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Graphing Linear Relationships

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


mm indicates the slope, and bb is the yy-intercept.

Graphing a Linear Function in Slope-Intercept Form

To graph a line in slope-intercept form, the slope, m,m, and the yy-intercept, b,b, are both needed. Consider the linear function y=2x3.y=2x-3. Since the rule is written as y=mx+b,y=mx+b, it can be seen that m=2=21andb=-3. m=2 = \dfrac{2}{1} \quad \text{and} \quad b=\text{-} 3. To graph the line, plot the yy-intercept, then use the slope to find another point on the line. Specifically, from (0,-3),(0,\text{-} 3), move up 22 units and right 11 unit.

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


In Clear Lake, Iowa, during a particular evening, there is a 33-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 66 AM.

Show Solution

To begin, we can define the quantities that xx and yy represent.

  • Let xx represent the number of hours it's been snowing.
  • Let yy represent the number of inches of snow on the ground.

It's been given that there is a 33-inch layer of snow on the ground before it begins to snow. Thus, when x=0,y=3.x=0, y=3. In other words, the yy-intercept is b=3.b=3. It is also given that the amount of snow on the ground increases by 11 inch every 11 hour. Thus, the slope of the line is 11.\frac{1}{1}. We can write the rule for this function as f(x)=11x+3f(x)=x+3 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),(0,3), then move up 11 unit and right 11 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.f(x)=x+3. It can be seen that, at 66AM, there is a total of 99 inches of snow on the ground.


Standard Form of a Line

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


Here, A,A, B,B, and CC are real numbers and AA and BB cannot both equal 0.0. Several combinations of A,A, B,B, and CC 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. 4x2y=7 4x-2y=7

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 xx-values into the equation to find the corresponding yy-values. Let's start with x=0.x=0.
402y=74\cdot {\color{#0000FF}{0}} - 2y=7
-2y=7\text{-} 2y=7
y=7-2y=\dfrac{7}{\text{-} 2}
y=-3.5y=\text{-} 3.5
One point on the line is (0,-3.5).(0,\text{-}3.5). We can use the same process for finding other points.
xx 4x2y=74x-2y=7 yy
1{\color{#0000FF}{1}} 412y=74 \cdot {\color{#0000FF}{1}}-2y=7 -1.5\text{-}1.5
2{\color{#0000FF}{2}} 422y=74 \cdot {\color{#0000FF}{2}}-2y=7 0.50.5
3{\color{#0000FF}{3}} 432y=74 \cdot {\color{#0000FF}{3}}-2y=7 2.52.5
4{\color{#0000FF}{4}} 442y=74 \cdot {\color{#0000FF}{4}}-2y=7 4.54.5

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


Finding the Intercepts of a Graph

The intercepts of a graph share an important feature. For all xx-intercepts, the yy-coordinate is 0,0, and for all yy-intercepts, the xx-coordinate is 0.0. x-int:(x,0)y-int:(0,y)\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 2x+5y=10


Finding the xx-intercept

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

2x+5y=102x+50=10 2x+5y=10 \quad \Rightarrow \quad 2x+5\cdot {\color{#0000FF}{0}} =10 Next, solve the equation for x.x.
2x+50=102x+5\cdot0 =10
The xx-intercept is (5,0).(5,0).

Finding the yy-intercept

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

20+5x=102\cdot {\color{#0000FF}{0}}+5x =10
The yy-intercept is (0,2).(0,2).

The amusement park ride "Spinning Teacups" has two different sizes of cups, large and small. Large cups fit 66 people and small cups fit 44 people. Maximum capacity for each ride is 4848 people. The equation 4x+6y=48 4x+6y=48 models this situation, where xx is the number of small cups and yy 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 xx-intercept, we can substitute y=0y=0 into the rule and solve for x.x.
The xx-intercept is (12,0).(12,0). To find the yy-intercept we can substitute x=0x=0 and solve for y.y.
Solve for yy
The yy-intercept is (0,8).(0,8).

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 xx and yy represent the numbers of different cups, negative numbers should not be included.


Interpreting the intercepts

We can interpret the intercepts in terms of what xx and yy represent. The xx-intercept is (12,0).(12,0). This means a ride with 1212 small cups can not have any large cups, because the maximum capacity of people has already been met. Similarly, the yy-intercept of (8,0),(8,0), tells us that a ride with 88 large cups will not allow for any small cups.


Linear Inequality

A linear inequality is an inequality involving a linear relation in one or two variables, usually xx and y.y. An example of a linear inequality is 9x+3y6. 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.

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+3y6, 9x+3y\leq6, write the inequality in slope-intercept form, draw the boundary line, and shade the region that contains the solutions.


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.y.
3y-9x+63y\leq \text{-}9x+6
y-9x+63y \leq \dfrac{\text{-} 9x+6}{3}
y-9x3+63y \leq \text{-} \dfrac{9x}{3} + \dfrac{6}{3}
y-3x+2y\leq\text{-} 3x+2
Written in slope-intercept form, the inequality becomes y-3x+2. y\leq \text{-} 3x+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=-3x+2. 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 yy-intercept and the slope.


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)(0,0) is preferable.
y-3x+2y\leq \text{-} 3x+2
0?-30+2{\color{#009600}{0}}\stackrel{?}{\leq} \text{-} 3\cdot{\color{#0000FF}{0}} + 2
020\leq 2
Since 020 \leq 2 makes a true statement, it is a solution to the inequality.


Shade the correct region

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)(0,0) is a solution to the inequality.

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

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