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Chapter 3: Problem 50

What does it mean if two quantities vary directly?

Short Answer

Expert verified
Direct variation means that two quantities change at the same rate. This means that as one quantity increases, the other quantity also increases at the same rate and as one decreases, the other decreases at the same rate. The mathematical formula for direct variation is \(y = kx\), where k is the constant ratio.

Step by step solution

01

Definition of Direct Variation

In mathematics, direct variation is a relationship between two variables where their ratio is constant. This means, when one variable increases, the other variable also increases at the same rate or when one variable decreases, the other also decreases at the same rate. The formula for direct variation is \(y = kx\), where k is the constant of variation.
02

Example of Direct Variation

As an example, consider the relationship between the cost of a product and its quantity. If a pen costs $2, then two pens would cost $4, three pens would cost $6, and so on. Here we see that as the quantity of pens (x) increases, the total cost (y) also increases. In this case, the constant of variation (k) is 2, which is the cost of one pen. Hence the relationship between cost and quantity can be represented as \(y = 2x\). This fulfills the definition of direct variation, thus we can say that cost and quantity vary directly.

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Most popular questions from this chapter

In Exercises \(21-26,\) use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=-5 x^{4}+7 x^{2}-x+9$$

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=-2 x^{4}+4 x^{3}$$

If you are given the equation of a rational function, how can you tell if the graph has a slant asymptote? If it does how do you find its equation?

In your own words, explain how to solve a variation problem.

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of \(p\) and \(q\) are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has no vertical. horizontal, or slant asymptotes, and no \(x\) -intercepts.
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