4| Step 3: Slope

4| Step 3: Slope

By Maria Leon

What is slope?

​Slope measures the steepness and direction of a line.

Slope measures the steepness of a line. The bigger the slope, the steeper the line is. The smaller the slope, the flatter it gets.

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Slope also tells us the direction of the line - if it goes up, down, or if it's horizontal or vertical.

​Slope's special Variable

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There are 4 different types of slope, depending on the direction of the line. If the line goes up to the right, the slope is positive. If the line goes down to the right, the slope is negative. A horizontal line has a slope of 0. A vertical line has an undefined slope. We'll look at each type in more detail next.

Four types of Slope

If the y-values are increasing as the x-values increase, the line has a positive slope.

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If you trace the line with your finger from left to right (the same order you read a book), the line will go up to the right.

Think of a situation that has a positive rate of change. Suppose you were adding $100 to your bank account each week (that's a positive change!). As time goes by, the amount of money you have in the account is increasing. If you graphed this situation, the line would go up to the right to show that the amount of money is increasing as time goes by.

If the y-values are decreasing, the line has a negative slope.

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If you trace the line with your finger from left to right (the same direction that you read a book), the line will go down to the right.

Think of a situation with a negative rate of change. Suppose instead of putting money into your bank account, you're spending $100 each week (this is a negative change!).

If you graphed this, the line would go down to the right to show that the amount of money you have in your account is decreasing.

If the y-values are not changing as x increases, the line will have a slope of 0. Anytime the line is horizontal (flat from left to right), the slope is zero. This would indicate a situation where there isn't any change.

For example, if you just left your bank account alone and didn't put money in or take money out (and no interest was added). This is an example of a situation with a rate of change equal to zero, nothing is happening! To show that there is 0 amount of change, the graph would be a flat horizontal line.

A vertical line has an undefined slope. In this situation, the y-values are changing, but the x-value always stays the same. If you look at the definition of slope, the amount of horizontal change is in the denominator of a fraction. In math, you can't have a 0 in the denominator. It doesn't make sense to divide by 0 so we say that the slope of a vertical line is undefined. There isn't a slope for these types of lines.

Think about the bank account example. It wouldn't make sense if the graph was a vertical line. That would you mean you had all sorts of different amounts in the bank at the same exact moment, which isn't possible.

Multiple Choice

A line with $$m=0$$   is _______ .

Multiple Choice

What is the slope in the picture?

• Positive

• Negative

• Zero

• Undefined

Multiple Choice

What is the slope in the picture?

• Positive

• Negative

• Zero

• Undefined

Multiple Choice

What type of slope does the following graph have?

• Positive

• Negative

• Zero

• Undefined

Multiple Choice

What type of slope does the following graph have?

• Positive

• Negative

• Zero

• Undefined

Multiple Choice

What type of slope does the following graph have?

• Positive

• Negative

• Zero

• Undefined

Using a Graph to Find Slope

Multiple Choice

What is the rise of the line?

• $$2$$  

• $$3$$  

• $$0$$  

• Undefined

Multiple Choice

What is the run of the line?

• $$2$$  

• $$3$$  

• $$0$$  

• Undefined

Multiple Choice

What is the trend of the line (i.e. increasing, decreasing, horizontal, or vertical)?

Remember: The trend of the line will determine if you should write the slope as positive or negative.

•  increasing

• decreasing

• horizontal

• vertical

Fill in the Blank

What is the slope of the line?

Note: Write your answer as in the example below.

m=-1/2

Multiple Choice

What is the rise of the line?

Note: If you notice that the line is going down instead of up, then count the rise as negative.

Example: -5

• $$3$$  

• $$-3$$  

• $$1$$  

• $$-1$$  

Multiple Choice

What is the run of the line?

• $$3$$   

• $$-3$$  

• $$1$$   

• $$-1$$   

Fill in the Blank

What is the slope of the line?

Note: Always write the slope in simplest form.

Don't forget to follow the same format as in the previous question!

Fill in the Blank

What is the rise of the line?

Note: This time, write your answers down by putting only the number.

Fill in the Blank

What is the run of the line?

Fill in the Blank

What is the slope of the line?

Note: Don't forget that slope questions must always be answered as in the example.

m=-5/4

Fill in the Blank

What is the rise of the line?

Don't forget to watch out for the direction!

Fill in the Blank

What is the run of the line?

Fill in the Blank

What is the slope of the line?

Fill in the Blank

What is the rise of the line?

Fill in the Blank

What is the run of the line?

Fill in the Blank

What is the slope of the line?

Fill in the Blank

What is the rise of the line?

Fill in the Blank

What is the run of the line?

Fill in the Blank

What is the slope of the line?

Fill in the Blank

What is the rise of the line?

Fill in the Blank

What is the run of the line?

Fill in the Blank

What fraction does the rise/run form?

Fill in the Blank

What is the slope of the line?

Given two points, you don't have to count rise over run. You can just plug in the values in the formula:

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Finding Slope with Two Points

The steps can be remembered as LWPC:

1. Label the points.

2. Write down the general formula.

3. Plug in (or substitute) the values in their proper places.

4. Chug out the formula using your integer and algebra skills.

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Take a look at the example below. Notice the color coding!​

Example

Multiple Choice

Here we will label $$\left(-2,-1\right)$$  as Point 1 and $$\left(2,2\right)$$  as Point 2. Thus:

$$x_1=-2,\ y_1=-1$$  

$$x_2=2,\ y_2=2$$  

Please choose the proper formula below. Be very careful.

• $$m=\frac{-2-\left(-1\right)}{2-2}$$

• $$m=\frac{2-\left(-1\right)}{2-\left(-2\right)}$$  

• $$m=\frac{2-\left(-2\right)}{2-\left(-1\right)}$$  

• $$m=\frac{2-2}{-1-\left(-2\right)}$$  

Fill in the Blank

What is the slope of this line?

Don't forget to write it as m=___ .

Fill in the Blank

Substitute values into the slope formula.

Example: m=2+2/3-1

Note: It doesn't matter which point you choose to be Point 1 or Point 2, as long as you stay consistent once you do label. I hope the color coding will help you.

Make sure you have the general formula written on a piece of paper beside you, and LABEL your points first.

Fill in the Blank

What is the slope of this line?

Don't forget to write it as m=___ . And don't forget to simplify the fraction if possible!

Fill in the Blank

Substitute values into the slope formula.

Example: m=2+2/3-1

Note: Refer back to the general formula on your piece of paper, and LABEL the points first on this paper before you begin trying to substitute.

Fill in the Blank

What is the slope of the line?

Fill in the Blank

Substitute values into the slope formula.

Fill in the Blank

What is the slope of the line?

Fill in the Blank

Substitute values into the slope formula.

Fill in the Blank

What is the slope of the line?

Fill in the Blank

Substitute values into the formula.

Fill in the Blank

What is the slope of the line?

Fill in the Blank

Substitute values into the formula for the slope of a line that goes through points $$\left(2,5\right)$$  and $$\left(3,6\right)$$ .

Fill in the Blank

What is the slope of the line that goes through points $$\left(2,5\right)$$  and $$\left(3,6\right)$$ ?

Fill in the Blank

Substitute values into the formula for the slope of a line that goes through points $$\left(-3,2\right)$$  and $$\left(-3,-7\right)$$ .

Fill in the Blank

What is the slope of the line that goes through points $$\left(-3,2\right)$$  and $$\left(-3,-7\right)$$ in fraction form?

Fill in the Blank

What is the final slope of the line that goes through points $$\left(-3,2\right)$$  and $$\left(-3,-7\right)$$ ?

Fill in the Blank

Substitute values into the formula for the slope of a line that goes through points $$\left(-4,5\right)$$  and $$\left(-8,-5\right)$$ .

Fill in the Blank

What is the slope of the line that goes through points $$\left(-4,5\right)$$  and $$\left(-8,-5\right)$$ ?