Problem 64 - Algebra A Combined Function

Question

Solve. See the Concept Check in this section. Choose the type of variation that each equation represents. a. Direct variation b. Inverse variation c. Joint variation \(y=\frac{0.6}{x}\)

Step-by-Step Solution

Answer

The equation represents an inverse variation.

Step 1: Understand the Types of Variation

Different types of variation describe how two or more variables relate to each other. Direct variation means that as one variable increases, the other also increases by a constant factor: it follows the general form \( y = kx \). Inverse variation means that as one variable increases, the other decreases by a constant factor, following the form \( y = \frac{k}{x} \). Joint variation involves more than two variables and follows a combination of direct and inverse relations.

Step 2: Identify the Equation Form

Look at the given equation \( y = \frac{0.6}{x} \). This equation is of the form \( y = \frac{k}{x} \), where \( k \) is a constant (here, \( k = 0.6 \)). This resembles the general form for inverse variation.

Step 3: Determine the Type of Variation

Since the given equation \( y = \frac{0.6}{x} \) matches the inverse variation form, we conclude that this represents an inverse variation.

Key Concepts

Direct Variation

Direct variation is one of the straightforward types of variation in algebra. It is when the relationship between two variables can be expressed by the equation \( y = kx \). Here, \( k \) represents a constant, and it determines

• how strongly one variable affects the other, and
• in what direction this effect occurs.

Whenever you see that as one variable increases, the other one also increases (or decreases together), it is a sign of direct variation. This relationship maintains the ratio between \( x \) and \( y \) constant, meaning if you divide\( y \) by \( x \), you will always get the same number: \( k \). This can be represented as \( \frac{y}{x} = k \). For example, if the equation is \( y = 2x \), for \( x = 1, 2 \, \text{or} \, 3 \), the \( y \) values would be \( 2, 4, \text{and} \, 6 \) respectively. See how the ratio remains the same: 2.

Inverse Variation

Inverse variation describes a relationship where as one variable increases, the other decreases, and it can be represented with the formula \( y = \frac{k}{x} \). In this case, when the magnitude of \( x \) grows,

• \( y \) becomes smaller, because they are inversely proportional.

Unlike direct variation, the product \( xy \) is constant for all points on the graph, and it equals \( k \). For example, if \( k \) is 3, and one point on the curve is \( (1,3) \), then increasing \( x \) to 3 must decrease \( y \) to 1 to maintain\( xy = 3 \).
In practical terms, think about the scenario with speed and travel time: if the speed (\( x \)) increases, the time you need (\( y \)) decreases. This particular pattern is what you find in equations similar to \( y = \frac{0.6}{x} \), just like in the exercise solution provided, where \( k = 0.6 \), confirming it shows inverse variation.

Joint Variation

As its name implies, joint variation combines two or more types of variations, typically by involving multiple variables in a relationship. It is expressed when a variable is directly proportional to one variable and inversely proportional to another.
For example, the equation \( z = k \frac{x}{y} \) indicates that

• \( z \) increases with \( x \), and
• decreases with \( y \).

Here, \( z \) varies directly with \( x \) and inversely with \( y \). Joint variation is useful to define scenarios in which several factors

• affect a dependent variable,
• often observed in real life applications like physics or economics,

where complex systems are explained by relatively simple equations with multiple factors.“