Problem 67 - Calculus

Question

If \(y\) is a linear function of \(x,\) then the ratio \(y / x\) is constant for all points on the graph at which \(x \neq 0\)

Step-by-Step Solution

Answer

The ratio \( y/x \) is constant only if the function is of the form \( y = mx \) (i.e., \( c = 0 \)).

Step 1: Define a Linear Function

A linear function is typically defined as \[ y = mx + c \]where \( m \) is the slope and \( c \) is the y-intercept.

Step 2: Express the Ratio y/x

For a linear function of the form \( y = mx + c \), express the ratio \( \frac{y}{x} \) as \[ \frac{y}{x} = \frac{mx + c}{x} \]. This simplifies to \[ \frac{y}{x} = m + \frac{c}{x} \].

Step 3: Consider Cases where x ≠ 0

To determine when \( \frac{y}{x} \) is constant, observe the expression \( m + \frac{c}{x} \). For the ratio to remain constant for all \( x eq 0 \), the term \( \frac{c}{x} \) should not change with \( x \), which occurs only if \( c = 0 \).

Step 4: Conclusion on Linearity and Constant Ratio

If \( c = 0 \), then our function becomes \( y = mx \), and \( \frac{y}{x} = m \), which is indeed constant. For a general linear function \( y = mx + c \) with non-zero \( c \), \( \frac{y}{x} \) is not constant as depends on \( x \). Thus, for \( \frac{y}{x} \) to be constant for all points \( x eq 0 \), the linear function must be of the form \( y = mx \).

Key Concepts

Slope

The slope of a linear function is a crucial concept to understand the graph of the function. Imagine the slope as a measure of how steep a line is on a graph. It tells us how much the value of \( y \) changes for every increase of 1 unit in \( x \). The formula for a linear function is given by \( y = mx + c \), where \( m \) represents the slope.

• If \( m \) is positive, the line rises as it moves from left to right.
• If \( m \) is negative, the line falls as it moves from left to right.
• If \( m \) equals zero, the line is completely horizontal, indicating no change in \( y \) as \( x \) changes.

Grasping how the slope works will help you predict and understand the behavior of the linear function in different scenarios.

y-intercept

The y-intercept is another important element in the equation of a linear function \( y = mx + c \). The y-intercept, represented by \( c \), tells us where the line crosses the y-axis.

• This intersection point occurs when \( x = 0 \).
• The value of \( c \) gives the starting position of the line on the y-axis.

Understanding the concept of the y-intercept helps in visualizing the initial starting point and in graphing the linear equation accurately. If \( c = 0 \), the line will pass through the origin (0,0). This means our line equation is simply \( y = mx \), simplifying analysis regarding the line's behavior.

Constant Ratio

When dealing with linear functions, the term "constant ratio" refers to the condition where the ratio of \( y \) to \( x \), expressed as \( \frac{y}{x} \), remains the same across all points on the graph.

• If \( c = 0 \), the function becomes \( y = mx \), making \( \frac{y}{x} = m \) a constant.
• This constant nature implies that the relationship between \( y \) and \( x \) is directly proportional.

The constant ratio characteristic provides a method to predict \( y \) values for any given \( x \) value, as long as \( x eq 0 \). This helps greatly in real-world applications where such linear relationships are observed.

x not equal to zero

In the context of linear functions, the condition \( x eq 0 \) is critical for the analysis of the ratio \( \frac{y}{x} \). When \( x = 0 \), you potentially face undefined expressions in mathematics since you can’t divide by zero.

• Thus, the condition \( x eq 0 \) prevents this problem, ensuring the expression \( \frac{y}{x} = m + \frac{c}{x} \) is well-defined.
• This condition is crucial for verifying if the ratio of \( \frac{y}{x} \) remains constant.

In practical terms, always keeping \( x \) not equal to zero allows the examination and application of linear functions without mathematical conflict, making sure calculations remain valid wherever the function applies.