Using two ordered pairs and the slope formula to find slope of the line Slope of a Straight Line Stapel, Elizabeth. "Slope of a Straight Line." Purplemath.

Using two ordered pairs and the slope formula to find slope of the

line

Slope of a Straight Line

Stapel, Elizabeth. "Slope of a Straight Line." Purplemath. Available from

    http://www.purplemath.com/modules/slope.htm. Accessed 19 January 2009

Description One of the most

important properties of a straight line is its angle from the horizon or the slope

Sometimes called it’s steepness

Question?Where is

the horizon on a

coordinate graph?

Let’s look at the equation and the graph of a straight line

y = ( 2/3 )x – 4

Or y = 2/3x –

4

Finding SLOPE using the slope

formula

Why the m for slope?

Answer: There are no definite answers by mathematicians. Some believe it comes from the French word ‘monter’, to climb. Most

believe that this is an urban legend. JUST DON’T FORGET THAT m STANDS FOR SLOPE IN OUR FORMULA!

Analyze the formula. It uses x and y. What do we know about x and y? If we pick an x (input) can we get the matching y (output)? We need two ordered pairs (or points) on the line. HOW?

If you are asked to find slope… you could use the slope formula

Question?What could

you do to find two ordered

pairs? (x,y)

Complete the table

x y = 2/3x - 4 y (x,y)

0 ( , )

3 ( , )

Your ordered pairs should be (0,-4) and (3,-

2)

NOTE: When the coefficient is a fraction use 0, the denominator, and the opposite of the denominator as the inputs (domain).

Let’s check the graph

Let’s use those points in the

formula

Δ y = (-4) – (-2)

Δ x = (0) – (3)

x yf(x)

0 -4

3 -2= -2 = 2

-3 3

Our slope is 2/3

Slope as a ratio

The ratio that describes the tilt of a line is the slope of that line Remembering ratios: a comparison of two quantities by division

Ratios can be written three ways 2 to 3 2:3 2/3

We will use the fraction form 2/3

You try it…

x yf(x)

1. y = -2x +3 2. y = 3/5x - 2

x yf(x)

The slopes are -2 and 3/5.

What do you notice about the slope and the equation?

Find the slope of each line.

m= _______ m= _______

Slope-Intercept Form

Look at those equations again.

Each slope that we found (using the slope formula) appears just before the x in the equation.

This will help you understand the slope-intercept form.

1. y = -2x +3 2. y = 3/5x - 2

y = mx + b

Let’s examine this equation. X means inputs. Y means outputs. That’s the T-chart numbers.M means the slope (tilt of the line) and b means y-intercept. We’ll discuss the y-intercept another day.

Let’s look at the graphs of these

lines

1. 2.

1. y = -2x +3 2. y = 3/5x - 2

This relationship is always true: Increasing lines have positive slopes, and decreasing lines have negative

slopes. Always!

This fact can help you check your calculations: if you calculate a slope as being negative, but you can see from the graph

that the line is increasing (so the slope must be

positive), you know you need to re-do your

calculations. Being aware of this connection can

save you points on a test because it will enable you to check your work before

you hand it in.

Special equations, graphs, and slopes

Increasing lines have positive slopes; decreasing lines have negative slopes. With this in mind, consider the following horizontal line:y = 4Its graph is shown to the right.

What’s the slope (or, tilt of the line)?

Is the horizontal line going up; that is, is it an increasing line? No, so its slope won't be positive. Is the horizontal line going down; that is, is it a decreasing line? No, so its slope won't be negative. What number is neither positive nor negative? Zero! So the slope of this horizontal line is zero. Let's do the calculations to confirm this value. Using the points (–3, 4) and (5, 4), the slope is:

This relationship is true for every horizontal line: a slope of zero means the line is horizontal, and a horizontal line means you'll get a slope of zero. (By the way, all horizontal lines are of the form "y = some number", and the equation "y = some number" always graphs as a horizontal line.)

Describe this airplane’s slope (or, tilt with the horizon)?

What’s the slope (or, tilt of the line)?

Now consider the vertical line x = 4: 

 Is the vertical line going up on one end? Well, kind of. Is the vertical line going down on the other end? Well, kind of. Is there any number that is both positive and negative? Nope.   Verdict: vertical lines have NO SLOPE. In particular, the concept of slope simply does not work for vertical lines. The slope doesn't exist! Let's do the calculations. I'll use the points (4, 5) and (4, –3); the slope is:

(We can't divide by zero, which is of course why this slope value is "undefined".)

Describe the slope of these

airplanes (or, tilt with the horizon)?

This relationship is always true: a vertical line will have no slope, and "the slope is

undefined" means that the line is vertical. (By the way, all vertical lines are of the

form "x = some number", and "x = some number" means the line is vertical. Any

time your line involves an undefined slope, the line is vertical, and any time the line is

vertical, you'll end up dividing by zero if you try to compute the slope.)

WARNING

It is very common to confuse these two lines and their slopes, but they are very

different. Just as "horizontal" is not at all the same as

"vertical", so also "zero slope" is not at all the same as "no slope". The number "zero"

exists, so horizontal lines do indeed have a slope. But

vertical lines don't have any slope; "slope" just doesn't

have any meaning for vertical lines. It is very common for tests to contain questions regarding horizontals and

verticals. Don't mix them up!

y =

Slope = ______

x =

Slope = ______

The 4 Types of Slopes 

 Negative Slope  Positive Slope

Undefined Slope/No slope

Zero SlopeReview

Find slope from a graph

1.Locate two or three good points on the line.

2.Write each ordered pair.

3.Use the slope formula to calculate the slope of the line.

4.Check the tilt of the line and the slope that you calculated for any mistakes.

Find slope from a graph

Find slope from a graph

y-intercept and x-intercept

The y-intercept means where a line will cross the y-axis.

What is the y-intercept of this line? The variable b is used to represent this intercept in the slope-intercept form of an equation.y = mx + b The x-intercept is where a line will cross the x-axis. It is not indicated in the slope-intercept form.

Identify the slope (m) and y-intercept (b) of each line

1. y= 5x +32. y = -1/2x – 83. y = 3/2x 4. y = -x + 25. y = 36. x = -3

1. m= 5 b=32. m= -1/2 b=-83. m=3/2 b=0 4. m=-1 b=25. m=0 b=36. m=no slope

b=no intercept