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# Graphing Linear Relationships

The most common way to write equations of linear functions is in slope-intercept form.

$y=mx+b$

$m$ indicates the slope, and $b$ is the $y$-intercept.
Method

## Graphing a Linear Function in Slope-Intercept Form

To graph a line in slope-intercept form, the slope, $m,$ and the $y$-intercept, $b,$ are both needed. Consider the linear function $y=2x-3.$ Since the rule is written as $y=mx+b,$ it can be seen that $m=2 = \dfrac{2}{1} \quad \text{and} \quad b=\text{-} 3.$ To graph the line, plot the $y$-intercept, then use the slope to find another point on the line. Specifically, from $(0,\text{-} 3),$ move up $2$ units and right $1$ unit.

Next, draw a line through both points to create the graph of the linear function.

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Exercise

In Clear Lake, Iowa, during a particular evening, there is a $3$-inch layer of snow on the ground. At midnight, it begins to snow. Each hour, one inch of snow falls. Graph a function that shows the amount of snow on the ground from midnight to $6$ AM.

Show Solution
Solution

To begin, we can define the quantities that $x$ and $y$ represent.

• Let $x$ represent the number of hours it's been snowing.
• Let $y$ represent the number of inches of snow on the ground.

It's been given that there is a $3$-inch layer of snow on the ground before it begins to snow. Thus, when $x=0, y=3.$ In other words, the $y$-intercept is $b=3.$ It is also given that the amount of snow on the ground increases by $1$ inch every $1$ hour. Thus, the slope of the line is $\frac{1}{1}.$ We can write the rule for this function as $f(x)=\dfrac{1}{1}x+3 \quad \Rightarrow f(x)=x+3$ To graph the function, we can plot a point at $(0,3),$ then move up $1$ unit and right $1$ unit to find another point. The line that connects these points is the graph of the function.

The graph above shows the function $f(x)=x+3.$ It can be seen that, at $6$AM, there is a total of $9$ inches of snow on the ground.

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.
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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 $x=0.$
$4x-2y=7$
$4\cdot {\color{#0000FF}{0}} - 2y=7$
$\text{-} 2y=7$
$y=\dfrac{7}{\text{-} 2}$
$y=\text{-} 3.5$
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.

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 $x.$
$2x+5\cdot0 =10$
$2x=10$
$x=5$
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 $y.$

$2\cdot {\color{#0000FF}{0}}+5x =10$
$5y=10$
$y=2$
The $y$-intercept is $(0,2).$
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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 $x.$
$4x+6y=48$
$4x+6\cdot{\color{#0000FF}{0}}=48$
$4x=48$
$x=12$
The $x$-intercept is $(12,0).$ To find the $y$-intercept we can substitute $x=0$ and solve for $y.$
$4x+6y=48$
$4\cdot{\color{#0000FF}{0}}+6y=48$
Solve for $y$
$6y=48$
$y=8$
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.

Concept

## Linear Inequality

A linear inequality is an inequality involving a linear relation in one or two variables, usually $x$ and $y.$ An example of a linear inequality is $9x+3y\leq6.$

Linear inequalities are similar to linear equations, but, whereas the solutions to a linear equation are all the coordinates that lie on the line, the solution set to a linear inequality is a region containing one half of the coordinate plane.
Method

## Graphing a Linear Inequality

The method to graph a linear inequality is similar to graphing a linear equation in slope-intercept form, but instead of a line, the graph of a linear inequality is an entire region.

To graph the linear inequality $9x+3y\leq6,$ write the inequality in slope-intercept form, draw the boundary line, and shade the region that contains the solutions.

### 1

Write the inequality in slope-intercept form
To find the boundary line of the region, start by writing the inequality in slope-intercept form. In other words, solve for $y.$
$9x+3y\leq6$
$3y\leq \text{-}9x+6$
$y \leq \dfrac{\text{-} 9x+6}{3}$
$y \leq \text{-} \dfrac{9x}{3} + \dfrac{6}{3}$
$y\leq\text{-} 3x+2$
Written in slope-intercept form, the inequality becomes $y\leq \text{-} 3x+2.$

### 2

Graph the boundary line

The boundary line of the inequality is the line corresponding to the equation produced if the inequality symbol is replaced by an equals sign. In this case, this is the line $y=\text{-} 3x+2.$ If the inequality symbol is $<$ or $>$, the boundary line is dashed. If the symbol is $\leq$ or $\geq$, the line is solid. Here, the line will be solid. The boundary line can be graphed using the $y$-intercept and the slope.

### 3

Test a point
The region either to the left or the right of the boundary line contains the solution set. To determine which, substitute an arbitrary test point (not on the boundary line) into the inequality to determine if it is a solution. Using $(0,0)$ is preferable.
$y\leq \text{-} 3x+2$
${\color{#009600}{0}}\stackrel{?}{\leq} \text{-} 3\cdot{\color{#0000FF}{0}} + 2$
$0\leq 2$
Since $0 \leq 2$ makes a true statement, it is a solution to the inequality.

### 4

If the test point is a solution to the inequality, the region in which it lies contains the entire solution set. If not, the other region contains the solutions. To show the set, shade the appropriate region.

Here, test point $(0,0)$ is a solution to the inequality.

The region containing $(0,0)$ lies to the left of the boundary line. Thus, this region shows the solution set of the inequality.