Problem 43 - College Algebra with Modeling and Visualization

Question

Exercises \(39-44\) : Average Rate of Change Find the average rate of change of \(f\) from \(-2\) to \(2 .\) What is the average rate of change of \(f\) from \(x_{1}\) to \(x_{2}\), where \(x_{1} \neq x_{2} ?\) $$ f(x)=4-3 x $$

Step-by-Step Solution

Answer

The average rate of change from \(-2\) to \(2\) is \(-3\) and it's \(-3\) for any \(x_1\) to \(x_2\).

Step 1: Recall the formula for the average rate of change

The average rate of change of a function between two points, \(x_1\) and \(x_2\), is given by the formula:\[\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}\]

Step 2: Identify values and the function

We need to find the average rate of change of the function \(f(x) = 4 - 3x\) from \(x_1 = -2\) to \(x_2 = 2\).

Step 3: Calculate \(f(x_1)\) and \(f(x_2)\)

Substitute \(x_1 = -2\) into the function:\[f(-2) = 4 - 3(-2) = 4 + 6 = 10\]Substitute \(x_2 = 2\) into the function:\[f(2) = 4 - 3(2) = 4 - 6 = -2\]

Step 4: Compute the average rate of change

Use the formula to find the average rate of change:\[\frac{f(2) - f(-2)}{2 - (-2)} = \frac{-2 - 10}{2 + 2} = \frac{-12}{4} = -3\]

Step 5: Answer the general case

For a general \(x_1\) and \(x_2\), where \(x_1 eq x_2\), the average rate of change is:\[\frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{4 - 3x_2 - (4 - 3x_1)}{x_2 - x_1} = \frac{-3x_2 + 3x_1}{x_2 - x_1} = -3\]

Key Concepts

Linear Functions

Linear functions are one of the simplest forms of functions in algebra. They are represented by the equation\[ f(x) = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. The slope \( m \) determines how steep the line is, and the y-intercept \( b \) is where the line crosses the y-axis. The equation \( f(x) = 4 - 3x \) is a perfect example of a linear function with a slope of \(-3\) and a y-intercept of \(4\).

Linear functions have a constant rate of change, which means their output values increase or decrease by the same amount as the input values change. This characteristic makes them easy to work with when calculating the average rate of change, as it remains constant regardless of the points chosen on the line.

Understanding how to work with linear functions is crucial because they model many real-life situations such as speed, cost, or any other scenario where a constant rate applies. Recognizing that a function is linear helps simplify many problems in algebra and calculus.

Function Notation

Function notation is a compact and efficient way to describe functions. Instead of writing out a rule every time, function notation allows us to say \( f(x) \) to indicate the output of a function \( f \) for an input \( x \). It's like a shorthand that makes it easier to perform operations and substitutions.

For example, when we write \( f(x) = 4 - 3x \), we're expressing a relationship between \( x \) and \( f(x) \) where \( f(x) \) is calculated by subtracting \( three times \) any \( x \) value from \( 4 \).

Using function notation, you can substitute any number for \( x \) to find out what \( f(x) \) equals. This is important in exercises like the one we're discussing, where you might need to find \( f(-2) \) or \( f(2) \) to determine the rate of change. When you see function notation, think of it as an instruction to perform the defined operation on whatever numerical value \( x \) represents.

Algebraic Calculations

Algebraic calculations follow specific rules and operations to solve problems using equations and expressions. They often involve substitutions, arithmetic operations like addition, subtraction, multiplication, and division, and rearranging terms to isolate variables.

For finding the average rate of change as in our example, you must substitute values into the given function equation. First, determine \( f(x_1) \) and \( f(x_2) \) by substituting respective \( x_1 \) and \( x_2 \) values into the function \( f(x) = 4 - 3x \).

This involves:

• Substitute \( x_1 = -2 \) into \( f(x) \), resulting in \( f(-2) = 10 \).
• Substitute \( x_2 = 2 \) into \( f(x) \), giving \( f(2) = -2 \).

After substituting, the formula for the average rate of change is applied:

\[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]
Doing these calculations step by step ensures you correctly find the average rate, which in our case is \(-3\). This represents the constant rate of change in the linear function between the points \( x_1 \) and \( x_2 \). Mastery of these calculations underpins much of algebra and analytical geometry.