If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

8th grade (Illustrative Mathematics)

Course: 8th grade (Illustrative Mathematics)>Unit 3

Lesson 8: Extra practice: Slope

Graphing slope-intercept form

Learn how to graph lines whose equations are given in the slope-intercept form y=mx+b.
If you haven't read it yet, you might want to start with our introduction to slope-intercept form.

Graphing lines with integer slopes

Let's graph y, equals, 2, x, plus, 3.
Recall that in the general slope-intercept equation y, equals, start color #ed5fa6, m, end color #ed5fa6, x, plus, start color #0d923f, b, end color #0d923f, the slope is given by start color #ed5fa6, m, end color #ed5fa6 and the y-intercept is given by start color #0d923f, b, end color #0d923f. Therefore, the slope of y, equals, start color #ed5fa6, 2, end color #ed5fa6, x, plus, start color #0d923f, 3, end color #0d923f is start color #ed5fa6, 2, end color #ed5fa6 and the y-intercept is left parenthesis, 0, comma, start color #0d923f, 3, end color #0d923f, right parenthesis.
In order to graph a line, we need two points on that line. We already know that left parenthesis, 0, comma, start color #0d923f, 3, end color #0d923f, right parenthesis is on the line.
Additionally, because the slope of the line is start color #ed5fa6, 2, end color #ed5fa6, we know that the point left parenthesis, 0, start color #ed5fa6, plus, 1, end color #ed5fa6, comma, start color #0d923f, 3, end color #0d923f, start color #ed5fa6, plus, 2, end color #ed5fa6, right parenthesis, equals, left parenthesis, 1, comma, 5, right parenthesis is also on the line.
A coordinate plane. The x- and y-axes each scale by one. The graph of a line goes through the points zero, three and one, five. Both of these points are plotted and labeled.

Check your understanding

Problem 1
Graph y, equals, 3, x, minus, 1.

Problem 2
Graph y, equals, minus, 4, x, plus, 5.

Graphing lines with fractional slope

Let's graph y, equals, start color #ed5fa6, start fraction, 2, divided by, 3, end fraction, end color #ed5fa6, x, start color #0d923f, plus, 1, end color #0d923f.
As before, we can tell that the line passes through the y-intercept left parenthesis, 0, comma, start color #0d923f, 1, end color #0d923f, right parenthesis, and through an additional point left parenthesis, 0, start color #ed5fa6, plus, 1, end color #ed5fa6, comma, start color #0d923f, 1, end color #0d923f, start color #ed5fa6, plus, start fraction, 2, divided by, 3, end fraction, end color #ed5fa6, right parenthesis, equals, left parenthesis, 1, comma, 1, start fraction, 2, divided by, 3, end fraction, right parenthesis.
While it is true that the point left parenthesis, 1, comma, 1, start fraction, 2, divided by, 3, end fraction, right parenthesis is on the line, we can't plot points with fractional coordinates as precisely as we draw points with integer coordinates.
We need a way to find another point on the line whose coordinates are integers. To do that, we use the fact that in a slope of start color #ed5fa6, start fraction, 2, divided by, 3, end fraction, end color #ed5fa6, increasing x by start color #ed5fa6, 3, end color #ed5fa6 units will cause y to increase by start color #ed5fa6, 2, end color #ed5fa6 units.
This gives us the additional point left parenthesis, 0, start color #ed5fa6, plus, 3, end color #ed5fa6, comma, start color #0d923f, 1, end color #0d923f, start color #ed5fa6, plus, 2, end color #ed5fa6, right parenthesis, equals, left parenthesis, 3, comma, 3, right parenthesis.
A coordinate plane. The x- and y-axes each scale by one. The graph of a line goes through the points zero, three and two, zero. Both of these points are plotted and labeled.

Check your understanding

Problem 3
Graph y, equals, start fraction, 3, divided by, 4, end fraction, x, plus, 2.

Problem 4
Graph y, equals, minus, start fraction, 3, divided by, 2, end fraction, x, plus, 3.

Want to join the conversation?

• How come if the negative sign is next to the fraction it causes the rise to be negative but not the run
(24 votes)
• Think about the fraction as division... How do you get a negative number when dividing:
a negative divided by a positive = a negative
a positive divided by a negative = a positive

As you can see, only one of the 2 numbers can be negative. Thus, for a slope like -4/5, you can apply the negative sign to the numerator which would tell you to go down 4 units, then right 5 units. Or, you can apply the negative to the denominator which would make you go up 4 units and left 5 units.

If you make both numbers negative, then you are doing: negative divided by negative = positive. And, you would have a positive slope.

Hope this helps.
(44 votes)
• i don't really get it why in the last exercise the slope is -3/2 you ad plus 2 for the change in x but minus 3 for the change in y.
(32 votes)
• Because -3/2 is basically equal to minus 3 by PLUS 2
(7 votes)
• im having some trouble... anybody have some helpful tips hehehe
(12 votes)
• How do I graph a line if the slope isn't provided? Here is what I mean:

y=-x+6

How do I graph it if I do not know the slope? Thanks!
(12 votes)
• When a variable doesn't have a variable, it's safe to assume the variable is 1. So, -x would be -1x or -1/1x.
Hope that makes sense!
(8 votes)
• I don't have a clue on how to do this
(11 votes)
• If you have an equation in slope-intercept form, you know both a point (the y intercept) and the slope, so it should be relatively easy to graph especially with a little practice. So if you have y=3x-4, the slope is 3=3/1, the y intercept is (0,-4). We can plot the point by starting at the origin and counting down 4 to get to (0,-4) and put a dot at this point. With a slope of rise (up) 3 over run (right) 1, you get to (0+1,-4+3) which is (1,-1), and a second time (1+1,-1+3) which is (2,2) and you have three points to draw a line through. One more example, if you have y=-3/4x + 2, you have a point (0,2) and a slope of -3/4 (rise down 3 right 4). This gives a second point of (0+4,2-3) or (4,-1) and (4+4,-1-3) or (8,-4) to draw a line. So start with the y intercept, and count the slope from that point.
(9 votes)
• i don't really get it why in the last exercise the slope is -3/2 you ad plus 2 for the change in x but minus 3 for the change in y.
(9 votes)
• brah can sum1 help me I no understand um, Mahaloz
(8 votes)
• what is the difference between zero slope and no slope.
(1 vote)
• Zero slope means the slope is defined as zero, but no slope means the slope is undefined. A horizontal line has zero slope, but a vertical line has no slope.

Have a blessed, wonderful day!
(12 votes)
• Someone please help me. Im not undertsnding
(6 votes)
• Okay so, for example, the question,
y=1/3x - 2
is a demonstration of the Slope-Intercept Form. Which is written,
y=mx + b
the -2 is obviously b which is the y-intercept. You would graph -2 as (0,-2). You would go 3 left and 1 up. Or better known as rise/run. And you would continue to get the line, by connecting the dots. Hope that helped you!
(1 vote)
• Not to be that person but like When am I reallyyyyyyyyyyyy going to use this in everyday life?
(4 votes)
• That all depends on what your choice of careers is. Many fields of Math, science, construction, business, finance, etc. use graphs.
(3 votes)