Problem 41 - 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 \(x_{1} \neq x_{2} ?\) $$ f(x)=-\frac{1}{4} x $$

Step-by-Step Solution

Answer

The average rate of change of \(f(x) = -\frac{1}{4}x\) from \(-2\) to \(2\) is \(-\frac{1}{4}\).

Step 1: Understand the Formula

The average rate of change of a function \(f\) over the interval \([x_1, x_2]\) is given by the formula \(\frac{f(x_2) - f(x_1)}{x_2 - x_1}\). This formula is similar to the slope of a line connecting two points \((x_1, f(x_1))\) and \((x_2, f(x_2))\) on the graph of \(f\).

Step 2: Determine \(f(x_1)\) and \(f(x_2)\)

For the function \(f(x) = -\frac{1}{4}x\), we first need to find \(f(-2)\) and \(f(2)\). Calculating these gives: \(f(-2) = -\frac{1}{4}(-2) = \frac{1}{2}\) and \(f(2) = -\frac{1}{4}(2) = -\frac{1}{2}\).

Step 3: Substitute into the Formula

Replace the values found in Step 2 into the average rate of change formula: \(\frac{f(2) - f(-2)}{2 - (-2)} = \frac{-\frac{1}{2} - \frac{1}{2}}{2 - (-2)}\).

Step 4: Simplify the Expression

Simplify the expression from Step 3: \(\frac{-\frac{1}{2} - \frac{1}{2}}{2 + 2} = \frac{-1}{4} = -\frac{1}{4}\). Therefore, the average rate of change from \(-2\) to \(2\) is \(-\frac{1}{4}\).

Key Concepts

Functions

Functions are fundamental to mathematics, particularly in understanding relationships between variables. A function is a rule that assigns each input exactly one output. Mathematically, this is described as a relationship where each element from a set of inputs, usually called the domain, is associated with one element in a set of outputs, called the range.

In the context of our exercise, the function is a simple linear one: \(f(x) = -\frac{1}{4} x\). This means for any value of \(x\), you multiply it by \(-\frac{1}{4}\) to get the output. Understanding functions is crucial because it helps us predict behavior and analyze change, both essential skills in science and engineering.

Key characteristics of functions include:

• The domain: all the possible input values (in many cases, all real numbers).
• The range: the set of all possible output values.
• Function notation, where \(f(x)\) represents the value of the function at \(x\).

By grasping these elements, you'll better understand the behaviors and dynamics displayed in graphs and calculations.

Slope of a Line

The slope of a line is a measure of how steep the line is. Imagine you're walking along a hill: a steeper hill means a larger slope. In mathematics, it's the ratio of the vertical change to the horizontal change between any two points on a line.

This concept is mirrored in the average rate of change of a function. The average rate of change indicates how much the function's output value changes for a change in the input. Think about this as the slope of the straight line connecting two points on the graph of the function.

Calculating the slope involves a simple formula:

• Find two points on the line, \((x_1, y_1)\) and \((x_2, y_2)\).
• Use the formula: \(\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\).

In our exercise, the function \(-\frac{1}{4}\) also represents the slope of the line, as it is linear. This slope tells us that for each unit increase in \(x\), the function value decreases by \(\frac{1}{4}\). Thus, understanding slopes helps us describe the incline or decline behavior of functions.

Graph of a Function

Visualizing functions graphically provides insight into their behavior. The graph of a function plots its input-output pairs in a coordinate system. Each \(x\)-value in the domain is plotted against the corresponding \(f(x)\)-value in the range.

For linear functions like \(f(x) = -\frac{1}{4} x\), the graph is a straight line. Each point on this line represents a solution to the equation, showing how the function behaves over different inputs. This specific function decreases linearly, which is clear by the negative slope \(-\frac{1}{4}\).

When you draw the graph:

• Start with the y-intercept, where the line crosses the y-axis. For \(f(x) = -\frac{1}{4} x\), the intercept is at the origin \((0,0)\).
• Use the slope to find other points. From any point, move \(1\) unit right (increase \(x\)) and \(-\frac{1}{4}\) unit down to plot the next point.

The graph not only gives a visual of the function's behavior but also helps in estimating solutions and understanding real-world phenomena tied to the function's model.