Module 4, Lesson 17 Part 1

Module 4, Lesson 17 Part 1

Slope and the equation y = mx + b

Today's Objectives

- Today we will be able to find the slope of any line using its equation in the form y = mx + b.

- We will know that the slope connecting any two points of a line is always the same as the slope of the line.

The standard we are working towards:

- Use similar triangles to show that we can find slope from any 2 points

- Derive the equation y = mx + b

Remember:

• Slope represents the steepness of a line
  - To write the slope as a ratio, find 2 points on the line and calculate:

   $\frac{\text{change in y}}{\text{change in x}}\ or\ \frac{\text{rise}}{\text{run}}$  

• We read a graph like we read a sentence: left to right

  - If the line goes up, the slope is positive. If the line goes down, the slope is negative

Graph from science:

Lichen & Pollution

• What tells us if the slope is positive or negative?

• Negative because it is going down from left to right!

Finding slope

• From left to right, the line goes up (positive slope)

•  $\frac{\text{change in y}}{\text{change in x}}=\frac{3}{1}$  because to get between the points, we go up 3 and across 1

We can use any two points to find slope

• How many different ways can we calculate the slope of this line?

•  $\frac{rise}{run}=\frac{2}{1}\ or\ \frac{4}{2}$  

• Both of these fractions equal 2

Multiple Choice

What is the slope of this line?

 $\frac{\text{change in y}}{\text{change in x}}\ or\ \frac{\text{rise}}{\text{run}}$  

•  $\frac{3}{1}$  

•  $\frac{1}{3}$  

•  $\frac{3}{2}$  

Slope shows up in the equation of the line!

•  $y=2x$  is the equation of this line

• From last slide: the slope is 2

• Where does the slope show up in the equation?

This line: y = 3x - 1

•  $\frac{rise}{run}=\frac{3}{1}\ or\ \frac{6}{2}$  

• Where does the slope show up in the equation?

This line: y = 3x - 1

• How can we find other examples of points on this graph?

• Pick any x value and plug it into the expression: 3x - 1                        to find the matching y value.

For equations of the form

y = mx + b

The slope of the line is m.

Vocab: m is the number multiplied by x. What is that called?

For equations of the form

y = mx + b

The slope of the line is m.

Vocab: m is the coefficient on x in the equation.

Multiple Choice

What is the slope of y = 4x + 1?

• 1

• 4

•  $\frac{1}{4}$  

Multiple Choice

 $y=\frac{3}{4}x-8$  

What is the slope?

• 8

•  $\frac{3}{4}$  

• -8

•  $-\frac{3}{4}$  

Multiple Choice

What is the slope of y = 7x + 3?

• 7

• 3

• 10

Multiple Choice

What is the slope of y = -2x + 6?

• 6

• 2

• -2

Comparing slope:

y = 2x + 1

y = 4x + 1

• Compare: Which slope will be steeper? How do you know?

• The second equation has the steeper slope because 4 > 2

Poll

What is your favorite way to find slope?

• Use the number m from equations in the form

  y = mx + b

• Calculate from a graph:

   $\frac{\text{rise}}{\text{run}}\ or\ \frac{\text{change in y}}{\text{change in x}}$  

Summary

• The m of y = mx + b is the number that represents the slope of the line. For example, in y = 2x + 4, the slope is 2.

• The line joining any two points of the graph of the linear equation   y = mx + b has slope m.