1. Rate of Change & Slope Essential Question?
How do you find the rate of change (slope)?
8.F.4

2. Common Core Standard:
8.F.4 ─ Use functions to model relationships between quantities.
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

3. Objectives:
To find the slope (rate of change) of a linear function.

4. Curriculum Vocabulary
Rate of Change (tasa de cambio):
The ratio of vertical change (change in the dependent variable, y) to horizontal change (change in independent variable, x) in a function; the rate at which the quantity represented by y increases or decreases with respect to a change in the quantity represented by x.
Initial Value (valor inicial):
The starting value of a function; the first pair of x, y values for which a function is true.

5. Finding the Rates of Change
Step 1:
Identify the independent and dependent variables (input) (output)
Step 2:
Find the 𝒐𝒖𝒕𝒑𝒖𝒕 𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏𝒑𝒖𝒕 𝒄𝒉𝒂𝒏𝒈𝒆 for each jump on the table.
Step 3:
Simplify the fraction

6. Finding the Rates of Change
Find the rate of change for the following table:
Jump 1:
30 2 =15
Jump 2:
45 3 =15
Jump 3:
Input
Output 1 15 3 45 6 90 8
120 30 2 45 3 30 2
Is the rate of change constant?
YES

7. Finding the Rates of Change
Find the rate of change for the following table:
Jump 1:
=36
Jump 2:
13 1 =13
Jump 3:
−5 0.5 =-10
Input
Output
0.5 18
1.5 31 2 26 18
0.5 13 1 -5
0.5
Is the rate of change constant? NO
When the rate of change is NOT constant, we say there is a VARIABLE rate of change.

8. 𝒎= 𝒓𝒊𝒔𝒆 𝒓𝒖𝒏 = 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 = 𝚫𝒚 𝚫𝒙
SLOPE
Slope(pendiente):
The slope of a line is the ratio of the change in y-values (rise) to the corresponding change in x-values (run).
Slope uses the letter m.
𝒎= 𝒓𝒊𝒔𝒆 𝒓𝒖𝒏 = 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 = 𝚫𝒚 𝚫𝒙

9. Undefined Slope (No Slope)
KINDS of SLOPE
There are four kinds of slope.
Positive slope
Negative slope
Zero Slope
Undefined Slope (No Slope)

10. Positive Slope Slope = Rise over Run Positive slope goes uphill
These ratios are always positive.

11. Negative Slope Slope = Rise over Run Negative slope goes downhill
These ratios are always negative.

12. Zero Slope Slope = Rise over Run HORIZONTAL LINE.
These slope are always zero.

13. Undefined Slope (no slope)
Slope = Rise over Run
Undefined slope looks like a cliff (VERTICAL LINE)
These ratios are always have 0 in the denominator which is UNDEFINED.

14. Slope Practice: Identify the type.

15. How to Find Slope
1) Look at a graph and count RISE over RUN.

16. Slope from a Graph Slope - The rate of change that determines the
direction of a line and how steep it is. y x
vertical change
Slope =
horizontal change +2 +3 -6
rise
Slope =
run -4 +3 3 2 =
Slope = m = +2 -6 3
m = = -4 2

17. Find the slope of the line below.x y
Slope =
rise
run +1
Slope = +4 +4
m = 1 4 +1 -2 -2 = 1 4 -8
m = -8

18. Find the slope of the line below.x y
Slope =
rise
run +1 +3
Slope = +3 +1
m = 3

19. Find the slope of the line below.x y +4
Slope =
rise
run +3 +3
Slope = +4
m = 3 4

20. Find the slope of the line below.x y
Slope =
rise
run -1 +2 -1
Slope = +2
m = 1 2

21. Find the slope of the line below.x y
Slope =
rise
run -2 -2
Slope = +3 +3
m = 2 3

22. Find the slope of the line below.x y
Slope =
rise
run -2
Slope = +1
m = -2 -2 +1

23. Identify from a graph Identify the type of slope. Find the slope.

24. Identify from a graph Identify the type of slope. Find the slope.

25. Identify from a graph Identify the type of slope. Find the slope.

26. Identify from a graph Identify the type of slope. Find the slope.

27. Identify from a graph Identify the type of slope. Find the slope.

28. Identify from a graph Identify the type of slope. Find the slope.

29. USING THE SLOPE FORMULA
Step 1:
Identify two points on the line.
Point 1: (2,6)
Point 2: (3,9)
Step 2:
LABEL each point
(x1, y1)
Point 1: (3,9)
(x2, y2)
Step 3:
Write the slope formula
𝒎= 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏
Step 4:
Substitute and solve (SIMPLIFY!)
𝒎= 𝟗−𝟔 𝟑−𝟐 = 𝟑 𝟏 =𝟑

30. Try These

31. Try These Find the slope of the line between the two points:
(6,5) and (3,2)
(-7,5) and (2,-4)
(9,-7) and (-3,-7)
(4,8) and (3,-2)
(11,15) and (11,-2)