Algebra Honors ( Period 4)

Chapter 3-3 Rate of Change & Slope

We’ve already looked at the slope (m) of lines—today we will connect slope to the RATE of the CHANGE of the linear function (the line). the rate of change for a line is a CONSTANT… it is the same value EVERYWHERE on the line

This change, also know as the slope, is found by  finding the rise over the run between ANY 2 points.  rise/run

The rise is the change in y and the run is the change in x.

In a real world example, the rate of change is the UNIT RATE

If you are buying video games that are all the same price on BLACK FRIDAY, two data points might be

# of computer           Total
games                          cost

4                                  $156
6                                  $234

The slope or rate of change  is the  change in y/ the change in x

(234- 156)/ 6-4

78/2

or $39/ video game

Again, as long as the function is linear, or one straight line, it has a constant rate of change, or slope between ANY TWO POINTS

The constant rate of change, or slope, is the rise over the run—or the change in y over the change in x

or
y₂ – y₁/ x₂-x₁ 

Slope = rise/run ( rise over run)
=change in the y values/ change in the x values =
Difference of the y values/ Difference of the x values

Mrs Sobieraj uses “Be y’s first!” Be wise first!  meaning always start with the y vales on top (in the numerator)

TWO WAYS OF CALCULATING on a graph:

       1) Pick 2 points and use the following formula
Difference of the 2 y –values/ Difference of the 2 x-values
The formal is restated with SUBSCRIPTS on the x’s and y’s below: (memorize this) y₂ – y₁/ x₂-x₁ 
The subscripts just differentiate between point one and point two. You get to decide which point is point one or two. I usually try to keep the difference positive, if I can—but often, one of them will be negative and the other will be positive.

EXAMPLE:   ( 3, 6)  and (2, 4)    y₂ – y₁/ x₂-x₁       6-4/3-2 = 2/1 = 2

    2)Count the slope on the GRAPH using rise over run.
From the point (2,4) count the steps UP ( vertically) to (3,6): I get 2 steps
Now count how many steps over to the right (horizontally): 1 step
Rise = 2 and Run = 1 or 2/1 = 2

HORIZONTAL LINES  have only a y intercept (unless it’s the line y = 0 and then that is the x-axis) The equation of a horizontal line is y = b where b is a constant. Notice that there is NO X in the equation. For example y = 4 is a horizontal line parallel to the x-axis where the y value is always 4 What is the x value? All real numbers! Your points could be ( (3, 4) or ( 0, 4) or ( -10, 4)
Notice y is always 4! The constant rate of change  or slope is 0

If you take any 2 points on a horizontal line the y values will always be the same so the change ( or difference) in the numerator = 0.

EXAMPLE  y = 4

Pick any two points Let’s us ( 3,4) and (-10, 4)

(4 - 4)/ (3 - ^-10) becomes ( 4-4)/ 3 + 10 = 0/13 = 0

VERTICAL LINES ( which are NOT functions)  have only an x intercept ( unless it is the line x = 0 and then it is the y-axis) The equation of a vertical line is x = a, where a is a constant. Notice that there is NO Y in this equation.

EXAMPLE: x = 4

This is a vertical line parallel to the y axis 4 steps to the right of it. Pick any two points on this line Let’s use ( 4, -1) and (4, 7)

This time the change in y is -1  - 7 = -8

and the change in x is 4 -4 = 0

BUT -8/0 is UNDEFINED

Make sure you write undefined for the slope!

Finding a Missing Coordinate if you know 3 out of 4 values and the Slope

Say you know the following:
(1,4) and (-5, y) and the slope is given as 1/3

Find the missing y value

Use the slope formula

Change in y/ change in x

(y – 4)/- 5 – 1  and you know that the slope is 1/3

That means

(y – 4)/- 5 – 1   = 1/3

(y – 4)/-6 = 1/3

Solve

3(y -4)= -6

3y – 12 = -6

 y = 2

Or you could have divide both sides by 3 FIRST

y - 4 = -2

y = 2