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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=CAx+By=C

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.
Exercise

Graph the linear function given by the equation using a table of values. 4x2y=7 4x-2y=7

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.
4x2y=74x-2y=7
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.

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Theory

Graphing Linear Functions using Intercepts

A function's xx- and yy-intercepts are the points where the graph of a function intersects with the xx- and yy-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 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

Method

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
2x=102x=10
x=5x=5
The xx-intercept is (5,0).(5,0).
Method

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
5y=105y=10
y=2y=2
The yy-intercept is (0,2).(0,2).
Exercise

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.

Solution
Example

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.
4x+6y=484x+6y=48
4x+60=484x+6\cdot{\color{#0000FF}{0}}=48
4x=484x=48
x=12x=12
The xx-intercept is (12,0).(12,0). To find the yy-intercept we can substitute x=0x=0 and solve for y.y.
4x+6y=484x+6y=48
40+6y=484\cdot{\color{#0000FF}{0}}+6y=48
Solve for yy
6y=486y=48
y=8y=8
The yy-intercept is (0,8).(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 xx and yy 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 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.


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