Math 8 ( Period 1)

Chapter 3-4 Slope-intercept Form

In a proportional linear relationship, the line ( and equation) go through the origin. When linear relationships are NOT PROPORTIONAL, the equation and graph go through a point on the y axis OTHER THAN (0,0)

The x-value will always be 0 because you are on the y-axis. The y value can be any value other than 0.

We call this the y-intercept and the variable we give this is b.
Mrs Sobieraj calls b your “Home Base” because  you will graph this point first before using the slope ( rise/run) to find additional points on the line.

y = mx + b  the most used form of a linear equation

The Slope- Intercept Form

y = mx + b

 where m = slope and b = y intercept (where this line hits the y-axis)

All you do is solve the equation for “y”  meaning isolate the y on one side of the equal sign.

For example:

-3y = -2x – 6 is NOT in slope –intercept form because y is NOT by itself.

Restate

-3y = -2x – 6 into slope intercept form
divide both sides by -3

y = (2/3)x + 2
So m = 2/3
and b = 2

If you are looking to find a point by  plugging into the equation:
Look at the coefficient for x

What x values will give you INTEGER ANSWERS for y? ( That will be easier to graph) In this example, they need to be multiples of 3. Now look at the graph of y = (2/3)x + 2
Notice that the   + 2     at the end is the y-intercept
(without having to do ANY WORK).
Use the counting method for slope on your graph.
You should have counted : UP 2 and then to the RIGHT 3

The slope therefore is 2/3 Look at the equation.
It told you 2/3 WITHOUT ANY WORK!!!

Graph when line is in SLOPE INTERCEPT form
So if you have the slope- intercept form of the equation, it is really easy to graph the line.

           1)Graph the intercept on the y-axis
    (That is the+/- constant at the end of the equation y= mx + b)

 “   2)Count” the next point by using the slope of the x-coefficient as a fraction.

For the equation
 y = (2/3)x + 2

1   1)Put a dot at (0,2)

    2)From (0,2) count UP 2 and over to the RIGHT 3 to find the next coordinate ( 3,4)

Remember slope is y over x  so the numerator is the change in y and the denominator is the change in x.

If it is positive you count UP (positive)  and over to the RIGHT (positive)
OR you can count DOWN (negative) AND to the LEFT (negative) because 2 negatives make a positive

If it is negative you count DOWN (negative) and over to the RIGHT (positive)
OR you can count UP (positive) AND to the LEFT (negative) because a positive and a negative make a negative

If you are given the slope and the y intercept, you can write the equation of any line.

Just use: y = mx + b

So if m = ¾ and b = -9

the equation is

y = (3/4)x – 9

It is easy to create the equation if you are either given BOTH m and b or you can find them by looking at the graph

 EXAMPLE:
m = -2/3 and b = -12

 The line would be y = (-2/3)x – 12

If you are given the graph of this line

Read the y- intercept by looking at the  y-axis ( 0, -12)
Count the slope from there to get another point that you can read on the graph using rise/run.

Another example:
Given 3x + 4y = 10

Solve for y
subtract 3x from both  sides
4y =-3x + 10
Now divide both sides by 4

y = (-3/4)x + 10/4
or
y = (-3/4)x + 5/2

Remember: The slope is the coefficient of the x
m = -3/4 ( so you are sliding DOWN at a little less than a 45 degree angle)

the y- intercept ( b) is the constant
b = 5/2
So the line crosses the y axis at  2½   

Notice that in this case the “b” is a fraction
That makes it harder to graph the slope from this point!
When this happens, the slope intercept form may not be the best from to graph the line

You must start at 2 ½ on the y axis and count down 3 from there and right 4. That is hard to do accurately!
Neither 3 nor 4 is a factor of the constant 10 so the intercepts will be fractions!

We need the x term to end up with ½ so that when we add that to the b ( 5/2) we will get an integer. so let’s make x = 2 and plug in

That will cross cancel with -3/4 slope to get

y = (-3/4)(2) + 5/2 = -3/2 + 5/2 carefully use your additional rules to get 2/2 = 1

So we found a coordinate that has just integers  ( 2, 1)
Now count the slope ( -3/4) from this point  instead of from the y-intercept

Real world meanings of the y intercept (b) and SLOPE (m)

Remember that the slope is still the constant rate of change, which means it is the unit rate.
They y-intercept is the initial value of the real world problem.

Make sure to watch the T-shirt video example in your McGraw Portal
In the example, the initial design of the shirt costs $20 and then each shirt costs $5 to print. Even before printing one shirt, you need to pay $20 so this is the initial value of the y-intercept (b)
The unit rate of constant change is the slope of $5/shirt.