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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.

A linear equation is in standard form if all terms for the x and y variables are on one side of the equation, and the constant is on the other side of the equation.

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 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,-3.5). We can use the same process for finding other points.

4x−2y=7

Substitute

x=0

4⋅0−2y=7

ZeroPropMult

Zero Property of Multiplication

-2y=7

DivEqn

$LHS/(-2)=RHS/(-2)$

$y=-27 $

CalcQuot

Calculate quotient

y=-3.5

x | 4x−2y=7 | y |
---|---|---|

1 | 4⋅1−2y=7 | -1.5 |

2 | 4⋅2−2y=7 | 0.5 |

3 | 4⋅3−2y=7 | 2.5 |

4 | 4⋅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. ### Method

### Finding the x-intercept

To find the x-intercept, y=0 can be substituted into the equation.
Next, solve the equation for x.
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.
The y-intercept is (0,2).

$x-inty-int :(x,0):(0,y) $

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

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

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.

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