Question

Find the average rate of change of \(f(x)=3 x-5\) from \(x=0\) to \(x=4\).

Short answer

The average rate of change is 3.

Step-by-step solution

- Understand the formula for average rate of change

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

- Substitute the given points into the formula

In this exercise, \( f(x)=3x-5 \), \( a = 0 \), and \( b = 4 \). We need to find \( f(0) \) and \( f(4) \): \[ f(0) = 3(0) - 5 = -5 \] \[ f(4) = 3(4) - 5 = 12 - 5 = 7 \]

- Calculate the difference in function values

Now, find the difference \( f(4) - f(0) \): \[ f(4) - f(0) = 7 - (-5) = 7 + 5 = 12 \]

- Substitute into the average rate of change formula

Using the formula from Step 1 and our computed values: \[ \text{Average Rate of Change} = \frac{f(4) - f(0)}{4 - 0} = \frac{12}{4} = 3 \]

Key concepts

linear functions

Linear functions are a foundational concept in algebra and calculus. A linear function is one of the form \( f(x) = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. The graph of a linear function is always a straight line.

The function given in the exercise, \( f(x) = 3x - 5 \), is a perfect example of a linear function. Here, \( m = 3 \) and \( b = -5 \).

Understanding linear functions is crucial because they are simple yet powerful tools for modeling real-world situations. They help you understand rates of change and relationships between variables. This exercise specifically asks for the average rate of change, which is synonymous with the slope in a linear function.

algebraic formulas

Algebraic formulas allow us to solve problems methodically. The average rate of change formula is an essential tool in algebra and calculus.

The formula \( \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \) helps determine how one quantity changes in relation to another. In the context of linear functions, this is equivalent to finding the slope.

Breaking down the problem using algebraic formulas helps simplify complex tasks. Here are the steps we used in this exercise:

• Identify the points \( a \) and \( b \).
• Calculate \( f(a) \) and \( f(b) \).
• Find the difference \( f(b) - f(a) \).
• Divide the result by \( b - a \).

This step-by-step approach helps ensure accuracy and clarity, especially when dealing with more complex functions.

calculus fundamentals

Calculus fundamentally revolves around the concepts of change and motion. One of its core notions is the rate of change, which is precisely what this exercise is about.

The average rate of change gives you a broad view of how a function behaves between two points. For our function \( f(x) = 3x - 5 \), the average rate of change between \( x = 0 \) and \( x = 4 \) tells us how the function changes over that interval.

To break this down further, we calculate:

• \( f(0) = 3(0) - 5 = -5 \)
• \( f(4) = 3(4) - 5 = 7 \)
• Difference: \( f(4) - f(0) = 7 - (-5) = 12 \)
• Average rate of change: \( \frac{12}{4} = 3 \)

This average gives us the constant rate at which \( f(x) \) changes between \( x = 0 \) and \( x = 4 \), which is essential for understanding the general behavior of functions.

Grasping these fundamentals is key to excelling in both algebra and calculus.