The simplest function we can graph on an -plane is a linear function. Even though they are simple, linear functions are still important! In AP Calculus, we study lines that are tangent to (or touching) curves, and when we zoom in enough on a curve, it looks and behaves like a line!
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Sign up for freeThe slope of a linear function is also called...
Increasing linear functions have...
Decreasing linear functions have...
Horizontal linear functions have...
Some examples of nonlinear functions are:
What two ways can we use to graph a linear function?
In our study of calculus, we will have to deal with linear functions that may not be uniformly defined throughout their domains. It could be that they are defined in two or more ways as their domains are split into two or more parts.In these cases, these are called...
The slope of a linear function is also called...
Increasing linear functions have...
Decreasing linear functions have...
Horizontal linear functions have...
Some examples of nonlinear functions are:
What two ways can we use to graph a linear function?
In our study of calculus, we will have to deal with linear functions that may not be uniformly defined throughout their domains. It could be that they are defined in two or more ways as their domains are split into two or more parts.In these cases, these are called...
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In this article, we discuss in detail what a linear function is, its characteristics, equation, formula, graph, table, and go through several examples.
What is a linear function?
A linear function is a polynomial function with a degree of 0 or 1. This means that each term in the function is either a constant or a constant multiplied by a single variable whose exponent is either 0 or 1.
When graphed, a linear function is a straight line in a coordinate plane.
By definition, a line is straight, so saying "straight line" is redundant. We use "straight line" often in this article, however, just saying "line" is sufficient.
When we say that is a linear function of
, we mean that the graph of the function is a straight line.
The slope of a linear function is also called the rate of change.
A linear function grows at a constant rate.
The image below shows:
The graph and table of sample values of a linear function, StudySmarter Originals
Notice that when increases by 0.1, the value of
increases by 0.3, meaning
increases three times as fast as
.
Therefore, the slope of the graph of , 3, can be interpreted as the rate of change of
with respect to
.
A linear function can be an increasing, decreasing, or horizontal line.
Increasing linear functions have a positive slope.
Decreasing linear functions have a negative slope.
Horizontal linear functions have a slope of zero.
The y-intercept of a linear function is the value of the function when the x-value is zero.
This is also known as the initial value in real-world applications.
Linear functions are a special type of polynomial function. Any other function that does not form a straight line when graphed on a coordinate plane is called a nonlinear function.
Some examples of nonlinear functions are:
When we think of a linear function in algebraic terms, two things come to mind:
The equation and
The formulas
A linear function is an algebraic function, and the parent linear function is:
Which is a line that passes through the origin.
In general, a linear function is of the form:
Where and
are constants.
In this equation,
There are several formulas that represent linear functions. All of them can be used to find the equation of any line (except vertical lines), and which one we use depends on the available information.
Since vertical lines have an undefined slope (and fail the vertical line test), they are not functions!
The standard form of a linear function is:
Where are constants.
The slope-intercept form of a linear function is:
Where:
is a point on the line.
is the slope of the line.
Remember: slope can be defined as , where
and
are any two points on the line.
The point-slope form of a linear function is:
Where:
is a point on the line.
is any fixed point on the line.
The intercept form of a linear function is:
Where:
is a point on the line.
and
are the x-intercept and the y-intercept, respectively.
The graph of a linear function is pretty simple: just a straight line on the coordinate plane. In the image below, the linear functions are represented in slope-intercept form. (the number that the independent variable,
, is multiplied by), determines the slope (or gradient) of that line, and
determines where the line crosses the y-axis (known as the y-intercept).
The graphs of two linear functions, StudySmarter Originals
What information do we need to graph a linear function? Well, based on the formulas above, we need either:
two points on the line, or
a point on the line and its slope.
To graph a linear function using two points, we need to either be given two points to use, or we need to plug in values for the independent variable and solve for the dependent variable to find two points.
If we are given two points, graphing the linear function is just plotting the two points and connecting them with a straight line.
If, however, we are given a formula for a linear equation and asked to graph it, there are more steps to follow.
Graph the function:
Solution:
The graph of a line using two points, StudySmarter OriginalsTo graph a linear function using its slope and y-intercept, we plot the y-intercept on a coordinate plane, and use the slope to find a second point to plot.
Graph the function:
Solution:
Using the slope and y-intercept to graph a line, StudySmarter OriginalsSo, why do we extend the graph of a linear function past the points we use to plot it? We do that because the domain and range of a linear function are both the set of all real numbers!
Any linear function can take any real value of as an input, and give a real value of
as an output. This can be confirmed by looking at the graph of a linear function. As we move along the function, for every value of
, there is only one corresponding value of
.
Therefore, as long as the problem doesn't give us a limited domain, the domain of a linear function is:
Also, the outputs of a linear function can range from negative to positive infinity, meaning that the range is also the set of all real numbers. This can also be confirmed by looking at the graph of a linear function. As we move along the function, for every value of , there is only one corresponding value of
.
Therefore, as long as the problem doesn't give us a limited range, and , the range of a linear function is:
When the slope of a linear function is 0, it is a horizontal line. In this case, the domain is still the set of all real numbers, but the range is just b.
Linear functions can also be represented by a table of data that contains x- and y-value pairs. To determine if a given table of these pairs is a linear function, we follow three steps:
Calculate the differences in the x-values.
Calculate the differences in the y-values.
Compare the ratio for each pair.
If this ratio is constant, the table represents a linear function.
We can also check if a table of x- and y-values represents a linear function by determining if the rate of change of with respect to
(also known as the slope) remains constant.
Typically, a table representing a linear function looks something like this:
| x-value | y-value |
| 1 | 4 |
| 2 | 5 |
| 3 | 6 |
| 4 | 7 |
To determine if a function is a linear function depends on how the function is presented.
If a function is presented algebraically:
then it is a linear function if the formula looks like: .
If a function is presented graphically:
then it is a linear function if the graph is a straight line.
If a function is presented using a table:
then it is a linear function if the ratio of the difference in y-values to the difference in x-values is always constant. Let's see an example of this
Determine if the given table represents a linear function.
| x-value | y-value |
| 3 | 15 |
| 5 | 23 |
| 7 | 31 |
| 11 | 47 |
| 13 | 55 |
Solution:
To determine if the values given in the table represent a linear function, we need to follow these steps:
Let's apply these steps to the given table:
Determining if a table of values represents a linear function, StudySmarter Originals
There are a couple of special types of linear functions that we will likely deal with in calculus. These are:
Linear functions represented as piecewise functions and
Inverse linear function pairs.
In our study of calculus, we will have to deal with linear functions that may not be uniformly defined throughout their domains. It could be that they are defined in two or more ways as their domains are split into two or more parts.
In these cases, these are called piecewise linear functions.
Graph the following piecewise linear function:
The symbol ∈ above means "is an element of".
Solution:
This linear function has two finite domains:
Outside of these intervals, the linear function does not exist. So, when we graph these lines, we will actually just graph the line segments defined by the endpoints of the domains.
Notice in the domain of x+2 that there is a parenthesis instead of a bracket around the 1. This means that 1 is not included in the domain of x+2! So, there is a "hole" in the function there.
| x-value | y-value |
| -2 | |
| 1 |
| x-value | y-value |
| 1 | |
| 2 |
The graph of a piecewise linear function, StudySmarter OriginalsLikewise, we will also deal with inverse linear functions, which are one of the types of Inverse Functions. To briefly explain, if a linear function is represented by:
Then its inverse is represented by:
such that
The superscript, -1, is not a power. It means "the inverse of", not "f to the power of -1".
Find the inverse of the function:
Solution:
If we graph both and
on the same coordinate plane, we will notice that they are symmetric with respect to the line
. This is a characteristic of Inverse Functions.
The graph of an inverse linear function pair and their line of symmetry, StudySmarter Originals
There are several uses in the real world for linear functions. To name a few, there are:
Distance and rate problems in physics
Calculating dimensions
Determining prices of things (think taxes, fees, tips, etc. that are added to the price of things)
Say you enjoy playing video games.
You subscribe to a gaming service that charges a monthly fee of $5.75 plus an additional fee for each game you download of $0.35.
We can write your actual monthly fee using the linear function:
Where is the number of games you download in a month.
Write the given function as ordered pairs.
Solution:
The ordered pairs are: and
.
Find the slope of the line for the following.
Solution:
Find the equation of the linear function given by the two points:
Solution:
The relationship between Fahrenheit and Celsius is linear. The table below shows a few of their equivalent values. Find the linear function representing the given data in the table.
| Celsius (°C) | Fahrenheit (°F) |
| 5 | 41 |
| 10 | 50 |
| 15 | 59 |
| 20 | 68 |
Solution:
Let's say that the cost of renting a car can be represented by the linear function:
Where is the number of days the car is rented.
What is the cost to rent the car for 10 days?
Solution:
So, the cost of renting the car for 10 days is $320.
To add onto the last example. Let's say we know how much someone paid to rent a car, using the same linear function.
If Jake paid $470 to rent a car, how many days did he rent it?
Solution:
We know that , where
is the number of days the car is rented. So, in this case, we replace
with 470 and solve for
.
Determine if the function is a linear function.
Solution:
We need to isolate the dependent variable to help us visualize the function. Then, we can verify whether it is linear by graphing it.
The graph of a line, StudySmarter OriginalsDetermine whether the function is a linear function.
Solution:
The graph of a nonlinear function, StudySmarter OriginalsSign up for free to gain access to all our flashcards.
What is a linear function?
A linear function is an algebraic equation in which each term is either:
The graph of a linear function is a straight line.
For example, the function: y = x is a linear function.
How do I write a linear function?
How do you determine a linear function?
To determine if a function is a linear function, you need to either:
Which table represents a linear function?
Considering the following table:
x: 0, 1, 2, 3
y: 3, 4, 5, 6
From this table, we can observe that the rate of change between x and y is 3. This can be written as the linear function: y = x + 3.
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