Problem 68 - Calculus

Question

If \(y=f(x)\) is a linear function, then increasing \(x\) by 2 units adds \(m+2\) units to the corresponding \(y,\) where \(m\) is the slope.

Step-by-Step Solution

Answer

The slope \( m \) is 2.

Step 1: Understand the Definition of a Linear Function

A linear function can be expressed in the form \( y = f(x) = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept. The function graph is a straight line. The slope \( m \) indicates the rate of change of \( y \) with respect to \( x \).

Step 2: Analyze the Effect on \( y \) when \( x \) is Increased by 2

When the input \( x \) is increased by 2, from \( x \) to \( x + 2 \), the output \( y \) changes from \( y = mx + b \) to \( y = m(x + 2) + b \).

Step 3: Calculate the Change in \( y \)

Find the difference between the new value \( y = m(x + 2) + b \) and the original value \( y = mx + b \):\(\Delta y = m(x + 2) + b - (mx + b) = mx + 2m + b - mx - b = 2m.\)

Step 4: Relate Change in \( y \) to Given Information

According to the problem, an increase of 2 in \( x \) results in an increase of \( m + 2 \) in \( y \), while calculation shows it is \( 2m \). Equating these two expressions:\[ 2m = m + 2 \]

Step 5: Solve for \( m \)

From the equation \( 2m = m + 2 \), solve for \( m \):\[ 2m - m = 2 \]\[ m = 2 \]

Key Concepts

Slope

The slope is a key concept when dealing with linear functions. It's a measure that provides a lot of information about the behavior of the line. Given a linear function in the form \( y = mx + b \), the slope \( m \) defines how steep the line is. It tells us how much \( y \) will change as \( x \) increases.

In essence, the slope represents:

• The "rise," or vertical change, over the "run," or horizontal change, between two points on the line
• Slope \( m \) of 2, for example, means that for every 1 unit increase in \( x \), \( y \) increases by 2 units

Understanding the slope helps one predict how changes in \( x \) will affect \( y \). If the slope is positive, \( y \) increases as \( x \) increases. If it is negative, \( y \) decreases as \( x \) increases. A slope of zero indicates a flat line, where \( y \) remains constant regardless of \( x \).

Rate of Change

The rate of change in a linear function serves a similar purpose to the slope but is focused more on the concept of change itself. It describes how quickly \( y \) changes in relation to \( x \). In our linear function, where \( y = mx + b \), this rate of change is constant and equal to the slope \( m \).

When we talk about the rate of change:

• It can be a real-world description of how something in a system increases or decreases over time
• In financial terms, it might express how a stock price changes with trading time
• In physics, it might describe how speed changes over distance

This consistent rate means the linear function graph is always a straight line. For example, in our problem, when \( x \) is increased by 2 units, \( y \) increases by an extra \( 2m = m + 2 \), which confirms \( m = 2 \), showing a consistent change rate.

y-intercept

The y-intercept is another fundamental element of a linear equation. It is represented by \( b \) in the linear equation \( y = mx + b \). The y-intercept is where the line crosses the y-axis. This means that if we set \( x = 0 \), \( y \) will equal \( b \).

Understanding the y-intercept involves:

• Knowing that it is the value of \( y \) when \( x \) is zero
• Providing a starting point for plotting the graph
• Giving insight into the initial value of the dependent variable before any change in \( x \)

In the problem exercise, although the focus is on the change in \( y \) with \( x \), the y-intercept remains crucial because it helps define the overall position of the line on the coordinate plane. It is a fixed value that is not affected by changes in \( x \).