Question

Related Rates Consider the linear function $$y=a x+b$$ If \(x\) changes at a constant rate, does \(y\) change at a constant rate? If so, does it change at the same rate as \(x ?\) Explain.

Short answer

Yes, \(y\) changes at a constant rate for the given linear function, but it does not necessarily change at the same rate as \(x\). It changes at a rate that is determined by the slope of the line, \(a\).

Step-by-step solution

Understand the function

Given a linear function, \(y = ax + b\). In this linear function, \(a\) is the slope and \(b\) is the y-intercept.

Derive the function

To understand the rate of change, the derivative of the function is required. The derivative of \(y = ax + b\) with respect to \(x\) is \(y' = a\). The derivative, \(y'\), gives the rate of change of \(y\) with respect to \(x\). Here it is constant and equals to the slope \(a\). This means that for each unit increase in \(x\), \(y\) increases by \(a\) units.

Answer the questions

From Step 2, we can conclude the following: the function \(y = ax + b\) does change at a constant rate, and that rate is \(a\). However, \(y\) does not necessarily change at the same rate as \(x\). The rate at which \(y\) changes is proportional to the change in \(x\), and is determined by the slope \(a\). If \(a = 1\), then \(y\) changes at the same rate as \(x\). If \(a > 1\), \(y\) changes faster than \(x\); if \(a < 1\), \(y\) changes slower than \(x\).

Key concepts

Linear Functions

Linear functions are foundational in mathematics and represent a straight line when plotted on a graph. They have the general form \(y = ax + b\), where \(a\) is the slope, and \(b\) is the y-intercept. The equation tells us how \(y\), the dependent variable, changes with \(x\), the independent variable.
In the context of a linear function, every time the value of \(x\) changes by one unit, \(y\) changes by the same amount, consistently dependent on the slope \(a\). It's this uniform behavior that makes linear functions simple and predictable for solving problems.
Understanding a linear function requires recognizing the impact of the slope and the y-intercept. They determine not just the line's angle and steepness but also its starting point on the graph.

Derivative

In calculus, the derivative is a crucial concept used to calculate the rate of change of a function. For a linear function \(y = ax + b\), the derivative with respect to \(x\) is \(y' = a\).
This tells us how \(y\) changes as \(x\) varies, specifically, the rate at which the dependent variable \(y\) changes per unit change in the independent variable \(x\).
Since the derivative of our linear function is simply the slope \(a\), this indicates that the rate of change is constant throughout the function, meaning \(y\) increases or decreases linearly as \(x\) does. This concept is pivotal in understanding how functions behave and is widely applied in different fields such as physics, engineering, and economics.

Rate of Change

The rate of change describes how one quantity changes in relation to another. In the case of the linear function \(y = ax + b\), the rate of change of \(y\) with respect to \(x\) is given by the slope \(a\).
This shows that for every unit increase in \(x\), \(y\) changes by \(a\) units. The reason the rate of change is always \(a\) in linear functions is because of their straight-line nature.
Understanding the rate of change is vital when predicting how modifying one variable will affect another. It’s used in various practical scenarios, like calculating velocity (which is the rate of change of distance with respect to time), economics, and biology.

Slope

The slope of a linear function \(y = ax + b\) is denoted by \(a\). It characterizes the steepness and direction of the line graph of the function. A positive slope means the function is increasing, while a negative slope indicates a decreasing function.
Slope is calculated as the "rise over run," meaning the vertical change divided by the horizontal change between two points on the line.
If the slope \(a\) is greater than 1, \(y\) changes faster than \(x\); if \(a\) is less than 1, \(y\) changes slower than \(x\). Understanding slope aids in comprehending how sensitive \(y\) is to changes in \(x\). For real-world applications, this can help in determining angles of ascent or descent, assessing risk versus reward in investments, and more.