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Linear Functions

Graphing Linear Functions in Standard Form

There are several different ways to graph a linear function. Sometimes, the way the rule of the function is written can dictate the simplest way to graph it. Below, the graphs of linear functions given in standard form will be explored.


Standard Form of a Line

One way to write linear function rules is in standard form.

Here, and are real numbers and and cannot both equal Several combinations of and can describe the same line, but representing them with the smallest possible integers is preferred.

Graph the linear function given by the equation using a table of values.

Show Solution
To graph the function, we can create a table of values giving different points on the line. To do this, we'll substitute arbitrarily-chosen -values into the equation to find the corresponding -values. Let's start with
One point on the line is We can use the same process for finding other points.

To draw the graph of the function, we can plot all five points in a coordinate plane and connect them with a line.


Graphing Linear Functions using Intercepts

A function's - and -intercepts are the points where the graph of a function intersects with the - and -axes, respectively. It's possible to use a linear function's intercepts to graph it.


Finding the Intercepts of a Graph

The intercepts of a graph share an important feature. For all -intercepts, the -coordinate is and for all -intercepts, the -coordinate is This can be used to find the intercepts of a graph when its rule is known. For example, consider the line given by the following equation.


Finding the -intercept

To find the -intercept, can be substituted into the equation. Next, solve the equation for
The -intercept is


Finding the -intercept

The -intercept can be found in a similar way. Substitute into the equation and solve for
The -intercept is

The amusement park ride "Spinning Teacups" has two different sizes of cups, large and small. Large cups fit people and small cups fit people. Maximum capacity for each ride is people. The equation models this situation, where is the number of small cups and is the number of large cups. Graph the situation and interpret the intercepts.

Show Solution


Finding the intercepts

To begin, we will find each of the intercepts. Starting with the -intercept, we can substitute into the rule and solve for
The -intercept is To find the -intercept we can substitute and solve for
Solve for
The -intercept is


Graphing the function

To graph the function, we can plot the intercepts in a coordinate plane, and connect them with a line.

Notice that the graph does not extend infinitely. This is because, since and represent the numbers of different cups, negative numbers should not be included.


Interpreting the intercepts

We can interpret the intercepts in terms of what and represent. The -intercept is This means a ride with small cups can not have any large cups, because the maximum capacity of people has already been met. Similarly, the -intercept of tells us that a ride with large cups will not allow for any small cups.

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