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Chapter 7: Problem 12

Determine whether each equation represents direct, inverse, joint, or combined variation. $$ y=2 x^{3} $$

Short Answer

Expert verified
Direct Variation.

Step by step solution

01

Understand the Types of Variation

There are four main types of variation: direct, inverse, joint, and combined. Direct variation is when one variable is a constant multiple of another (i.e., y = kx). Inverse variation is when one variable is a constant multiple of the reciprocal of another (i.e., y = k/x). Joint variation is when a variable varies directly as the product of two or more other variables (i.e., y = kxz). Combined variation involves both direct and inverse variations in the same equation.
02

Analyze the Given Equation

Look at the given equation: \( y = 2x^3 \). Notice that the equation relates \( y \) and \( x \) directly with a constant 2 and an exponent 3.
03

Determine the Type of Variation

Since the equation \( y = 2x^3 \) shows \( y \) directly proportional to \( x^3 \), it is a form of direct variation where the constant of proportionality is 2 and the exponent is 3.
04

Conclusion

Therefore, the equation \( y = 2x^3 \) represents direct variation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Direct Variation
Direct variation occurs when one variable is a constant multiple of another, represented by the equation \( y = kx \), where \( k \) is the constant of proportionality. In simpler terms, if you double one variable, the other variable doubles too.
For example:
  • The equation \( y = 5x \) shows direct variation, where \( k = 5 \).
  • If \( y = 2x^3 \), it still represents direct variation because \( y \) is directly proportional to \( x^3 \). In this case, the constant of proportionality is 2, and the relationship involves an exponent.
These equations illustrate how one variable directly changes with the other.
Inverse Variation
Inverse variation happens when one variable is a constant multiple of the reciprocal of another, expressed as \( y = \frac{k}{x} \), where \( k \) is the constant of proportionality. In this type of variation, if one variable increases, the other variable decreases.
For example:
  • In the equation \( y = \frac{10}{x} \), if \( x \) doubles, \( y \) is halved, demonstrating inverse variation.
  • Another example is \( y = \frac{3}{x^2} \); as \( x \) increases, \( y \) decreases, but the relationship involves \( x^2 \).
These equations show how one variable inversely changes in relation to the other.
Joint Variation
Joint variation is when a variable varies directly as the product of two or more variables. The general form is \( y = kxz \), where \( k \) is a constant of proportionality. Here, if either \( x \) or \( z \) increases, \( y \) increases.
Examples include:
  • In the equation \( y = 2xz \), if \( x \) or \( z \) doubles, then \( y \) also doubles.
  • Another example is \( y = kx^2z \), where \( y \) changes jointly with \( x^2 \) and \( z \).
These equations illustrate how \( y \) depends on the products of two or more other variables.
Combined Variation
Combined variation involves both direct and inverse variations in the same equation. This means a variable depends on multiple factors that vary in different ways. An example form is \( y = \frac{kx}{z} \), where \( k \) is a constant.
Examples include:
  • The equation \( y = \frac{4x}{z} \) shows that \( y \) varies directly with \( x \) and inversely with \( z \).
  • Another example is \( y = \frac{kx^2}{w} \, where \) y varies directly with \( x^2 \) and inversely with \( w \).
These examples help to understand how multiple types of variation can be combined in one equation.

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Most popular questions from this chapter

Determine whether each equation represents direct, inverse, joint, or combined variation. $$ y=\frac{4 x}{w z} $$

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