Problem 14 - College Algebra Essentials

Question

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\) \(f(x)=6 x\) from \(x_{1}=0\) to \(x_{2}=4\)

Step-by-Step Solution

Answer

The average rate of change of the function \(f(x) = 6x\) from \(x_{1}=0\) to \(x_{2}=4\) is 6.

Step 1: Identify the function and the intervals

The function given is \(f(x) = 6x\), and we are looking for the average rate of change from \(x_{1}=0\) to \(x_{2}=4\).

Step 2: Calculate the y-values

Substitute \(x_{1}\) and \(x_{2}\) into function \(f(x)\) to get the corresponding \(f(x_{1})\) and \(f(x_{2})\). So, \(f(x_{1}) = 6*0 = 0\), \(f(x_{2}) = 6*4 = 24\).

Step 3: Calculate the average rate of change

The formula for the average rate of change is \((f(x_{2})-f(x_{1}))/(x_{2}-x_{1})\). Substitute the above values into this equation, we have, \((24-0)/(4-0) = 24/4 = 6.\)

Key Concepts

Linear Functions

Linear functions are one of the simplest forms of functions in mathematics. A linear function is typically displayed in the format of a straight line when you graph it. The standard form of a linear function is given by the equation \(f(x) = mx + b\). Here:

• \(m\) represents the slope of the line
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

In the context of the given exercise, the linear function is \(f(x) = 6x\). Since there is no constant term (\(b = 0\)), the line passes through the origin.
Linear functions are essential because they model relationships with a constant rate of change. This makes them straightforward to work with and to analyze. The key characteristic of a linear function is its constant slope, which represents how much the function value (or y-value) changes with each unit increase in the x-value.

Calculating Slope

Understanding how to calculate the slope is critical for working with linear functions and understanding their properties. The slope of a line indicates the steepness or incline and is often described as "rise over run."
This can be calculated using the formula:

• \(\text{slope} = \frac{\Delta y}{\Delta x} = \frac{f(x_{2}) - f(x_{1})}{x_{2} - x_{1}}\)

The given exercise provides two points, defined by \(x_{1} = 0\) and \(x_{2} = 4\). By evaluating the linear function \(f(x) = 6x\), we find:

• At \(x_{1} = 0\), \(f(x_{1}) = 6 \times 0 = 0\)
• At \(x_{2} = 4\), \(f(x_{2}) = 6 \times 4 = 24\)

Plug these values into the slope formula:

• \(\text{Average rate of change} = \frac{24 - 0}{4 - 0} = \frac{24}{4} = 6\)

Thus, the slope or average rate of change is equal to 6, indicating that for every increase of 1 in \(x\), \(f(x)\) increases by 6.

Intervals in Functions

Intervals are sections of the domain or input values over which a function is analyzed. They are crucial in determining how a function behaves over specific ranges.
In this exercise, the interval is specified from \(x_{1} = 0\) to \(x_{2} = 4\). To find the average rate of change over this interval, we observe how the function values change from one endpoint to the other. This involves calculating how much the output (y-values) of the function changes as the input (x-values) changes within this defined interval.
Analyzing functions over intervals helps determine several important characteristics like:

• The rate of change or slope
• Intervals of increase or decrease
• Potential transformations of the function's graph

Understanding intervals enables students to observe how functions behave across different sections of their domains, making it easier to apply these concepts to various real-world applications.