Algebra Period 3 (Thursday)

Finding the Equation of a Line: 7-6
Review:
1) We know how to GRAPH a line by 3 points where we decide what to plug in and chug
Usually, we just try 0, 1, 2 first
2) We know how to GRAPH a line by intercepts...we plug in zero for y and x and chug
This works really well when the line is in STANDARD form and the coefficients are factors
of the constant on the other side of the equation.
3) We know how to GRAPH a line by using slope-intercept form...
We isolate y on one side
We read the y intercept (the b -> the constant on the other side)
We graph that value on the y axis
We COUNT to the next point by reading the slope, the coefficient of the x
The slope should be read what happens to the y value (+up or -down) and then what happens to the x value (+ right or - left)
If the slope is not a fraction, make it a fraction by putting the integer over 1

Oh mystery line,
What can you be?
If I could only find you,
y = mx + b

So first I find m
Then I find b.
Now put it all together
And you've found me!
y = mx + b

The rhyme has 3 steps and usually you will have 3 steps or questions to ask yourself:
1) Do I have the slope (m)? If not, find it by using the slope formula
2) Do I have the y intercept (b)? If not, find it by plugging in a point and the slope
3) Don't forget to put it all together in one equation at the end.

THERE ARE 5 CASES THAT YOUR BOOK INCLUDES:

First case:You're given the slope and the y intercept
(easiest case)
m = 3/2 b = -7/5
Just plug in to the generic slope intercept equation: y = 3/2 x - 7/5

Second case: You're given a point and the slope and need to find the intercept (b)
(3, 1) m = 2
Plug in the point and the slope and solve for b
1 = 2(3) + b
1 = 6 + b
b = -5
Now put it altogether with the given slope and the intercept you just found:
y = 2x -5

Third case: You're given a point and and the y intercept and need to find the slope
(3, 1) b = 2
Plug in the point and the y intercept and solve for slope
1 = 3m + 2
-1 = 3m
m = -1/3
Now put it altogether with the given intercept and the slope you just found:
y = -1/3 x + 2

Fourth case: You're given 2 points and need to find the slope and the intercept
(1 , 3) and (-2 , -3)
You need to first find the slope:
m = change in y / change in x = 3 - (-3)/ 1 - (-2) = 6/3 = 2
Now plug the slope in with one of the points and find the intercept b
3 = 2(1) + b
3 = 2 + b
b = 1
Finally, put it all together:
y = 2x + 1

Fifth case: You have a graph of a line and need to determine the equation
Look at the graph and find 2 easy points to use to find the slope (make sure they are integers!)
(If the y intercept is not an integer, then follow fourth case completely)
Put the information together in y = mx + b form


ANOTHER WAY TO FIND THE EQUATION
WITH 1 POINT & SLOPE: ( and it is MY favorite)
POINT SLOPE FORM OF THE EQUATION
You know one point and the slope. This is the same case as the SECOND CASE, but there is a ANOTHER WAY to solve it other using slope intercept form.
Most people use the slope intercept form for all cases.


Point-slope form of a line: You know one point and the slope. Use the following formula:
y - y1 = m (x - x1)
Using the same example from the second case above: (3 , 1) and m = 2
y - 1 = 2 (x - 3)
What you have now is point slope form of the line

If you simplify this, you will get the slope intercept form of the line!
y - 1 = 2 (x - 3)
y - 1 = 2x - 6
y = 2x - 5

If you're trying to link the slope-intercept form to the point slope form of the same line:
The point-slope version eliminates one step from using the slope intercept form.
In the slope-intercept form, you plug in the point and slope, solve for b, and then rewrite the equation using the intercept that you found.
In point-slope form, once you plug in the point and slope, you just simplify and the equation is already done!