Problem 61 - Calculus

Question

Explain what is wrong with the statement. Values of \(y\) on the graph of \(y=0.5 x-3\) increase more slowly than values of \(y\) on the graph of \(y=0.5-3 x\)

Step-by-Step Solution

Answer

The statement is incorrect since the first function's \(y\) increases and the second's \(y\) decreases due to their slopes.

Step 1: Understand the Functions

First, let's examine the two functions given. The functions are linear equations which can be written in the slope-intercept form: \( y = mx + c \). The first function is \( y = 0.5x - 3 \) and the second one is \( y = 0.5 - 3x \). Each function has a slope (\(m\)) and an intercept (\(c\)).

Step 2: Identify the Slopes

The rate at which \(y\) changes with respect to \(x\) is determined by the slope of the line. The slope of the first function, \( y = 0.5x - 3 \), is \(0.5\). The slope of the second function, \( y = 0.5 - 3x \), is \(-3\).

Step 3: Compare the Slopes

For linear functions, a higher slope means a steeper line, indicating faster increases or decreases in \(y\) per unit increase in \(x\). The slope of the first line is \(0.5\), which means \(y\) increases by \(0.5\) units for every one unit increase in \(x\). For the second line, the slope is \(-3\), meaning \(y\) decreases by \(3\) units for every unit increase in \(x\).

Step 4: Analyze the Statement

The statement claims that \(y\) values on the graph of \(y = 0.5x - 3\) increase more slowly than \(y = 0.5 - 3x\). This is incorrect, as the slope comparison shows that the first line's \(y\) values actually increase while the second line’s \(y\) values decrease. Therefore, it's incorrect to say the first increases more slowly than the second; they change in opposite directions.

Key Concepts

Slope-Intercept Form

Linear functions are often expressed in the slope-intercept form which is written as \( y = mx + c \). In this equation, \( m \) represents the slope of the line, and \( c \) is the y-intercept. The slope indicates how steep the line is and dictates how much \( y \) changes when \( x \) changes by one unit.
For example, in the function \( y = 0.5x - 3 \), the slope \( m \) is \( 0.5 \), and the y-intercept \( c \) is \( -3 \). This means that the line crosses the y-axis at \( -3 \) and for every increase of 1 in \( x \), \( y \) increases by \( 0.5 \).
The slope-intercept form makes it easy to quickly graph a line or understand its basic characteristics. Knowing just the slope and the y-intercept allows you to sketch the entire line, which is a powerful tool in graph analysis.

Slope Comparison

When you compare slopes from linear functions, you're determining how one function's rate of change compares to another's. The slope \( m \) can be positive, negative, or zero. A positive slope means that \( y \) increases as \( x \) increases. A negative slope means the opposite: as \( x \) increases, \( y \) decreases.
Consider these two functions from the exercise:

• \( y = 0.5x - 3 \) has a slope of \( 0.5 \).
• \( y = 0.5 - 3x \) has a slope of \( -3 \).

The positive slope in the first function indicates an upward trend as you move right along the x-axis. In contrast, the negative slope in the second function suggests a downward trend.
When determining which line increases or decreases faster, the absolute value of the slope is very informative. A slope of \( -3 \) (from the second function) has a greater magnitude than \( 0.5 \) (from the first function), indicating a steeper line, but the direction of change is downward for the negative slope.

Graph Analysis

Analyzing a graph of a linear function involves understanding how the slope and y-intercept affect the line's visual representation on a coordinate plane. When examining the graphs of \( y = 0.5x - 3 \) and \( y = 0.5 - 3x \), you should look at how each line trends as \( x \) changes.
The line \( y = 0.5x - 3 \) slopes gently upwards from left to right, indicating a gradual increase in \( y \). In contrast, the line generated by \( y = 0.5 - 3x \) descends steeply, showing a fast decrease in \( y \) as \( x \) increases.
Key components to focus on when analyzing these graphs include:

• The slope, which determines the angle of the line. A steeper angle appears with greater slope magnitudes.
• The y-intercept, showing where the line crosses the y-axis, is critical as it provides a starting point for plotting.

By understanding these elements, you can accurately interpret the behavior of linear functions and avoid misconceptions such as thinking a line with a positive slope increases more slowly than a line with a negative slope.