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

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?

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

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

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.

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

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

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

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:

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:

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

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.

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:

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

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:

8-1 Direct Variation and Proportion More examples

8-1 Direct Variation and Proportion HOMEWORK: P. 354 1-15 ALL

8-1 Direct Variation and Proportion