Slope-Intercept Form Basics

Slope-Intercept Form Basics

With graphing line basics

What is a line?

For our purposes we want to think of a line as a bunch of ordered pairs(dots) in a row. If we plot all the ordered pairs and connect the dots it would form a line. That line would have an infinite amount of ordered pairs, but normally we just look at a few. You only need 2 ordered pairs to see what the line will look like. Draw a line connecting the 2 points and continue the line past the points for the whole line.

Notice in the table, we have 4 ordered pairs listed.

Those 4 ordered pairs are on our line along with a bunch more we did not list. Notice also the equation y=2x+3, this is what a linear(line) equation looks like. It contains one Y variable and one X variable. All linear equations will only contain one X and one Y, or just one Y, or just one X, after combining like terms.

Y=MX+B Slope-Intercept Form

This is the most popular way to write a linear equation. Notice the one X and one Y, this represents all the ordered pairs on our line. If we input an X value, you will output one unique Y value(definition of a function) on our line.

The M represents the slope of our line and the B represents the

Y-intercept of our line. Let us examine M and B on the next few slides.

What is an Intercept?

Notice in the picture, the blue line in the coordinate plane. That is our line and it crosses the X axis, this is called the x-intercept and it crosses the Y axis, this is called the y-intercept. Since a line consists of an infinite number of ordered pairs, then the exact point where our line crosses the axises must be an ordered pair. For the x-intercept this ordered pair is (X,0), for the Y intercept this ordered pair is (0,Y).

B is also the y-intercept in y=mx+b

What is Slope?

Slope can be thought of how steep the line is, much like how steep a mountain is. The slope in Y=MX+B is the M. Notice M is the coefficient of X in our equation. Positive slopes(+M) will go upwards and negative slopes(-M) will go downwards. Zero slopes are horizontal lines and vertical lines are not functions, so their slope is undefined. This concludes the various parts of our linear equation Y=MX+B. try some questions...

Multiple Choice

What is the lowest number of ordered pairs you need to draw your line?

• One

• Two

• Zero

• Three

Multiple Choice

What does the B stand for in Y=MX+B?

• x-intercept

• y-intercept

• Slope

• Base

Multiple Choice

What does the M stand for in Y=MX+B

• Slope

• y-intercept

• x-intercept

• multiply

Can you use the x and y intercepts to draw your line?

Yes!!! notice the X-intercept is the ordered pair (-2,0) and the y-intercept is (0,4). You only need two points to make your line. But what about Y=MX+B? well B is the Y-intercept, so B=4. How can you find M? M is the slope and another way to find the slope is the distance between any two points on your line.

What is the slope of this picture of a line?

If the slope is the distance between two points on my line and i know the x-intercept is (-2,0) and the y-intercept is (0,4), all i have to do is count how many spaces away the two points are. But how do i do that? First you want to count how many spaces up or down the two points are away, then you want to count how many spaces left and right the two points are away.

Why am I counting this way?

Remember slope is the M in Y=MX+B and M is a coefficient of X. M can be any real number but we prefer it to be a fraction. As you can see in this picture, the slope is rise(up and down) divided by run(left and right). Rise/Run is a fraction and will tell us the distance between the X and Y coordinates of our line. With Y being in the numerator and X being in the denominator. Let us go back to our line we were working with.

So slope is Y/X...

This is why we counted the distance visually up and down first, we see from our x-intercept to our y-intercept is is a distance of +4 up. Next we counted the distance visually of left to right, we see from our x-intercept to our y-intercept is a distance of +2 right. we can express our Y/X fraction as +4/+2 or simplified as 2. this means m=2. Now we know m=2 and b=4 and we can substitute that into Y=MX+B: Y=2x+4. That is the equation of our line.

What if we did not have a picture.

The slope is still the distance between two points. How we find the distance between two points algebraically is by subtracting one point from the other. Our points were (0,4) and (-2,0). If we subtract the y values 4-0 we get 4. 4 goes in the Y spot of our slope, Y/X or 4/X. if we subtract the x values 0- -2(0+2) we get 2. 2 goes in our X spot of our slope Y/X, Y/2. Both would be 4/2, same as when we counted.

Notice how they found the slope in this picture.

Since the slope is the distance between Y values divided by the distance between X values, in the bottom part of the picture, you see they subtracted in the same order

4- -2(4+2)/3-0=6/3=2. so M=2 here. B=-2 thus you can write the slope-intercept equation as Y=2x-2.

Try some questions on what we learned...

Multiple Choice

What is the Y-intercept?

• (0,4)

• (4,0)

• (0,-6)

• (-6,0)

Multiple Choice

What is the slope?

• -6/4

• 4/6

• 6/4

• 4/-6

Multiple Choice

If the slope was 4/6 and the y-intercept was 4, how would you write the slope-intercept equation?

• Y=4X+(6/4)

• Y=(4/6)X+4

• Y=(6/4)X+4

• Y=4X+(4/6)

Multiple Choice

What is the slope?

• 3/1

• -3/1

• -3/-1

• 3

Multiple Choice

write the slope-intercept form if m=-3 and b=-7

• y=-3x-7

• y=-7x-3

• x=-3y-7

• x=-7y-3

Graphing a line in slope-intercept form.

Earlier we said you can graph a line using X and Y intercepts because you only need two points to connect to form your line. But what if we want to draw a line with Y=MX+B. Well B is the Y-intercept, we can plot that point easy(0,B). M(Y/X) is the distance between two points, so if we start at the y-intercept and move Y units up or down, then X units left or right, we will get to our next point.

Notice the slope is

Negative 3/4.

How do we know if the negative applies to the 3 or the 4? you can just pick one, but not both. So -3/4 or 3/-4. So if we plot our y-intercept point (0,2) and then use the slope to find the next point. Let us use -3/4 as the slope. From (0,2) we subtract 3 from our Y-intercept's y-coordinate then add 4 to the y-intercepts's x-coordinate. (0+4,2-3) = (4,-1). now we have plotted two points and we can draw our line.

How would you write Y=MX+B for this line?

You need two things, the slope and the y-intercept. The y-intercept is easy, because it is where our line crosses the Y-axis. Counting up from 0 we see our line crosses at (0,3), so B=3. From (0,3) the distance to the next point, our slope, is down 4 and right 5. Down 4 means we subtract 4 and right 5 means we add 5. so our slope(M) is -4/5. Now we can write our equation; Y=(-4/5)X+3.

What if our equation does not say Y=MX+B?

You can use what you have already learned to solve for Y. In this picture we see X+15Y=3. To solve for Y, we need to subtract X from both sides to get 15Y=-x+3. now we need to divide both sides by 15. Remember variables with no visible coefficient really have a coefficient of one. The equation is Y=(-1/15)X+3/15 or simplified Y=(-1/15)X+1/5. Try some...

Multiple Choice

What is the correct slope intercept equation for this graph?

• Y=3X+5

• Y=(4/6)X+5

• Y=(-2/3)X+5

• Y=5X+5

Multiple Choice

Which is the correct graph of Y=(-1/3)X - 3

Multiple Choice

Before you would graph this, Change this equation to slope-intercept form,

3X-2Y=5

• Y=(-2/3)X+5

• Y=(2/3)X+5

• Y=(-3/2)X+5

• Y=(3/2)X+5

Multiple Choice

Why did we want to change 3X-2Y=5,

to slope-intercept form before graphing?

• Because the X and Y intercepts

  would be fractions

• Because slope-intercept

  is my favorite form

• Because i need practice

  solving for Y

• Because all functions have to say Y equals