Problem 8 - College Algebra

Question

For the following exercises, find the average rate of change of each function on the interval specified for real numbers \(b\) or \(h\) in simplest form. $$ k(x)=4 x-2 \text { on }[3,3+h] $$

Step-by-Step Solution

Answer

The average rate of change is 4.

Step 1: Understand the Formula for Average Rate of Change

The average rate of change of a function \(k(x)\) over an interval \([a, b]\) is given by the formula \(\frac{k(b) - k(a)}{b-a}\). In this problem, \(a = 3\) and \(b = 3 + h\).

Step 2: Find Function Values at Interval Endpoints

Calculate \(k(a)\) and \(k(b)\):- \(k(a) = k(3) = 4(3) - 2 = 12 - 2 = 10\).- \(k(b) = k(3+h) = 4(3+h) - 2 = 12 + 4h - 2 = 10 + 4h\).

Step 3: Substitute into the Formula

Substitute \(k(a)\) and \(k(b)\) into the average rate of change formula:\[\text{Average Rate of Change} = \frac{k(b) - k(a)}{b-a} = \frac{(10 + 4h) - 10}{(3 + h) - 3}\].

Step 4: Simplify the Expression

Simplify the expression:\[\frac{(10 + 4h) - 10}{(3 + h) - 3} = \frac{4h}{h}\].Since \(heq 0\), this simplifies to 4.

Key Concepts

Linear Functions

Linear functions are mathematical equations that form straight lines when plotted on a graph. They take the general form of \(f(x) = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept.
In a linear function, the rate of change is constant. This means that for every unit increase in \(x\), \(f(x)\) changes by \(m\).

• The slope \(m\) indicates how steep or flat the line is.
• The y-intercept \(b\) is where the line crosses the y-axis.

Understanding how linear functions work is key to mastering the concept of the average rate of change, as it provides a real-world application of these equations.

Interval Notation

Interval notation is a way of writing subsets of the real number line. It is used to specify particular segments where a function's value is being evaluated. This helps in understanding the domain and range of a function over specific bounds.
For instance, the interval \([3, 3+h]\) includes all numbers from 3 to \(3+h\), inclusive and dependant on \(h\).

• Square brackets \([\ ,\ ]\) are used when the endpoint is included (closed interval).
• Round brackets \((\ ,\ ))\) exclude the endpoint (open interval).

In our problem, the interval notation \([3, 3+h]\) lets us explore the changes in function behavior from the point \(3\) to \(3+h\), capturing changes as \(h\) varies.

Simplifying Expressions

Simplifying expressions involves reducing them to a more concise form while retaining their value. This process often includes

• Combining like terms
• Eliminating unnecessary parentheses
• Reducing fractions

In the exercise, simplifying was necessary after substituting values into the average rate of change formula. For example, \(\frac{(10 + 4h) - 10}{(3 + h) - 3}\) simplifies to \(\frac{4h}{h}\). This step is crucial as it reveals the cleanest version of the average rate of change, making it easier to interpret.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to make them easier to work with. In solving for the average rate of change, algebraic manipulation was employed to simplify the expression \(\frac{4h}{h}\) to its simplest form.
This practice involves techniques such as:

• Cancelling common factors
• Isolating variables
• Performing operations like addition, subtraction, multiplication, and division

Understanding algebraic manipulation is key because it aids in breaking down complex problems into manageable, solvable pieces. In this case, simplifying \(\frac{4h}{h}\) when \(h eq 0\) efficiently leads us to the result of 4.