1. Chapter 8 Variation and Polynomial Equations
Algebra 2
Chapter 8 Variation and Polynomial Equations

2. 8-1 Direct Variation and Proportion
WARMUP
If y=4x and x=7, what is y?
If y/x = 6 and x=2, what is y?
If y=mx and x=12 and y=9, what is m?

3. 8-1 Direct Variation and Proportion
Goal: To solve problems involving direct variation.

4. 8-1 Direct Variation and Proportion
Water pressure at depth for scuba divers.
Depth
x (m)
Pressure
y (kPA)
(kPA/m) or
y/x 3
29.4
9.8 6
58.8 9
88.2 12
117.6

5. 8-1 Direct Variation and Proportion
In the previous chart, for each ordered pair
(x, y), the ratio of pressure to depth is constant:
If we solve for y, we get that y = 9.8x.
Because of this relationship, we say that the pressure varies directly as the depth.

6. 8-1 Direct Variation and Proportion
Definition:
A linear function defined by an equation of the form:
y = mx (m ≠ 0)
is called a direct variation, and we say that y varies directly as x.
The constant m is called the constant of variation.
Notice that

7. 8-1 Direct Variation and Proportion
Look at Example 1.

8. 8-1 Direct Variation and Proportion
What does the graph of any function y=mx look like?

9. 8-1 Direct Variation and Proportion
We can choose two ordered pairs (x1, y1) and (x2, y2) of the variation y = mx, (x1, x2  0). In that case, and
Therefore:

10. 8-1 Direct Variation and Proportion
Any equality of ratios like this is called a proportion. This is why that in direct variation situations, y is often said to be directly proportional to x, and m is called the constant of proportionality. You may see the proportion written:

11. 8-1 Direct Variation and Proportion
We can read this:
as “y1 is to x1 as y2 is to x2”. The numbers x1 and y2 are called the means. And y1 and x2 are called the extremes of the proportion.
means
extremes

12. 8-1 Direct Variation and Proportion
Multiplying both sides of this proportion by x1x2, we get:
So, in any proportion, the product of the extremes equals the product of the means.

13. 8-1 Direct Variation and Proportion
Example 2 from the book:
If y varies directly as x, and y=15 when x=24, find x when y=25.
Two ways to solve this one:

14. 8-1 Direct Variation and Proportion
First find m and write an equation of the direct variation.

15. 8-1 Direct Variation and Proportion
Since “y varies directly as x” means that y is directly proportional to x, a proportion can be used:

16. 8-1 Direct Variation and Proportion
More examples

17. 8-1 Direct Variation and Proportion
HOMEWORK:
P ALL

18. 8-1 Direct Variation and Proportion