Graphing and Equations
Please view this tutorial and answer the follow-up questions on loose leaf to
turn in to your teacher.
Important Information
• When graphing a linear equation, you need to know the slope and the y-intercept
• The slope is the rate of change in your graph or how the y values change as x increases
• The y-intercept is the place where the graph crosses the y-axis
• The general form of a linear equation is y = a + bx where a is the y-intercept and b is the slope
Slope
• There are many different ways to represent the slope.
change in y; ; change in x
• If the line is increasing from left to right, the slope is positive
• If the line is decreasing from left to right, the slope is negative
rise
run
ΔyΔx
Slope• The numerator (top number in the fraction) will tell you how much to move up or down• The denominator (bottom number in the fraction) will tell you how much to move left or right• If the slope is positive, start at the y-intercept and move up and to the right or down and to the left• If the slope is negative, start at the y-intercept and move up and to the left or down and to the right
SlopeLook at the following fractions to see why you
move in specific directions for positive or negative slopes.
down(negative)
right(positive)=negative
OR
OR
Y-Intercept
• When looking for the y-intercept, you should follow the line until it hits the y-axis
• If the line crosses above the x-axis, the y-intercept is positive
• If the line crosses below the x-axis, the y-intercept is negative
Making the Graph Given the Equation
Let’s take a look at the following
equation.
y=2 +35x
What are the slope and y-intercept?
The slope will always
be the value with the
x-term so in this case,
the slope
is… 3
5
Making the Graph Given the Equation
Let’s take a look at the following
equation.
y=2 +35x
What are the slope and y-intercept?
The y-intercept will be
the number that
stands alone (not with
the x term) so in this
case the y-intercept
is… 2
Making the Graph Given the Equation
Now that we know the slope and y-
intercept, we can graph the equation.
y=2 +35x
Making the Graph Given the Equation
First you’ll need to plot the y-intercept.
y=2 +35x
Put a point at 2 on the y-axis.
Making the Graph Given the Equation
Next, you’ll need to find another
point on the line using the slope.
y=2 +35x
The numerator is the change in y
and the denominator is the
change in x.
Making the Graph Given the Equation
Since the slope is positive, you can
find the next point by moving up and
to the right or down and to the
left.
y=2 +35x
Making the Graph Given the Equation
Start at the y intercept and go up
3
y=2 +35x
then to the right 5 and put a point.
Making the Graph Given the Equation
You can also start at the y intercept and
go down three
y=2 +35x
then to the left 5 and put a point.
Making the Graph Given the Equation
Now connect these points with a line.
y=2 +35x
Your line should go through all the points you
plotted and continues through the entire graph.
Making the Graph Given the Equation
y=2 +35x
Be sure to label the line with your y=
equation.
Making the Graph Given the Equation
Let’s try another example…
y=4 −2x
Again, our first step is to find the slope
and y-intercept.
What is the slope?
In this case, the slope is -2 because that is the
value with the x-term. Be very careful to
always include the sign when you are finding
slope.
Making the Graph Given the Equation
Let’s try another example…
y=4 −2x
Again, our first step is to find the slope
and y-intercept.
What is the y-intercept?
The y-intercept is 4 because that is the number that stands
alone.
Making the Graph Given the Equation
Now we can graph the equation!
Plot a point at the y-intercept.
y=4 −2x
Making the Graph Given the Equation
Since the slope is a whole number, we
need to change it to a fraction.
How would you write -2 as a
fraction?
y=4 −2x
Making the Graph Given the Equation
-2 is equal to
Now we need to plot the next point.
y=4 −2x
−21
Making the Graph Given the Equation
-2 is equal to
Start at the y-intercept and go
down 2
y=4 −2x
−21
then to the right 1 and plot a point
Making the Graph Given the Equation
-2 is equal to
You could also go up 2
y=4 −2x
−21
then to the left 1 and plot a point
Making the Graph Given the Equation
Now draw a line to connect the points.
y=4 −2x
Making the Graph Given the Equation
Again, be sure to label the line with
the equation.
y=4 −2x
Making the Equation Given the Graph
When you are given then graph, you’ll need to find the y-intercept and the
slope.
Where does the line touch the y-axis?
Making the Equation Given the Graph
The graph crosses the y-axis at 5.
Therefore, the y-intercept is 5.
Making the Equation Given the Graph
Next, you’ll need to find the slope.
Remember to look for “points in
corners”
Making the Equation Given the Graph
Find another point in a “corner”.
The first point in a corner is at your y-
intercept (0, 5).
Making the Equation Given the Graph
Now find the slope between these two
points.
The next point in a corner is (1, 1).
Making the Equation Given the Graph
You can use the slope formula above
to find the slope.
(0, 5) and (1, 1)
y2 −y1x2 −x1
Making the Equation Given the Graph
(0, 5) and (1, 1)
y2 −y1x2 −x1
1−51−0
=−41
Making the Equation Given the Graph
Now you know the slope and y-intercept so you can find your
equation.
Remember the general form of a line
is
y=a+bx
Making the Equation Given the Graph
So, the equation of the line is
y=5 −4x
Making the Equation Given the Graph
Let’s try another example.
First, you’ll need to find the y-intercept.
In this case, the y-intercept is -2.
Making the Equation Given the Graph
Next, you’ll need to find the slope.
You can use the slope formula or you can use the graph to find the change in y and the change in x.
Making the Equation Given the Graph
Now we need to find another point in a
“corner”.
We already have the y-intercept as our
first point in a corner.
Making the Equation Given the Graph
Let’s use the triangle method to find the
slope between these two points.
(3, 1) is a point in a “corner”.
Making the Equation Given the Graph
Starting from the y-intercept, you need to draw a triangle between your two points in “corners”.
Making the Equation Given the Graph
What is the difference in your y
values?
33
What is the difference in your x
values?
Making the Equation Given the Graph
So your slope is
33
3
3=1
Making the Equation Given the Graph
We can plug the slope and y-intercept
into the general equation of a line.
y =−2 +1x
Follow-Up Questions
Answer the following questions on loose leafand hand them in to your teacher.
Follow-Up Questions
1. Graph the following equations. Be sure to label each line.
y=7 −34x
y=−3+13x
y =5x
y =−1−4x
y =x+ 6
a)
c)
b)
d)
e)
Follow-Up Questions
2. Find the equations to the given graphs.
a) b)
Follow-Up Questions
c) d)
Follow-Up Questions
e)