Activity 6 - Slope of a Line

Activity 6 - Slope of a Line

​Lesson Target​

By the end of this lesson, you will be able to complete the following skills.

• Find the slope of a line passing through given points.

• Recognize and interpret positive, negative and undefined slopes.

• Compare and interpret relative slopes of given lines.

• Identify and distinguish between direct and inverse variation.

Activity 6 - Slope of a Line

​1. Find the Slope of a Line Passing Through Given Points​

​- We have described linear graphs by where the cross the x- and y-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.

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- 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.

Let’s look at an example.

Activity 6 - Slope of a Line

​1. Find the Slope of a Line Passing Through Given Points​

​- 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.

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​Here is a graph where the slope is highlighted.

Activity 6 - Slope of a Line

​1. Find the Slope of a Line Passing Through Given Points​

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.

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​Here is a graph where the slope is highlighted.

Multiple Choice

How do you calculate slope using a graph?

• rise over run
• run over rise
• look at the y intercept
• look at the x intercept

Multiple Choice

What type of slope does the following graph have?

• Postive
• Negative
• Zero Slope
• Undefined

Multiple Choice

Categorize the slope of the line

• Positive

• Negative

• Zero

• Undefined

Multiple Choice

Categorize the slope of the line

• Positive

• Negative

• Zero

• Undefined

Multiple Choice

Categorize the slope of the line

• Positive

• Negative

• Zero

• Undefined

Multiple Choice

Categorize the slope of the line

• Positive

• Negative

• Zero

• Undefined

Multiple Choice

What type of slope does this graph have?

• Positive

• Negative

• Zero

• Undefined

Multiple Choice

Find the slope

• 4/5

• 5/4

• -5/4

• -4/5

Multiple Choice

Find the slope.

• 7/3

• -7/3

• 3/7

• -3/7

Multiple Choice

Find the slope

• 2/3

• -3/2

• 3/2

• -2/3

Multiple Choice

Find the slope of the line.

• -5/4

• 5/4

• -4/5

• 4/5

Activity 6 - Slope of a Line

​1. Find the Slope of a Line Passing Through Given Points​

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.

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Here is the way to find the slope if given 2 O.P's.

Activity 6 - Slope of a Line

​1. Find the Slope of a Line Passing Through Given Points​

​Example:​

- 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 or y1 the key is that you are consistent in your choices. Here is how these values can be substituted into the formula.​

​

Here is the way to find the slope if given 2 O.P's.

Activity 6 - Slope of a Line

​1. Find the Slope of a Line Passing Through Given Points​

****Special Note***

- Note: Sometimes, you will also see the letter “m” used in place of the word “slope”.​

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Here is a simple setup for finding slope:

Multiple Choice

Find the slope of a line that passes through the given points:

(5,1) and (8,3)

• $$\frac{2}{3}$$

• $$\frac{-2}{3}$$

• $$\frac{3}{-2}$$

• $$\frac{3}{2}$$

Multiple Choice

Find the slope of a line that passes through the given points:
(6,3) and (1,4)

• $$\frac{1}{5}$$

• $$\frac{-1}{5}$$

• $$-5$$

• $$1$$

Multiple Choice

Find the slope of a line that passes through the given points:
(2,-2) and (5,7)

• $$\frac{1}{3}$$

• $$\frac{-1}{3}$$

• $$3$$

• $$-3$$

Multiple Choice

Find the slope of a line that passes through the given points:
(1,-6) and (9,-8)

• $$\frac{1}{-4}$$

• $$\frac{-1}{-4}$$

• $$-4$$

• $$4$$

Multiple Choice

Find the slope of the line that passes through:

(-4, 7) and (-6, -4)

• 2/11

• 13/2

• 3/10

• 11/2

Activity 6 - Slope of a Line

II. Calculate the Slopes and y⎯⎯ – Intercepts of Linear Equations in a Variety of Forms, Recognize Slope – Intercept Form as Equivalent to Function Form

• ​For any equation written in the form y=mx+b, m is the slope and b is the y-intercept. For that reason, y=mx+b is called the slope-intercept form. Using the properties of equations, you can write any equation in this form.

• Because we can use slope – intercept form, we can rewrite equations in standard form into slope – intercept form. Then we can easily determine the slope and y – intercept of each equation.

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Activity 6 - Slope of a Line

II. Calculate the Slopes and y⎯⎯ – Intercepts of Linear Equations in a Variety of Forms, Recognize Slope – Intercept Form as Equivalent to Function Form

​

Example:​ Slope intercept form

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Activity 6 - Slope of a Line

II. Calculate the Slopes and y⎯⎯ – Intercepts of Linear Equations in a Variety of Forms, Recognize Slope – Intercept Form as Equivalent to Function Form

​- Example

Write 4x+2y=6 in slope – intercept form. Then determine the slope and the y – intercept by using the equation.

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Activity 6 - Slope of a Line

II. Calculate the Slopes and y⎯⎯ – Intercepts of Linear Equations in a Variety of Forms, Recognize Slope – Intercept Form as Equivalent to Function Form

​- Now we can determine the slope and the y – intercept from the equation.​

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Activity 6 - Slope of a Line

II. Calculate the Slopes and y⎯⎯ – Intercepts of Linear Equations in a Variety of Forms, Recognize Slope – Intercept Form as Equivalent to Function Form

• Think back to our work with functions. Remember how we could write a function in function form? Well take a look at function form compared with slope – intercept form.

Function form =f(x)=2x+1

Slope – Intercept Form =y=2x+1

• Yes! The two are the same. These two equations are equivalent!

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Multiple Choice

What is the letter that represents slope?

• y

• b

• m

• x

Multiple Choice

Which letter represents the y-intercept?

• y

• m

• x

• b

Multiple Choice

What number represents the slope ?

y = -2/3x + 2

• -2/3

• 2

• -2

• -2/3x

Multiple Choice

Which ordered pair represents the Y-Intercept?

y = -2/3x + 2

• (0,2)

• (2,0)

• (0,-2)

• (0, -2/3)

Multiple Choice

re-write the following equation from standard form to slope-intercept form:

$$2x\ -5y\ =15$$  

• $$y\ =\ -\frac{2}{5}x\ -5$$  

• $$y\ =\ \frac{2}{5}x\ -3$$  

• $$y\ =\ \frac{5}{2}x\ -3$$  

• Your Mom!!!!!

Multiple Choice

Re-write the following equation from standard form to slope-intercept form:

$$5x\ +\ 10y\ =\ 20$$  

• $$y\ =\ \frac{1}{2}x\ -2$$  

• $$y\ =\ -\frac{1}{2}x\ +\ 2$$  

• $$y\ =\ 2x\ +\ 2$$  

• $$y\ =\ -2x\ -\ 2$$  

Activity 6 - Slope of a Line

3. Graph Linear Equations Given in a Variety of Forms Using Slope and y⎯⎯ – Intercept

​- Do you see how useful the slope-intercept form, y=mx+b, is to find the slope and y-intercept?

​

- Using this form, graphing is going to be easy, too. Since we know the slope and we know the y-intercept, then instead of using a table of values, we can plot the y-intercept on the coordinate plane and find our next point using the slope.​

Activity 6 - Slope of a Line

3. Graph Linear Equations Given in a Variety of Forms Using Slope and y⎯⎯ – Intercept

​- Do you see how useful the slope-intercept form, y=mx+b, is to find the slope and y-intercept?

​

- Using this form, graphing is going to be easy, too. Since we know the slope and we know the y-intercept, then instead of using a table of values, we can plot the y-intercept on the coordinate plane and find our next point using the slope.​

Activity 6 - Slope of a Line

3. Graph Linear Equations Given in a Variety of Forms Using Slope and y⎯⎯ – Intercept

​- We can also graph lines in a different form. First, we will need to rewrite them into slope – intercept form. Then we can graph the equation.

Activity 6 - Slope of a Line

3. Graph Linear Equations Given in a Variety of Forms Using Slope and y⎯⎯ – Intercept

​- We can also graph lines in a different form. First, we will need to rewrite them into slope – intercept form. Then we can graph the equation.

Activity 6 - Slope of a Line

3. Graph Linear Equations Given in a Variety of Forms Using Slope and y⎯⎯ – Intercept

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Multiple Choice

Which is the graph of y= -x + 2?

• A
• B
• C
• D

Multiple Choice

Graph x + y = 3

• A
• B
• C
• D

Multiple Choice

Graph the following equation:

Multiple Choice

What is the slope in the equation:  y=4x+3

• 3
• 4x
• 4
• x

Multiple Choice

What is the y-intercept in the following equation:  y=-4x-5

• -4
• -4x
• 5
• -5

Multiple Choice

Write the equation for the line.

• y= -1x+3
• y= 4x+3
• y=1/4x+3
• y= -4x+3