Graphing and Equations Please view this tutorial and answer the follow-up questions on loose leaf to...

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)