Rate of Change and Slope

Rate of Change and Slope

Remember the Domain and the Range? We can also refer to these as the dependent and independent variables.

Where the Domain (x) is the Independent Variable

The the Range (y) is ur dependent variable.

Rate of change measures how much one quantity changes with respect to another.

We can calculate rate of change using the formula above.

Consider this table.

Our dependent variable is the total cost of the games. Our independent variable is the number of games we buy.

We can measure the rate of change.

Usinng our formula, we can divide the change in cost by the change in the number of games.

We can chose two values to measure the rate of change between two points

In this example, we are measuring the rate of change between 2 games and 4 games.

Lets Practice.

See the table to the right. This table shows how a tiled surface area changes with the number of tiles.

Find the rate of change and explain its meaning.

We can use our formula.

Change in area of tiles/Change in number of tiles

(96-48)/(6-3)

The rate of change is 16.

This means that 16in² of floor is tiled for every tile that is used.

So, thats easy enough. But...

What is the rate of change is not constant?

This graph shows the amount of spend on pets in recent years.

Find the rates of change for 2003-2007 and again for 2011-2015

How can we calculate the change in sales for these time periods?

2003-2007

We can see by the graph that the sales in 2003 where 32.4 (billion dollars).

In 2007 they where 41.2 (billion dollars)

So the change in sales, divided by the change in time gives us our rate of change.

(41.2-32.4)/(2007-2003)

(41.2-32.4)/(2007-2003)

What is the rate of change?

2.2 Billion dollars.

We can say that over this 4-year period (2003-2007), sales increased by 8.8 billion at a rate of change of 2.2 billion dollars per year.

Now calculate for 2011-2015

In 2011, the spend wad 50.96 billion dollars and by 2015, the spend was 60.28 billion dollars.

What is our equation to calculate rate of change?

(60.28-50.96)/(2015-2011)

Over this period, sales increased by 9.32 billion in total, for a rate of change of 2.33 billion per year.

Explaining the meaning of rate of change

• For 2003-2007, on average, 2.2 billion more was spent on pets each year than the previous.

• For 2011-2015, on average, 2.33 billion more was spent on pets each year than the previous.

The amount of spend begins to increase over time.

The slope is steeper, so we can know just by looking that the rate of change is higher.

Without calculating, which time period shows the greatest rate of change?

2007-2011

Calculate to verify.

Spend increased by 2.44 billion per year.

Linear functions have a constant rate of change.

The rate of change is constant between any pair of points on the graph.

We can determine if a function table demonstrates a linear equation by calculating the rate of change.

For table a, the rate of change is constant.

Therefore, the equation is linear.

For table b, the rate of change is not constant.

Therefore, this equation is non-linear.

Got it? Try these...

Calculate if the tables show linear or non-linear functions.

First, calculate the rate of change between the first two points.

(15-11)/((-2)(-3)) = 4

Then, between another two.

(23-19)/(1-(-1) = 2

This equation is not linear.

The rate of change on this table is constant for all values, therefore the equation is linear.

Got it? Lets solve some

Page 165, Questions 1-5

Finding Slope!

The slope of a nonvertical line is the ratio of the change.

y coordinate is called the rise, because it rises up.

x coordinate is called the run, because you run horizontally.

Ratio between what?

It is the ratio of change between the rise (y coordinate) and run (x coordinate) as you move from one point to another along the graph.

This is called the slope of the line. The slope is how steep the line is.

Why slope?

We use slope to describe the rate of change.

The greater the absolute value of the slope, the steeper the line.

Lets use an example.

This graph shows..

A line that passes though the coordinates of

(-1,3) and (2, -2)

We work out the slope by working out the rate of change of the graph.

So the slope is the line is -5/3

At first glance, you might ask yourself 'Why is the calculation -2-3/2-(-1) if the x coordinates are (-1,3) and the y coordinates are (2,-2)?'

Good question

And you might wanna write this down...

the slope (m) = y₂-y₁ / x₂-x_{1 ...} Meaning the 2nd coordinate of y, minus the first coordinate of y divided by the second coordinate of x minus the first coordinate of x.

the variable m represents the slope

And that will be important later on, but for now, just acknowledge that m = slope

Lets practice.

Find the slope for a line that passes through (-2,0) and (1,5)

(-2,0) and (1,5)

We don't even need to see the graph to be able to interoperate the slope.

Here it is anyway

For this one, find m

Finally... find m

Page 163 and 165

Questions 4a, 4b and questions 6-11.

We can summarize different slopes

We use slope to explain rate of change, so when you say things like 'it is increasing over time'

or

'it is decreasing over time'

or

'it stays the sane, it neither decreases or increases'

You are describing slope!

Positive slope and Negative slope are obvious, and a horizontal line is a slope of zero. So, in one of our previous examples we saw a slope of zero.

Note how this works out to 0/5?

Calculate this...

when m= any value over zero, this is called 'undefined'... and remember how an equation must pass the verticle line test to be a function? This relation is not a function.

P 164

Questions 5a, 5b.

We can find the slope (m) when given the coordinates!

I keep saying slope = m because it does, and thats important later.

Remember

m = slope

m = slope

What if we know m, but are missing coordinates?

For example, the slope of a line is 1/3

and the coordinates are (1,4) and (-5,r)

Remember the slope formula

So we will let (1,4) be (x₁,y₁) and (-5,r) be (x2,y2)

This gives us

 $\frac{1}{3}=\frac{r-4}{-5-1}$  

And then simplify

 

Page 164

Questions 6a and 6b

Page 165

Questions 12 and 13

Now, lets combine our knowledge.

Page 165 and 166, Questions 14-36

If you get stuck, ask! We can refer back to remind ourselves how to calculate things.