Problem 87 - A Graphical Approach to Precalculus with Limits

Question

For \(k>0,\) if \(y\) varies directly with \(x,\) when \(x\) increases, \(y\) ________. and when \(x\) decreases, \(y\) _________.

Step-by-Step Solution

Answer

Y increases as x increases, and y decreases as x decreases.

Step 1: Understanding Direct Variation

When we say a variable y varies directly with another variable x, it means that y is proportional to x, or can be expressed in the form \( y = kx \), where \( k \) is a positive constant. The direction of change in y depends on the change in x.

Step 2: Analyzing y as x increases

Since we have the equation \( y = kx \), if \( x \) increases and \( k > 0 \), then the product \( kx \) also increases because you are multiplying an increasing x by a positive k. Thus, \( y \) increases when \( x \) increases.

Step 3: Analyzing y as x decreases

In the same equation \( y = kx \), if \( x \) decreases and \( k > 0 \), then the product \( kx \) decreases because decreasing x is multiplied by a positive k, which reduces the overall product. Therefore, \( y \) decreases when \( x \) decreases.

Key Concepts

Proportionality

Proportionality is a fundamental mathematical concept, especially when studying relationships between two variables. In direct variation, when we say that a variable \( y \) is directly proportional to another variable \( x \), it means that as \( x \) changes, \( y \) changes in a specific way. This relationship can be captured using a simple equation: \( y = kx \).
Here, \( k \) is a constant that remains the same for the entire situation, ensuring the relationship between \( y \) and \( x \) stays consistent. The key idea is that for any given pair of \( x \) and \( y \), the ratio \( \frac{y}{x} \) always equals \( k \). This means that both variables will increase or decrease together, as long as \( k \) remains positive.
Keep in mind that this relationship only holds true for direct variation. Other relationships might involve more complex interactions between \( y \) and \( x \). Understanding proportionality helps in predicting how changes in one variable cause changes in another.

Linear Equations

Linear equations form the backbone of understanding direct variation. In this context, the equation \( y = kx \) is a classic example of a linear equation, which represents a straight line when plotted on a graph. This line goes through the origin (0,0), presenting a clear, one-for-one increase between \( y \) and \( x \).
With direct variation, the equation is straightforward. You can predict \( y \) for any value of \( x \), using the positive constant \( k \). This simplicity makes linear equations very powerful in mathematical modeling and predictions.
Here's how linear equations can be visualized:

• As \( x \) increases, \( y \) increases if \( k > 0 \).
• The graph is a straight line through the origin, making it easy to predict values.
• This equation is scalable; no matter how big or small \( x \) gets, the relationship remains linear.

Thus, linear equations in direct variation are a handy tool for understanding how variables behave with respect to each other.

Positive Constant

The concept of a positive constant \( k \) is crucial in direct variation. In the equation \( y = kx \), \( k \) captures the rate of change between \( y \) and \( x \). When we say the constant is positive, we mean \( k > 0 \). This ensures that both \( y \) and \( x \) move in the same direction:

• As \( x \) increases, so does \( y \).
• As \( x \) decreases, \( y \) decreases as well.

The positive constant ensures a "direct" interaction; there's no reversal of direction between the variables. Its positivity guarantees that changes in one lead to proportional changes in the other, maintaining a straightforward and predictable relationship.
This characteristic of the positive constant is what makes direct variation a simple yet powerful mathematical model. Understanding the role of \( k \) allows us to appreciate how direct variation can model real-world scenarios where relationships are clear and linear.