** Drawing Lines from Given Equation**

**GPS 18: Graphing Lines in Slope-Intercept Form**

To see why it is so easy to graph a line in slope-intercept form, we first revisit an earlier formulation of the slope.

**Slope Formula **

The * slope* of the (nonvertical) line passing through any two points is

𝑚𝑚=subtract _{subtract }𝑦𝑦 −_{𝑥𝑥 −}coordinates_{coordinates}

**PB**

When we subtract the 𝑦𝑦-coordinates, geometrically we are calculating the distance between their values on the

𝑦𝑦-axis. This tells us how far up or down the 𝑦𝑦-axis we have moved. Similarly, when we subtract the 𝑥𝑥-

coordinates, geometrically we are calculating the distance between their values on the 𝑥𝑥-axis. This leads to the final interpretation of the slope.

**Slope Formula Geometrically **

The * slope* of the (nonvertical) line passing through any two points is

𝑚𝑚=subtract _{subtract }𝑦𝑦 −_{𝑥𝑥 −}coordinates_{coordinates =}rise_{run =}distance up/down_{distance right/left}

106 GPS 18: Graphing Lines in Slope-Intercept Form

**Problem 18. 1 **

Graph the line.

𝑦𝑦=2_{3}𝑥𝑥+ 1

1) Identify the slope and 𝑦𝑦-intercept.
a. The slope is 𝑚𝑚=2_{3}.
b. The 𝑦𝑦-intercept is (0,1).
2) Plot the 𝑦𝑦-intercept.

3) Use the slope to plot a 2nd_{ point. }

a. If the numerator (top) of the slope is

* positive*, go

*that many units. If it is*

**up*** negative*, go

*that many units. b. If the denominator (bottom) of the*

**down**slope is * positive*, go

*that many units. If it is*

**right***, go*

**negative***that many units.*

**left**Our slope tells us

𝑚𝑚=2_{3 =}_{right 3}up 2

4) Draw the line passing through the two points.

Even though plotting two points is good enough to graph the line, it is often helpful to plot more points. To do this, we can simply repeat the moving up/down and moving right/left as directed by the slope.

It is easy to see how to use this technique to plot a 3rd

point. But if we examine this line closely, we can spot

4th_{ and }_{5}th_{ points on the line. It turns out that we can }

also find these two points by remembering a fact from arithmetic.

The slope can be written as

𝑚𝑚=2_{3}

But since a negative divided by a negative is positive, we can also write the slope as

𝑚𝑚=2_{3 =}−_{−}2_{3 =}down 2_{left 3}

By doing this, we can easily plot the 4th_{ and }_{5}th_{ points! }

𝑥𝑥 𝑦𝑦 Up 2 Right 3 2nd point Start here 𝑥𝑥 𝑦𝑦 2nd point 1st Point 3rd point 4th point 5th point

Recall with fractions we can write the negative in *any* one of three places.
For example:

−3_{2 =}−_{2 =}3 _{−}3_{2}

**AA**

**Problem 18.2 **

Graph the line.

𝑦𝑦=−3_{2}𝑥𝑥+ 4

The slope is 𝑚𝑚=−3_{2}.
The 𝑦𝑦-intercept is (0,4).
We start by plotting (0,4)

Using the slope, we find our directions

−3_{2 =}−_{2 =}3 down 3_{right 2}

We can also write the negative in the denominator to get

−3_{2 =}_{−}3_{2 =}_{left 2}up 3

Doing this repeatedly, we can plot more than 2 points quite easily.

** Problem 18.3 **

Graph the line.

𝑦𝑦=−1_{2}𝑥𝑥+ 3

The slope is 𝑚𝑚=______________.

Using the slope, write down the directions to follow to find additional points.

The 𝑦𝑦-intercept is ______________.

Plot additional points on the line by repeatedly moving up/down and right/left as directed by the slope.

𝑥𝑥 𝑦𝑦 Down 3 Right 2 2nd point Start here 𝑥𝑥 𝑦𝑦

108 GPS 18: Graphing Lines in Slope-Intercept Form

Whole numbers can be written as a fraction by rewriting with denominator 1. For example:

3 =3_{1}

**AA**

** Problem 18.4 **

Graph the line.

𝑦𝑦= 3𝑥𝑥 −2

The slope is 𝑚𝑚=______________.

Using the slope, write down the directions to follow to find additional points.

The 𝑦𝑦-intercept is ______________.

Plot additional points on the line by repeatedly moving up/down and right/left as directed by the slope.

** Problem 18.5 **

Graph the line.

3𝑥𝑥 −5𝑦𝑦= 10

The slope is 𝑚𝑚=______________.

Using the slope, write down the directions to follow to find additional points.

The 𝑦𝑦-intercept is ______________.

Plot additional points on the line by repeatedly moving up/down and right/left as directed by the slope.

𝑥𝑥 𝑦𝑦

𝑥𝑥 𝑦𝑦

** Problem 18.6 **

Graph the line.

2𝑥𝑥+ 3𝑦𝑦= 0

The slope is 𝑚𝑚=______________.

Using the slope, write down the directions to follow to find additional points.

The 𝑦𝑦-intercept is ______________.

** Problem 18.7 **

Graph the line.

−𝑥𝑥+ 4𝑦𝑦= 8

The slope is 𝑚𝑚=______________.

Using the slope, write down the directions to follow to find additional points.

The 𝑦𝑦-intercept is ______________.

𝑥𝑥 𝑦𝑦

𝑥𝑥 𝑦𝑦

110 GPS 18: Graphing Lines in Slope-Intercept Form

** Problem 18.8 **

Graph the line.

−6𝑥𝑥 −5𝑦𝑦= 15

The slope is 𝑚𝑚=______________. The 𝑦𝑦-intercept is ______________.

** Problem 18.9 **

Graph the line.

𝑦𝑦= 4𝑥𝑥

The slope is 𝑚𝑚=______________. The 𝑦𝑦-intercept is ______________.

** Problem 18.10 **

Graph the line.

𝑥𝑥+𝑦𝑦= 4

The slope is 𝑚𝑚=______________. The 𝑦𝑦-intercept is ______________.

Plot additional points on the line by repeatedly moving up/down and right/left as directed by the slope. 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦