be able to see the type of curve that best fits the data.

The curve resembles a parabola. Therefore, the points are best described by a quadratic function.

We will start by connecting the points like we did in Part A. By doing this, we will be able to see the type of curve that best fits the data.

Now we can see that the curve that best fits the given points is an exponential function.

Just as we did before, we will connect the points to figure out which type of function is the best fit.

We can see that we can draw a straight line through the given points. Therefore, the points represent a linear function.

Do the points appear to represent a linear, exponential or quadratic function?

(- 2, 0), (- 1, 1), (1, 3), (2, 4)

(0, \frac{1}{4}), (1, 1), (2, 4), (3, 16)

(- 2, 9), (- 1, 0), (0, - 3), (1, 0), (2, 9)

To determine the most appropriate model for the given data set, we will start by recalling how linear, quadratic, and exponential graphs look like.

Let's now consider the given points. (- 2,0), (- 1, 1), (1,3), (2,4) Next, we will plot the points in a coordinate plane and connect them with a line or a smooth curve. Doing this makes it easier to tell the type of curve that best fits the points.

We see that a straight line passes through the points. This means the data set represents a linear function.

Let's consider the given points. (0, 14), (1, 1), (2,4), (3,16) Just like in Part A, we will plot the points in a coordinate plane and connect them with a line or smooth curve.

We obtained an exponential curve. Therefore, the curve that best fits the points is represented by an exponential function.

Finally, let's consider the last data set. (-2, 9), (- 1,0), (0, - 3), (1,0), (2,9) As we did in both previous parts, we will plot them in a coordinate plane and then connect them with a curve.

As we can see, the best curve we can fit to the points is a parabola, which is produced by a quadratic function.

Which graph does not belong with the other three?

After examining the graphs, we can see that the graphs given in options A, B, and D are parabolas. Therefore, these three graphs represent quadratic functions.

Conversely, the graph of opction C presents the characteristic shape of an exponential function. Therefore, this graph does not belong with the other three.

This time, if we examine the graphs we will see that the graphs given in options A, B, and C increase for all the points in their domain. Therefore, each of these graphs represents an increasing function.

On the other hand, the graph in option D is a decreasing graph and therefore represents a decreasing function. This is the graph that does not belong with the other three. Note that, although the graphs in options A, B, and C represent different types of functions, they can be grouped together because they all increase in their entire domain.

Kriz takes the bus to the school every morning. Their school is four stops away from their house. The table below shows the time it takes the bus to arrive at each of the four stops. Is this data modeled by a linear, exponential, or quadratic function?

a table of values showing time and distance

We want to know if the data can be modeled by a linear, exponential, or quadratic function. To do, so we will examine how the distance and time change between consecutive data points.

The difference in distance and time between the second and third data points is the same as the difference in distance and time between the third and fourth data points. Let's investigate the difference between the first and second data points by expanding the ratio of Δ d to Δ t by 4. This way we match the difference in t and can tell if the difference in d is the same.

After expanding the first rate of change, we can see that all the rates of change are constant. Therefore, the data can be modeled by a linear function.

LaShay is selling circular rugs. The table shows the cost of the rugs $c,$ in dollars, depending on their diameter $d,$ in feet.

a table of values showing diameter and cost of circular rugs

Determine whether the data represents a linear, quadratic, or exponential function.

If the data represents a linear function, the differences between consecutive data points, or their first differences, should be constant. Note that the difference in the independent variables should also be constant between consecutive points. Let's see if this is the case.

The first differences are not constant even though the differences in the independent variable are constant. Therefore, the data cannot be modeled by a linear function. Let's check whether it can be modeled by a quadratic function. We can do this by checking the second differences. If they are constant, the data can be modeled by a quadratic function.

Since the second differences are constant, we know that the data can be modeled by a quadratic function.

Comparing Linear, Exponential, and Quadratic Functions

Catch-Up and Review

Distinctive Characteristics of a Quadratic Function

Classifying Functions

Hint

Solution

Determining if a Function Is Linear, Quadratic, or Exponential

Finding the Pattern of a Table of Values to Determine Its Model

Determining the Pattern Describing Real-Life Data Sets

Answer

Hint

Solution

Saving for a New Computer

Answer

Hint

Solution

Finding the Optimal Increase to Get the Maximum Revenue

Answer

Hint

Solution

Determining the Storage Needed for a School Assignment

Answer

Hint

Solution

Identifying Some Remarkable Characteristics of Quadratic Functions

Answer

Hint

Solution

First Characteristic

Second Characteristic

Third Characteristic

Conclusion

Comparing Linear, Exponential, and Quadratic Functions

Recommended exercises

	10 Theory slides
	9 Exercises - Grade E - A
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