Algebra (Period 1)

Equations and Slope 7-5



Chapter 7-5: SLOPE INTERCEPT FORM



WE HAVE COVERED GRAPHING
 THE FOLLOWING 2 WAYS:


3 random x points using a table (Ch 7-2)


Most pick 0, 1, 2 and just plug and chug to find the y value


This works well when you don't end up with fraction answers!



EXAMPLE: -3y =-2x - 6


x y

0 2

3 4

-3 0
Then we learned a second way to graph:


Graph the 2 special points that are the y and x intercepts (Ch 7-3)


x intercept: where the line intersects the x axis


y intercept: where the line intersects the y axis


If it's in standard form, this way works great if both the x and y coefficients
 are factors of the constant on the other side of the equal sign.
EXAMPLE: 2x - 3y = -6


If x = 0, y = 2 so the y intercept is (0, 2)


If y = 0, x = -3 so the x intercept is (-3, 0)


x y

0 2

-3 0


The x intercept is the one that has the x value (y is 0)


The y intercept is the one that has the y value (x is 0)



TODAY, WE WILL COVER THE MOST USED FORM OF A LINEAR EQUATION:

THE SLOPE INTERCEPT FORM




You must restate the equation to get it into the following format:


y = mx + b


where m = slope and b = y intercept (where the 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!
Look at the example I gave you above:

-3y =-2x - 6
If you tried the x points 0, 1, 2, only 0 gave an integer for y.
All the other y values were fractions.
The slope intercept form provides the explanation on why.
RESTATE -3y =-2x - 6 INTO SLOPE INTERCEPT FORM:
Divide both sides by -3:
y = 2/3 x + 2


Look at the coefficient for x. WHAT X VALUES WILL GIVE YOU INTEGER ANSWERS FOR Y????
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 doing any work!)
Use the counting method for slope on your graph.
You should have counted: UP 2, 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's really easy to graph the line:


1) Graph the intercept on the y axis (that's the +/- constant at the end)


2) "Count" the next point by using the slope or x coefficient as a FRACTION
For the equation y = 2/3 x + 2
) Put a dot at (0, 2)

3) 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 y change and the denominator is the x change!


If it's positive, you're counting up (positive) and to the right (positive)
or you can count down (negative) and to the left (negative) because 2 negatives make a positive.



If it's negative, you're counting down (negative) and to the right (positive)
or you can count up (positive) and to the left (negative) because you would have a positive and negative = negative




If you're given the slope and the y intercept,
you can write the equation of any line!


Just use: y = mx + b

EXAMPLE: m = -2/3 and b = -12


The line would be y = -2/3 x - 12



Restate Standard Form to Slope Intercept Form:


Another example: 3x + 4y = 10 is the STANDARD FORM of a line
(x and y on the same side, x is positive, no fractions, constant alone)

Restate into Slope Intercept Form:


Solve for y

first 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


The slope is the coefficient of the x
 m = -3/4 (so you're sliding down at a little less than a 45 degree angle)


The y intercept is the constant
b = 5/2 (so the line crosses the y axis at 2 1/2)
Notice that the "b" is a fraction.

When this happens, the slope intercept form may not be the best form to graph the line.

You must start at 2 1/2 on the y axis and count down 3, right 4. THAT'S HARD!
Neither 3 nor 4 is a factor of the constant 10 so intercepts will be fractions.

We need the x term to end up with 1/2 so that when we add that to the b of 5/2, we'll get an integer!
So let's make x = 2 because that will cross cancel with the -3/4 slope to halves:
y = (-3/4)(2) + 5/2 = -3/2 + 5/2 = 2/2 = 1
So we found a coordinate that has just integers: (2, 1)


Now graph that point.


Now count the slope from that point (instead of from the y intercept)!