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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.

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

$Ax+By=C$

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

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 $x$-values into the equation to find the corresponding $y$-values. Let's start with $x=0.$
One point on the line is $(0,\text{-}3.5).$ We can use the same process for finding other points.

$4x-2y=7$

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

$y=\text{-} 3.5$

$x$ | $4x-2y=7$ | $y$ |
---|---|---|

${\color{#0000FF}{1}}$ | $4 \cdot {\color{#0000FF}{1}}-2y=7$ | $\text{-}1.5$ |

${\color{#0000FF}{2}}$ | $4 \cdot {\color{#0000FF}{2}}-2y=7$ | $0.5$ |

${\color{#0000FF}{3}}$ | $4 \cdot {\color{#0000FF}{3}}-2y=7$ | $2.5$ |

${\color{#0000FF}{4}}$ | $4 \cdot {\color{#0000FF}{4}}-2y=7$ | $4.5$ |

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

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.

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. $2x+5y=10$

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 $y$-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 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

$4x+6y=48$

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

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

MultiplyMultiply

$4x=48$

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

$x=12$

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.

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.

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