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Lesson 6: Graphing slope-intercept form

# 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 $y=2x+3$.
Recall that in the general slope-intercept equation $y=mx+b$, the slope is given by $m$ and the $y$-intercept is given by $b$. Therefore, the slope of $y=2x+3$ is $2$ and the $y$-intercept is $\left(0,3\right)$.
In order to graph a line, we need two points on that line. We already know that $\left(0,3\right)$ is on the line.
Additionally, because the slope of the line is $2$, we know that the point $\left(0+1,3+2\right)=\left(1,5\right)$ is also on the line.

Problem 1
Graph $y=3x-1$.

Problem 2
Graph $y=-4x+5$.

## Graphing lines with fractional slope

Let's graph $y=\frac{2}{3}x+1$.
As before, we can tell that the line passes through the $y$-intercept $\left(0,1\right)$, and through an additional point $\left(0+1,1+\frac{2}{3}\right)=\left(1,1\frac{2}{3}\right)$.
While it is true that the point $\left(1,1\frac{2}{3}\right)$ 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 $\frac{2}{3}$, increasing $x$ by $3$ units will cause $y$ to increase by $2$ units.
This gives us the additional point $\left(0+3,1+2\right)=\left(3,3\right)$.

Problem 3
Graph $y=\frac{3}{4}x+2$.

Problem 4
Graph $y=-\frac{3}{2}x+3$.

## 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
• 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 negative

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.
• 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.
• When you have a negative slope, like -3/2 then one of the numbers must be negative (remember, negative divided by positive = negative; and positive divided by negative = negative).

So, when you interpret the slope of -3.2...
1) You can do -3 for change in Y (move down 3 units) and +2 for X (move right 2 units). OR... 2) You can do +3 for change in Y (up 3 units) and -2 for X (move left 2 units).
Hope this helps.
• im having some trouble... anybody have some helpful tips hehehe
• place you first point on the y axis +/-. Then turn the slope into a fraction.
Slope: The positive or negative sign determine if the line goes up or down from the y intercept. Based on that, going left to right, if it is a negative travel down the numerator, travel right the denominator.

Y=27/3x+1 Place first point on y axis at positive 1. Then travel up 27, then go right 3. Simplified, you would go up 9, and right 1.
• 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!
• 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!
• Not to be that person but like When am I reallyyyyyyyyyyyy going to use this in everyday life?
• my teacher says yes but he is a goober so I don't know
• 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.
• I can't understand how to graph an equation with a fraction y-intercept. Ex: y=2x-1/2
• Put a point at (0, -1/2). It is half-way between 0 and -1.
Since the slope is 2, you move up 2 units and right 1.
-- Up 1 unit takes you to 1/2, up 2 units takes you to 1 1/2 (halfway between 1 and 2).
-- Then, go right 1 unit. You should now be at the point 1 1/2, 1)

Hope this helps.
• what if the question is y=x+4