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Chapter 2:Variations

Variations and Graphs

What are some examples of a constant of variation? r=5c, r=10. They are all a form of y=kx^n where k is a nonzero constant and n is a positive number.
Ex: The weight w of an adult animal of a given species is known to vary directly with the cube of its height h. a. Write an equation relating w and h. b. Identify the dependent and independent variables. Solution: An equation for the direct varation is w=kh^3 b. Because w is given in terms of h, The dependent variable is w and the independent variable is h.
What is an inverse-variation function? Its a function with a formula of the form y=k/x^n, with k not equaling 0 and n being greater than 0.
Ex: The number n of oranges you can pack in a box is approximentely inversely proportional to the cube of the average diameter d of oranges. Write an equation to express this relation. Solution: The cube of the diameter of d^3. So, n is = k/d^3
The Fundamental Theorem of Variation a. If y varies directly as x^n( That is, y=kx^n), and x is multiplied by c, then y is multiplied by c^n. b. If y varies inversely as x^n( That is, y=k/x^n), and x is multiplied by a nonzero constant c, then y is divided by c^n.
Formula for Slope of a Line =changes in vertical distance/ change in horizontal distance = change in dependent variable/ change in independent variable =rise/run
Domain and range k>0 The domain of the function with equation y=kx^2 is the set of all real numbers. When K > 0, the range is the set of nonnegative real numbers, and the parabola opens up.
Domain and range k<0 The range is thet set of nonpositive real numbers and the parabola opens down. That is, the vertex of the parabola is its maximum point.
Created by: kmalone30