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# Slope

## Understand slope as the steepness of a line.

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Finding the Slope of a Line

Do you know the difference between a positive slope and a negative slope? How can you tell whether the slope of a line will be positive or negative by looking at the equation?

To answer these questions, you will need to understand slope, be able to identify slope and know how to identify different types of slopes.

You will be able to answer these questions by the end of the Concept.

### Guidance

We have described linear graphs by where the cross the x\begin{align*}x\end{align*}- and y\begin{align*}y\end{align*}-axes. But, what about the steepness of the line? If you’ve ever been skiing, you know that the most important thing about a ski slope is how steep it is. In mathematics, we use numbers to quantify steepness so that you can describe graphs more precisely, predict its movement, and compare it with others.

When we have a line that has been graphed on the coordinate plane, we can calculate the steepness of the line. In mathematics, we call this steepness the slope of the line. The slope of the line is how steep the line is.

Take a look at this situation.

Now we want to calculate the steepness of this line. We want to calculate the slope. The slope of the line can be calculated by using a ratio. Remember that a ratio compares two quantities. In this case, we are going to compare the rise of the line with the run of the line.

Slope=riserun\begin{align*}Slope=\frac{rise}{run}\end{align*}

Here is a graph where the slope is highlighted.

You can see that the rise is 2 and the run is 1. It is moving up so it is a positive slope. We can write this as the following ratio.

Slope=21=2\begin{align*}Slope=\frac{2}{1}=2 \end{align*}

The slope of this line is 2.

Write this ratio down in your notebook.

Sometimes, you won’t have a graph to look at. We can also calculate the slope of a line when we have been given two sets of ordered pairs. Then we can use a formula to calculate the slope of the line.

Slope=y2y1x2x1\begin{align*}Slope=\frac{y_2-y_1}{x_2-x_1}\end{align*}

Write this formula down in your notebook.

Here is another one.

Calculate the slope of a line that passes through the points (0, -2) and (1, 2).

To start, we substitute the values of these coordinates into our formula. It doesn’t matter which value you use as y2\begin{align*}y_2\end{align*} or y1\begin{align*}y_1\end{align*} the key is that you are consistent in your choices. Here is how these values can be substituted into the formula.

SlopeSlope=2210=4

The slope of this line is 4.

Note: Sometimes, you will also see the letter “m\begin{align*}m\end{align*}” used in place of the word “slope”.

There are different types of slopes too.

Look at this graph of a positive slope.

We can also see what a graph with a negative slope would look like.

Notice that this line goes down from left to right. By looking at it, we can see that it is negative. We can also look at the slope of the line. Look at these arrows to see how this is a negative slope.

We can also have lines with a slope of zero.

Take a look at this graph.

This line doesn’t go up and doesn’t go down. It has a slope of zero.

Now use the slope formula on the line that goes through the points (2, 3) and (2, -3).

x1=2,y1=3,x2=2,y2=3\begin{align*}{x_1}=2,{y_1}=3,{x_2}=2,{y_2}=-3\end{align*}

mmm=y2y1x2x1=3322=60

Slope here is undefined since it has a denominator of zero, m\begin{align*}m\end{align*} is undefined. Graph this line.

This line is vertical!

All vertical lines have an undefined slope.

We’ve seen four types of slopes.

Positive slopes

Rise up and run right

Negative slopes

Rise up and run left

Zero slopes

Horizontal lines

Undefined slopes

Vertical lines

But we have yet to compare their steepness. If you consider the ski slopes, you know that some slopes are steeper than others. In the graph below, you can also see that some lines are steeper, or have a greater slope, than others.

In looking at these lines, we can determine that some lines are steeper than others. In the example above, lines with a greater slope are steeper than lines with a smaller slope.

Slope of 5 is > slope of 12\begin{align*}\frac{1}{2}\end{align*}

As you continue to work with slope, you will see how the steepness of a line can be measured and how to determine the slope by looking at the equation of a line.

Find the slope of each line by using each set of ordered pairs.

#### Example A

(0,3)(1,4)\begin{align*}(0,3)(1,4)\end{align*}

Solution: Slope of 1

#### Example B

(1,2)(3,6)\begin{align*}(-1,2)(-3,6)\end{align*}

Solution: Slope of -2

#### Example C

(4,2)(3,1)\begin{align*}(4,-2)(3,1)\end{align*}

Solution: Slope of -3

Now let's go back to the dilemma at the beginning of the Concept.

When the x-value is positive, the slope of the line will be positive. When the x-value is negative, then the slope will also be negative. When graphed, a positive slope is a line that goes up from left to right. When graphed, a negative slope is a line that goes down from right to left.

### Vocabulary

Slope
the steepness of the line, calculated with the ratio riserun\begin{align*}\frac{rise}{run}\end{align*}.
Ratio
a comparison between two quantities.
Positive Slope
a slope that goes up from the left to the right.
Negative Slope
a slope that goes down from the left to the right.
Zero Slope
the slope of a horizontal line
Undefined Slope
the slope of a vertical line

### Guided Practice

Here is one for you to try on your own.

Now use the slope formula on the line that goes through the points (1, 3) and (-1, -3).

x1=1,y1=3,x2=1,y2=3\begin{align*}{x_1}=1,{y_1}=3,{x_2}=-1,{y_2}=-3\end{align*}

mmm=y2y1x2x1=1313=22

The slope of the line is 1\begin{align*}-1\end{align*}.

### Practice

Directions: Answer true or false for each of the following questions.

1. True or false. The equation of a line is always linear.
2. True or false. A linear equation will be shown as a straight line on a graph.
3. True or false. The x\begin{align*}x\end{align*} – intercept is where the line crosses the y\begin{align*}y\end{align*} axis.
4. True or false. The y\begin{align*}y\end{align*} – intercept is where the line crosses the y\begin{align*}y\end{align*} axis.
5. True or false. A vertical line has an undefined slope.
6. True or false. A horizontal line has a slope of 0.
7. True or false. Slope is the distance that the line travels on the coordinate graph.
8. True or false. Slope is found using a ratio.
9. True or false. You can figure out the slope of a line if you have been given one set of points.
10. True or false. You will need two sets of points that a line passes through to figure out the slope.

Directions: Figure out the slope of a line that passes through each of the following pairs of points.

1. (2, 3) (3, 4)
2. (4, 5) (2, 3)
3. (2, 1) (-1, 3)
4. (3, 1) (4, 3)
5. (5, 7) (3, 6)
6. (3, 0) (4, 1)
7. (6, 4) (2, 7)
8. (2, 0) (0, 1)
9. (6, 1) (1, 6)
10. (4, 4) (5, 0)

### Vocabulary Language: English

Horizontally

Horizontally

Horizontally means written across in rows.
Negative Slope

Negative Slope

A line with a negative slope will slant down from left to right.
Positive Slope

Positive Slope

A line with a positive slope will slant up from left to right.
Ratio

Ratio

A ratio is a comparison of two quantities that can be written in fraction form, with a colon or with the word “to”.
Slope

Slope

Slope is a measure of the steepness of a line. A line can have positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the $y$ over the change in the $x$.” The symbol for slope is $m$
Undefined

Undefined

An expression in mathematics is undefined when it has no meaning. Division by zero is undefined.
Vertically

Vertically

Vertically means written up and down in columns.