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.

Consider the following graphs.

Match each graph with the correct function.

By examining the functions, we can see that we have a linear function, two exponential functions, and one quadratic function. We know that this is the case because the linear function has the form y=mx+b, the degree of the x-variable in the quadratic function is 2, and in exponential functions, the variable x appears in an exponent. lcl y = 2x+1 & → & linear [0.4em] y = 2^x+2 & → & exponential [0.4em] y = 0.5^x+2 & → & exponential [0.4em] y = 2x^2+2 & → & quadratic Let's now start matching the functions with the graphs.

Linear Function

A linear function is graphically represented by a straight line. We can only see one graph that is a straight line and that is graph B. Therefore, we should match the linear function with B. B → y=2x+1

Quadratic Function

The graph of a quadratic equation has the form of a parabola, a curve that is symmetric along the vertical line that passes through its vertex. There is only one graph that exhibits these characteristics, graph D. D → y=2x^2+2

Exponential Functions

Of the two remaining graphs, both exponential functions, we can see that one represents an increasing function and the other represents a decreasing function.

An increasing exponential function has a base, or constant multiplier, greater than 1. Conversely, a decreasing exponential function has a base that is less than 1 but greater than 0. Increasing:& y = ab^x b>1 Decreasing:& y = ab^x 0

The graphs of two functions are shown on the first quadrant of a coordinate plane.

Neither of the graphs is a straight line. Select the most correct statement below.

We will analyze f(x) and g(x) one at a time.

f(x)

Let's identify three points on the graph of f(x).

Since we know that the graph is not a line, the first differences are not constant. Let's calculate the second differences.

The second differences are not constant. Therefore, f(x) is not a quadratic function. Let's see if it is an exponential function by checking whether the y-coordinates of these points have a common ratio.

The y-values of consecutive data points have a common ratio. This means that f(x) is an exponential function.

g(x)

Let's start by identifying some points on the graph of g(x).

We already know that g(x) is not a linear function. Just as we did for f(x), we will analyze the second differences between consecutive data points.

Here, the second differences are constant. Therefore, we do not need to analyze if there is a common ratio because we already know that g(x) is a quadratic function. The correct choice is A.

Diego wants to write a function that satisfies the following conditions.

The function that Diego wants to write has constant second differences but not constant first differences. Therefore, we know that this is a quadratic function.

Let's recall the standard form of a quadratic function. y = ax^2+bx+c We know that the y-intercept is 4. This means that the value of c is 4. y = ax^2+bx+4 Next, because we know that the parabola passes through ( 1, 6) and ( - 1, 4), we can form a system of equations with a and b as its variables. 6 = a( 1)^2+b( 1)+4 & (I) 4 = a( - 1)^2+b( - 1)+4 & (II) Let's solve the system by using the Elimination Method. To do so, we will first simplify the equations.

Now, we can add Equation (I) to Equation (II).

Finally, we will substitute 1 for a in Equation (I).

Now that we know that both a and b are equal to 1, we can write the quadratic function. y=1x^2+1x+4 ⇕ y=x^2+x+4

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

Linear Function

Quadratic Function

Exponential Functions

f(x)

g(x)

Comparing Linear, Exponential, and Quadratic Functions

Recommended exercises

	10 Theory slides
	9 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions