Problem 66 - Calculus

Question

If \(y=f(x)\) is a linear function, then increasing \(x\) by 1 unit changes the corresponding \(y\) by \(m\) units, where \(m\) is the slope.

Step-by-Step Solution

Answer

Increasing \( x \) by 1 unit changes \( y \) by \( m \), the slope value.

Step 1: Understanding Linear Functions

In a linear function, the equation is of the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This means that for any given change in \( x \), the change in \( y \) is directly proportional to the slope \( m \).

Step 2: Effect of Increasing x by 1

To understand how increasing \( x \) by 1 unit affects \( y \), substitute \( x+1 \) into the linear equation. So, the new equation becomes \( y = m(x+1) + b \).

Step 3: Simplify the New Equation

The equation \( y = m(x+1) + b \) simplifies to \( y = mx + m + b \). Comparing with the original equation \( y = mx + b \), we notice that the additional term is \( m \).

Step 4: Determine the Change in y

The increase in \( y \) when \( x \) is increased by 1 unit is the difference between the new \( y \) value and the original \( y \) value. Thus, the change in \( y \) is \( (mx + m + b) - (mx + b) = m \).

Key Concepts

Slope

The slope of a linear function tells us how steep the line is. It indicates the rate at which the value of the function changes with respect to the independent variable. In the equation of a line, written as \( y = mx + b \), the slope is represented by the coefficient \( m \).
The slope can be understood as:

• The change in \( y \) for a unit change in \( x \).
• "Rise over run" — meaning the vertical change (rise) divided by the horizontal change (run).

A positive slope means the line rises as it moves from left to right. Conversely, a negative slope means the line falls. Understanding slope is crucial in graphing linear equations and predicting how changes in \( x \) affect \( y \).

Equation of a Line

The equation of a line describes the relationship between two variables, typically \( x \) and \( y \), in a linear function. This relationship is expressed in the form \( y = mx + b \), where:

• \( m \) is the slope, determining the line's steepness.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

This form is called slope-intercept form. It allows for easy graphing of the line and simplifies understanding shifts and trends in data. The equation provides a straightforward way to find the output \( y \) for any given input \( x \), making it invaluable in linear modeling.

Change in Variables

In a linear function, changing the value of \( x \) results in a change in \( y \). This concept is crucial to understanding how linear functions behave. If you increase \( x \) by 1, according to the equation \( y = mx + b \), the corresponding increase in \( y \) will be exactly the value of the slope \( m \).
Consider how this plays out:

• Original \( y \) value: \( mx + b \)
• New \( y \) value after incrementing \( x \): \( m(x+1) + b = mx + m + b \)

Comparing these, the change in \( y \) when \( x \) increases by 1 is \( m \). This predictable change simplifies analyzing and forecasting outcomes in linear systems, whether in mathematics, economics, or the sciences.