Rate of Change and Slope

Rate of Change and Slope

The rate at which one variable changes in relation to another variable.

Finding Rate of Change in Tables

• Find the difference between the dependent variables

• Find the difference between the independent variables

• Express differences as a ratio

   $\frac{dependent\ variable}{independent\ variable}$  

Open Ended

How do you distinguish between the dependent and independent variables?

Multiple Choice

What is the rate of change in the table?

• 3
• 2
• 2/3
• 3/2

Multiple Choice

Find the rate of change.

• -2 minutes per gallon

• -20 minutes per gallon

• -2 gallons per minute

• -20 gallons per minute

Multiple Choice

Find the rate of change.

• 3

• -3

• 20

• Not a constant slope/Non-Linear

Slope

• The slope of a line is the amount of steepness the line has. The letter 'm' is used to represent slope.

• Slope can be found by dividing the difference in 'y', the dependent variable, by the difference in 'x', the independent variable.

   $m=\frac{\Delta y}{\Delta x}=\frac{y^2-y^1}{x^2-x^1}$  

• Another to determine slope is  $\frac{rise\ }{run}$  

Types of Slope

• Positive slopes rises upwards from left to right. Increasing.

• Negative slopes decline from left to right. Decreasing.

• Zero slopes are horizontal, they neither rise nor decline.

• Undefined slopes are vertical lines.

Multiple Choice

What is the slope in the picture?

• Positive

• Negative

• Zero

• Undefined

Multiple Choice

What direction is an undefined slope?

• horizontal
• vertical
• oblique
• straight

Multiple Choice

What type of slope in the picture?

• Positive
• Negative
• Zero
• Undefined

Determine Slope from Equations

Some equations are written in slope intercept form   $y=mx+b$ , where 'm' represents the slope and 'b' is the y-intercept. 
The y-intercept is where the graph crosses the y-axis.  

The standard form of a linear equation is  $Ax+By=C$  can be rewritten in slope-intercept form.

In the next slides you will identify the dependent and independent variables and write an equation to represent each situation.

Multiple Choice

Scott works on cars. He charges $35 for each car plus $7 per hour. Write an equation that represents this scenario if Kylie's car bill was $63.

• 7x + 35 = 63

• 7x = 63

• 7x + 63 = 35

• 35x + 7 = 63

Multiple Choice

Dona plays a game with her brother. The both start with 75 points. For each round that they win, they receive 30 points. The equation that models each person's score is

y = 30 x + 75, where x is the number of games a person wins. What is the meaning of the slope?

• the number of games won

• the amount of points that a player begins with

• the change in points for each game a person wins

• the number of points needed to win the game

Multiple Choice

Find the slope of the line.

• Undefined

• 0

• 1

• -1

Multiple Choice

When you find the slope of a line, what do you do first?

• Rise Up
• Run Out
• Find the origin
• Measure the length

Multiple Choice

Slope is written as a(n) ________.

• Ordered Pair (x, y)
• Fraction

Multiple Choice

If m=3 and b=6, what is the correct slope intercept equation?

• y=3-6

• y=3x-6

• y=3x+6

• y=6x+3

Multiple Choice

In the equation y=mx+b, what does the m represent?

• slope
• y-intercept
• linear
• point-slope

Multiple Choice

What is the y-intercept of the line with the equation
$$y\ =\ -2x\ +\ 4$$  

• $$-2$$  

• $$2$$  

• $$4$$  

• $$-4$$  

Multiple Choice

Find the slope of the line.

• -5/4

• 5/4

• -4/5

• 4/5

Multiple Choice

What is the slope of the line with the equation
$$y\ =\ -\frac{3}{2}x\ +\ 4$$  

• $$-\frac{3}{2}$$  

• $$\frac{3}{2}$$  

• $$4$$  

• $$-4$$  

Multiple Choice

Find the slope of

$$\left(-1,3\right)\ and\ \left(2,5\right)$$  

• $$\frac{2}{3}$$  

• $$\frac{3}{2}$$  

• $$-\frac{2}{3}$$  

• $$-\frac{3}{2}$$  

Poll

How confident do you feel in your ability to find the rate of change and slope from a table, graph or equation?

• Not confident at all

• A little confident

• Somewhat confident

• Very Confident