5.1 rate of change and slope
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5.1 Rate of Change and SlopeRate of Change: The relationship between two changing quantities
Slope: the ratio of the vertical change (rise) to the horizontal change (run).
Rate of Change =
Change in the dependent variable (y-axis)Change in the independent variable (x-axis)
Slope = Vertical Change (y) = rise Horizontal Change (x) run
Real World:
Rate of Change can be presented in many forms such as: = We can use the concept of change to solve the cable problem by using two sets of given data, for example:
A band practices their march for the parade over time as follows:
Choosing the data from:
Time and Distance 1min 260 ft. 2min 520 ft.
We have the following:
=
=
Choosing the data from:
Time and Distance 1min 260 ft. 3min 780 ft.
We have the following:
=
=
Choosing the data from:
Time and Distance 1min 260 ft. 4min 1040 ft.
We have the following:
=
=
NOTE:
When we get the same slope, no matter what date points we get, we have a CONSTANT rate of change:
YOU TRY IT:Determine whether the following rate of
change is constant in the miles per gallon of a car.
Gallons Miles
1 28
3 84
5 140
7 196
Choosing the data from:
Gallons and Miles 1 g 28 m 3g 84 m
We have the following:
=
=
Choosing the data from:Gallons and Miles 1g 28 m. 5g 140 m.
We have the following:
=
=
THUS: the rate of change is CONSTANT.
Once Again: Real World
Remember: Rate of Change can be presented in many forms:
= We can use the concept of change to solve the cable problem by using two sets of given data:
( x , y )A : Horizontal(x) = 20 Vertical(y) = 30 (20, 30)B : Horizontal(x) = 40 Vertical(y) = 35 (40, 35)
Using the data for A and B and the definition of rate of change we have: ( x , y )
A : Horizontal = 20 Vertical = 30 (20, 30)B : Horizontal = 40 Vertical = 35 (40, 35)
Rate of Change =
Rate of Change =
Rate of Change =
Rate of Change from A to B =
Using the data for B and C and the definition of rate of change we have: ( x , y )
B : Horizontal = 40 Vertical = 35 (40, 35)C : Horizontal = 60 Vertical = 60 (60, 60)
Rate of Change =
Rate of Change =
Rate of Change =
Rate of Change from B to C =
Using the data for C and D and the definition of rate of change we have: ( x , y )
C : Horizontal = 60 Vertical = 60 (60, 60)D : Horizontal = 100 Vertical = 70 (100, 70)
Rate of Change =
Rate of Change =
Rate of Change =
Rate of Change from B to C =
Comparing the slopes of the three:
As we can see right now the pole from B to C is the one with the biggest change of rate(steepest) =
Rate of Change from A to B =
Rate of Change from B to C =
Rate of Change from C to D =
However, we must find all the combination that we can do. Try from A to C, from A to D and from B to C.
Finally:A to B =
Finally we can conclude that the poles with the steepest path are poles B to C with slope of 5/4.
B to C =
C to D = A to C =
A to D = B to D =
Class Work:
Pages: 295-297
Problems: 1, 4, 8, 9,
Remember:
When we get the same slope, no matter what date points we get, we have a CONSTANT rate of change:
When we get the same slope, no matter what date points we get, we have a CONSTANT rate of change:We further use the concept of CONSTANT slope when we are looking at the graph of a line:
We further use the concept of rise/run to find the slope:
Make a right triangleto get from one point to another, that is your slope.
=
SLOPE= ris
e
run
CONSTANT rate of change: due to the fact that a line is has no curves, we use the following formula to find the SLOPE:
A(x1, y1)
B(x2, y2)Slope =
Slope =
Slope = =
y 2-y1
x2-x1
A = (1, -1) B = (2, 1)
YOU TRY: Find the slope of the line:
YOU TRY (solution):
Slope = =
Slope =
Slope =
Slope =
-42
(0,4)
(2,0)
Slope = =
YOU TRY IT:
Provide the slope of the line that passes through the points A(1,3) and B(5,5):
YOU TRY IT: (Solution)Using the given data A(1,3) and B(5,5) and the definition of rate of change we have:
Slope =
Slope =
Slope =
Rate of Change from A to B is =
A( 1 , 3 ) B(5 , 5) (x1, y1) (x2, y2)
YOU TRY: Find the slope of the line:
YOU TRY IT: (Solution)Choosing two points say: A(-5,3) and B(1,5) and the definition of rate of change (slope) we have:
Slope = Slope =
Slope =
Rate of Change (slope) from A to B is =
A( -2 , 3 ) B(1 , 3) (x1, y1) (x2, y2)
YOU TRY: Find the slope of the line:
YOU TRY IT: (Solution)Choosing two points say: A(-1,2) and B(-1,-1) and the definition of rate of change (slope) we have:
Slope =
Slope =
Slope =
We can never divide by Zero thus our slope = UNDEFINED.
A( -1 , 2 ) B(-1 , -1) (x1, y1) (x2, y2)
THEREFORE:
Horizontal ( ) lines have a slope of ZERO
While vertical ( ) lines have an UNDEFINED slope.
VIDEOS: Graphs
https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/slope-and-intercepts/v/slope-and-rate-of-change
Class Work:
Pages: 295-297
Problems: As many as needed to master the concept
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