Problem 33 - Algebra and Trigonometry

Question

Functions with Constant Rate of Change Are Linear If the function \(f\) has the same average rate of change \(c\) between any two points, then for the points \(a\) and \(x\) we have $$ c=\frac{f(x)-f(a)}{x-a} $$ Rearrange this expression to show that $$ f(x)=c x+(f(a)-c a) $$ and conclude that \(f\) is a linear function.

Step-by-Step Solution

Answer

Rearranging confirms that \(f(x)\) has the form of a linear function.

Step 1: Start with Given Expression

Starting with the given formula for the average rate of change: \[ c = \frac{f(x) - f(a)}{x - a} \] where \(c\) is the same constant rate of change for any two points.

Step 2: Cross Multiply to Clear the Fraction

Cross multiply to eliminate the fraction: \[ c(x - a) = f(x) - f(a) \] This simplifies to: \[ cx - ca = f(x) - f(a) \]

Step 3: Isolate \(f(x)\)

Add \(f(a)\) to both sides of the equation to isolate \(f(x)\): \[ f(x) = cx - ca + f(a) \]

Step 4: Simplify Expression

Simplify the expression: \[ f(x) = cx + (f(a) - ca) \] This shows the structure of a linear function with slope \(c\) and y-intercept \((f(a) - ca)\).

Step 5: Conclude with Linear Function Form

The expression \(f(x) = cx + (f(a) - ca)\) confirms that \(f(x)\) is a linear function because it is in the form \(y = mx + b\), where \(m = c\) and \(b = f(a) - ca\).

Key Concepts

Average Rate of Change

The average rate of change of a function gives us an idea of how much the function is changing between two points. It's the total change in the function's value divided by the change in the input variable. For any function, let's say we're measuring the rate of change from point \(a\) to point \(x\). We can use this formula: \[ c = \frac{f(x) - f(a)}{x - a} \]
This formula calculates how much \(f(x)\) changes with respect to \(x\). It's like finding the slope of the straight line that connects the two points on the graph of the function. In real-life terms, think of it as how fast you're moving if \(f(x)\) was the distance covered, and \(x\) was the time taken.

• If the average rate of change is positive, the function is increasing over that interval.
• If it's negative, the function is decreasing.
• If the rate is zero, there is no change; the function is constant over that interval.

It's a powerful way to get information about a function's behavior between any two input values.

Constant Rate of Change

When we say a function has a constant rate of change, it means that the average rate of change is the same for any two points on its graph. This is a hallmark feature of linear functions. If the rate \(c\) is the same between any two points, the graph of the function will be a straight line.
Here's what happens with linear functions:

• They maintain a consistent slope throughout. There's no sudden increase or decrease in the steepness of the graph.
• The line goes in one direction endlessly, either upwards, downwards, or remains flat if the rate of change is zero.
• This consistent behavior makes it easy to predict values and establish relationships in real-world situations.

In mathematical terms, we can see this constant rate of change in the equation \(c(x - a) = f(x) - f(a)\), which we derived from the average rate of change. This reinforces the concept that every two points on a linear function share the same slope, \(c\).

Linear Equations

Linear equations are expressions that describe straight lines when graphed. They're the simplest types of functions and can be expressed in the standard form \(y = mx + b\), where:

• \(m\) represents the slope or the rate of change, which tells us how steep the line is.
• \(b\) represents the y-intercept, the point where the line crosses the y-axis.

In our exercise, we found the function \(f(x) = cx + (f(a) - ca)\). This matches the form of a linear equation, confirming that \(f\) is a linear function.
To think about this in real-world terms:- Imagine a line showing the number of books read over a year. If you consistently read two books a month, your reading rate doesn't change, akin to a constant rate of change. - This regularity maps out a straight line if plotted monthly, creating a predictable pattern, just like a linear equation expects in its graph form.
Understanding linear equations is crucial because they simplify complex relationships into easily manageable parts and are foundational to more complex mathematical concepts.