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