**MATH**

Standard Form

This also very important.

**Slope-Intercept Form**

This is very important.

Point-Slope Form

**( y - y ) = m ( x - x )**

Writing Slope-Intercept, Standard, and Point-Slope Equations of a Line

by Ivy Ho

**y = m x + b**

OUTPUT

SLOPE

INPUT

Y-INTERCEPT

**a x + b y = c**

It can be used to find the x- and y- intercepts of a line.

In an actual problem, the a, b, and c would be filled in. To find the x-intercept, you would plug zero in for y, then solve. To find the y-intercept, you would plug zero in for x and solve.

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This is slightly less important, but STILL important.

SLOPE

COORDINATE PAIRS

COORDINATE PAIRS

WRITING FROM DATA

The x and y are the "real" variables of this general equation, so they are left as they are.

m is the slope, as well as the constant rate of change, so you would find the slope of the data and plug that value in for m.

b is the y-intercept, as well as the starting value. In the data given to you, find the coordinate pair where x = 0. The y-value of that will be b. If you cannnot find it, pick any random coordinate pair and plug the x-value in for x, the y-value in for y, and the slope of the line into the general equation. Then, solve for b.

Slope Formula

m =

( y - y )

( x - x )

_______

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WRITING FROM DATA

WRITING FROM A GRAPH

WRITING FROM A GRAPH

WRITING FROM DATA

WRITING FROM A GRAPH

BTW:

m =

rise

run

___

How to Find the SLOPE

Again, the x and y are "real "variables, so don't do anything to them.

To find the slope, pick any two points on the line of a graph and plug them into the slope formula.

For the y-intercept, use the y-value of the coordinate pair at which the line crosses the y-axis. If the graph does not show an intersection of the line with the y-axis, take the fraction form of a point on the line's graph, preferably one close to the y-axis and with an x-value that is a multiple or factor of the x-value of the slope, and the slope of the line in fraction form. Make it so that both fractions have a common denominator. Then, subtract the slope from the random point's fraction. The numerator of the difference is b, the y-intercept of the line.

First, look at the graph and find the x- and y- intercept. Plug the x-value of the x-intercept into ax = c. Then, plug the y-value of the y-intercept into by = c. These equations came from splitting the general equation of standard form in half. Both terms (ax and by) need to equal c by themselves when the other variable term equals zero. Next, input values for a, b, and c that keep both equations true.

If you can't find the x- and y- intercepts, use the method described earlier in Writing Standard Form from Data.

If you secretly (or openly) hate the method above, just write the equation of the line first in slope-intercept or point-slope form, then use inverse operations to re-order the terms into standard form.

From the data, find the x- and y- intercepts. If they are not listed, find them using a point from the data (that would be found in the first quadrant if graphed) and the slope. Put the two in fraction form and if you're finding the y-intercept, change the slope so it has a common denominator with the coordinate point and subtract it from the fraction of the coordinate point. The numerator of the difference is the y-intercept. If you're finding the x-intercept, changed the slope so it has a common numerator with the coordinate point and subtract it from the fraction of the coordinate point. For these methods, do not treat fractions like you usually do. Subtract the denominators. The denominator of the difference is the x-intercept.

Then, plug the x- and y- intercepts into the ax = c and by = c. You can now plug in any values for a, b, and c that make both equations true statements.

Of course, if you hate this method, just write the equation in slope-intercept or point-slope form first. Then use inverse operations to change the equation into standard form.

This is one of the easiest to write.

Just use the slope formula and two points from the data to find the slope and plug that value in for m.

Then, take any coordinate pair from the data table and plug them in for and x and y .

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THAT'S IT.

Now, you can simplify the equation by distributing the slope to x - x and using inverse operations the get y on the other side. This puts it into a slope-intercept form.

OPTIONAL:

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It's easy with a graph too.

First, take two points from anywhere on the line of the graph. Plug them into the slope formula and solve. Once you have the slope, put the value in the equation for m.

Take any coordinate pair on the line of the graph and plug them into the equation for x and y .

You can do whatever you want with it now.

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This graph represents the rate, in meters per minute, at which a fat penguin waddles across the Antarctic.

STANDARD

SLOPE-INTERCEPT

WRITE AN EQUATION FOR THIS GRAPH IN EACH OF THE THREE FORMS.

POINT-SLOPE

**EXAMP E QUESTION: GRAPHS**

STANDARD

POINT-SLOPE

STANDARD

SLOPE-INTERCEPT

SLOPE-INTERCEPT

This data table represents the speed, in words per minute, at which a person types.

SLOPE-INTERCEPT

WRITE EQUATIONS FOR THIS DATA TABLE IN EACH OF THE THREE FORMS.

**EXAMPLE QUESTION: DATA**

X

Y

Per Minute

# of Words

1

2

3

4

5

72

216

144

288

360

DONE.

So, you have this general equation:

y = m x + b

All you have to fill out are this and this.

POINT-SLOPE

SLOPE-INTERCEPT

SLOPE-INTERCEPT

STANDARD

STANDARD

STANDARD

POINT-SLOPE

POINT-SLOPE

POINT-SLOPE

m = --------

5 - 4

3 - 0

Find the slope by plugging two coordinate pairs into the slope formula.

The slope is 3.

b is -12.

If the graph had not supplied you with this information, you could have filled in the rest of the equation and solved that to get b.

Like This:

3 = 3 (5) + b

The y-intercept is (0, -12).

The x-intercept is (4, 0).

a (4) + b (-12) = c

a x = c

6 (4) = 24

When 4 is x, y is 0, taking +by out of the equation.

b y = c

-2 (-12) = 24

When y is -12, x is zero, taking ac out of the equation.

Now, put it all together.

6 x - 2 y = 24

A standard form equation.

First, let's find the slope using the slope formula.

m = -----

0 + 12

4 - 0

Using the x- and y- intercepts is easier.

Plug in a coordinate pair now.

(8, 12)

(y - 12) = 3 (x - 8)

Point-Slope Form.

You can change it into slope-intercept if you want.

(y - 12) = 3 (x - 8)

y - 12 = 3x - 24

+12 +12

y = 3x - 12

In the general equation, fill out this and this.

y = m x + b

Let's find the slope.

m = -----

144 - 72

2 - 1

The slope is 72.

To find the y-intercept, look at the table for the y-value at which x is 0.

If those coordinates are not present in the data table, just take any coordinate pair, plug it and the slope into the general equation, and solve for b.

216 = 72 (3) + b

216 = 216 + b

-216 -216

0 = b

Look for the intercepts in the data table.

The x-intercept is not listed.

The y-intercept is not there. But, we can take it from when we were making a slope-intercept equation. The y-intercept (0, 0), the origin.

To find it, you could use the alternate method described in previous slides, but, here, I'll use a simpler method. Use the slope intercept form of this equation and plug in 0 for y.

0 = 72x

y = 72x + 0

0 = x

Of course though, when a linear graph crosses the origin, (0, 0) is both the y- and x- intercept.

a (0) + b (0) = c

Obviously, you can't use the ax=c, by=c method because everything will just equal zero and it wlill make no sense.

THAT'S WHEN YOU GIVE UP.

...and convert the slope-intercept equation you just made into standard form.

y = 72x + 0

-72x -72x

-72x + y = 0

So much easier.

Finding the slope.

m = -----

5 - 4

360 - 288

The slope is 72, again.

Take any pair of coordinate points and plug them in for x and y .

(2, 144)

( y - 144) = 72 (x - 2)

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Almost done.

(y - 144) = 72 (x - 2)

Now, you can convert it.

OPTIONAL

y - 144 = 72x - 144

+144 +144

y = 72x + 0

or y = 72x

Green

June 1, 2014

A very common form of expressing straight lines

Used often for easier graphing

Graphing calculators usually use this form

INPUT

OUTPUT

PARAMETERS