Modeling Using Variation

How to Solve Direct Variation Problems

1.Write an equation that models the given English statement. 2. Substitute the given pair of values into the equation in step 1 and find the value of k, the constant of variation. 3. Substitute the value of k into the equation in step 1. 4. Use the equation from step 3 to answer the problem's question.

How to Solve Inverse Variation Problems?

1.Write an equation that models the given English statement. 2. Substitute the given pair of values into the equation in step 1 and find the value of k, the constant of variation. 3. Substitute the value of k into the equation in step 1. 4. Use the equation from step 3 to answer the problem's question.

Is the inverse variation equation a rational function?

Yes. y=k/x or f(x)=k/x is a rational function.

Direct variation with powers

y varies directly as the nth power of x if there exists some nonzero constant k such that y=kx^n We also say that y is directly proportional to the nth power of x.

Direct Variation

If a situation is described by an equation in the form y=kx, where k is a nonzero constant, we say that y varies directly as x or y is directly proportional to x. The number k is called the constant of variation or the constant of proportionality.

What is an Inverse Variation?

If a situation is described by an equation in the form: y= k/x where k is a nonzero constant, we say that y varies inversely as x or y is inversely proportional to x. The number k is called the constant of variation. When two quantities vary inversely, one quantity increases as the other decreases and vice versa.

What is Combined Variation?

In combined variation, direct variation, and inverse variation occur at the same time. For example, as the advertising budget A, of a company increases, its monthly sales, S, also increase. Monthly sales vary directly as the advertising budget: S=kA By contrast, as the price of the company's product, P, increases, its monthly sales, S, decrease. Monthly sales vary inversely as the price of the product: S= k/P We can combine these two variation equations into one combined equation.

What is Joint Variation?

Joint variation is a variation in which a variable varies directly as the product of two or more other variables. Thus, the equation y=kxz is read "y varies jointly as x and z."

Is the direct variation equation, y=kx, a special case of the linear function y=mx+b?

Yes. When m = k and b = 0, y = mx + b becomes y = kx. Thus, the slope of a direct variation equation is k, the constant of variation. Because b, the y-intercept, is 0, the graph of a variation equation is a line passing through the origin.