3.7 Variation and Applications Mon Oct 13 Do Now Solve the inequality

HW Review: p.327 #41-57

Direct Variation If a situation gives rise to a linear function f(x) = kx, or y = kx, where k is a positive constant, we say that y varies directly as x This is direct variation, and the number k is called the variation constant

Ex1 Find the variation constant and an equation of variation in which y varies directly as x, and y = 32 when x = 2

Ex2 The number of centimeters W of water produced from melting snow varies directly as S, the number of centimeters of snow. Meteorologists have found that 150 cm of snow will melt to 16.8 cm of water. To how many centimeters of water will 200 cm of snow melt?

Inverse Variation If a situation gives rise to a function f(x) = k/x, where k is a positive constant, we say that y varies inversely with x. This is called inverse variation, and the number k is called the variation constant

Ex3 Find the variation constant and an equation of variation in which y varies inversely as x, and y = 16 when x = 0.3

Ex4 The time T required to do a job varies inversely as the number of people P who work on the job (assuming all work at the same rate). If it takes 72 hr for 9 people to frame a house, how long will it take 12 people to complete the same job?

Combined Variation More variation examples: Y varies directly as the nth power of x Y varies inversely as the nth power of x Y varies jointly as x and z

Ex5 Find an equation of variation in which y varies directly as the square of x, and y = 12 when x = 2

Ex7 Find an equation in which y varies jointly as x and z and inversely as the square of w, and y = 105 when x = 3, z = 20, and w = 2

Closure What are the different types of variations? What do the key words (direct, inverse, joint) stand for? HW: p #9-37 odds

Review Tues Oct 14 Do Now Find an equation of variation in which y varies directly as x and inversely as z, and y = 4 when x = 12 and z = 15

HW Review: p.337 #9-37 odds ) The intensity I of light from light bulb varies inversely as the square of the distance d from the bulb. Suppose that I is 90 W/m^2 when the distance is 5m. How much farther would it be to a point where the intensity is 40 W/m^2

Quiz Review – 8 questions, 50 pts 3.5 Rational Functions (2) – Finding vertical, horizontal, and oblique asymptotes – Graphing rational functions 3.6 Polynomial and Rational Inequalities (3) – Solving for 0, then using sign tests 3.7 Variation and Applications – Writing the equation (2) – A word problem (1)

3.5 Rational Function Asymptotes – Vertical asymptote: any unique x values where denom = 0 – Horizontal asymptote: If denom power was greater, y = 0 – Horizontal asymptote: If the num and denom powers are equal, y = ratio of lead coefficients – Oblique asymptote: If num power is greater than denom by exactly 1, then y = long division Graph them – Find all asymptotes – Graph all asymptotes – Test each x-region

3.6 Polynomial and Rational Inequalities 1) Rewrite the inequality so 0 is isolated 2) Factor the non-zero side – If a rational inequality, combine with a common denom first 3) Set each factor = 0 – These are your endpoints for each interval 4) Test intervals for +/- 5) If “or equal to”, use brackets for numerator endpoints

3.7 Variation Writing equation: Key words: – Varies: equal sign – Directly: multiply variable with K – Inversely: divide K by the variable – Jointly: multiply all variables with K Q varies jointly as W and the sq root of E, and inversely as R and the cube of T Q = kw(sqrte) / R(T^3)

Closure What do rational functions and variation have in common? Quiz Wed Oct 15