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.

Rule

Standard Form of a Line

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

$Ax+By=C$

Here,

A,

B,

and

C

are real numbers and

A

and

B

cannot both equal

0.

Several combinations of

A,

B,

and

C

can describe the same line, but representing them with the smallest possible integers is preferred.

Graph the linear function in standard form

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Exercise

Graph the linear function given by the equation using a table of values. $4x-2y=7$

Show Solution

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

x

-values into the equation to find the corresponding

y

-values. Let's start with

x=0.

4x-2y=7

Substitute

x={\color{#0000FF}{0}}

4\cdot {\color{#0000FF}{0}} - 2y=7

ZeroPropMultZero Property of Multiplication

\text{-} 2y=7

DivEqn

\left.\text{LHS}\middle/(\text{-} 2)\right.=\left.\text{RHS}\middle/(\text{-} 2)\right.

y=\dfrac{7}{\text{-} 2}

CalcQuotCalculate quotient

y=\text{-} 3.5

One point on the line is

(0,\text{-}3.5).

We can use the same process for finding other points.

$x$	$4x-2y=7$	$y$
${\color{#0000FF}{1}}$	$4 \cdot {\color{#0000FF}{1}}-2y=7$	$\text{-}1.5$
${\color{#0000FF}{2}}$	$4 \cdot {\color{#0000FF}{2}}-2y=7$	$0.5$
${\color{#0000FF}{3}}$	$4 \cdot {\color{#0000FF}{3}}-2y=7$	$2.5$
${\color{#0000FF}{4}}$	$4 \cdot {\color{#0000FF}{4}}-2y=7$	$4.5$

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

Theory

Graphing Linear Functions using Intercepts

A function's

x

- and

y

-intercepts are the points where the graph of a function intersects with the

x

- and

y

-axes, respectively. It's possible to use a linear function's intercepts to graph it.

Method

Finding the Intercepts of a Graph

The intercepts of a graph share an important feature. For all $x$ -intercepts, the $y$ -coordinate is $0,$ and for all $y$ -intercepts, the $x$ -coordinate is $0.$ $\begin{aligned} x\text{-int} &: (x,0) \\ y\text{-int} &: (0,y) \end{aligned}$ 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. $2x+5y=10$

Method

Finding the $x$ -intercept

To find the $x$ -intercept, $y=0$ can be substituted into the equation.

2x+5y=10 \quad \Rightarrow \quad 2x+5\cdot {\color{#0000FF}{0}} =10

Next, solve the equation for

x.

2x+5\cdot0 =10

MultiplyMultiply

2x=10

DivEqn

\left.\text{LHS}\middle/2\right.=\left.\text{RHS}\middle/2\right.

x=5

The

x

-intercept is

(5,0).

Method

Finding the $y$ -intercept

The $y$ -intercept can be found in a similar way. Substitute $x=0$ into the equation and solve for $y.$

2\cdot {\color{#0000FF}{0}}+5x =10

MultiplyMultiply

5y=10

DivEqn

\left.\text{LHS}\middle/2\right.=\left.\text{RHS}\middle/2\right.

y=2

The

y

-intercept is

(0,2).

Graph the function given the situation

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Exercise

The amusement park ride "Spinning Teacups" has two different sizes of cups, large and small. Large cups fit $6$ people and small cups fit $4$ people. Maximum capacity for each ride is $48$ people. The equation $4x+6y=48$ models this situation, where $x$ is the number of small cups and $y$ is the number of large cups. Graph the situation and interpret the intercepts.

Show Solution

Solution

Example

Finding the intercepts

To begin, we will find each of the intercepts. Starting with the

x

-intercept, we can substitute

y=0

into the rule and solve for

x.

4x+6y=48

Substitute

y={\color{#0000FF}{0}}

4x+6\cdot{\color{#0000FF}{0}}=48

MultiplyMultiply

4x=48

DivEqn

\left.\text{LHS}\middle/4\right.=\left.\text{RHS}\middle/4\right.

x=12

The

x

-intercept is

(12,0).

To find the

y

-intercept we can substitute

x=0

and solve for

y.

4x+6y=48

Substitute

x={\color{#0000FF}{0}}

4\cdot{\color{#0000FF}{0}}+6y=48

Solve for

y

MultiplyMultiply

6y=48

DivEqn

\left.\text{LHS}\middle/6\right.=\left.\text{RHS}\middle/6\right.

y=8

The

y

-intercept is

(0,8).

Example

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 $x$ and $y$ represent the numbers of different cups, negative numbers should not be included.

Example

Interpreting the intercepts

We can interpret the intercepts in terms of what $x$ and $y$ represent. The $x$ -intercept is $(12,0).$ This means a ride with $12$ small cups can not have any large cups, because the maximum capacity of people has already been met. Similarly, the $y$ -intercept of $(8,0),$ tells us that a ride with $8$ large cups will not allow for any small cups.