Problem 17 - Algebra and Trigonometry

Question

\(17-28\) A function is given. Determine the average rate of change of the function between the given values of the variable. $$ f(x)=3 x-2 ; \quad x=2, x=3 $$

Step-by-Step Solution

Answer

The average rate of change is 3.

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

The average rate of change of a function between two points is calculated using the formula: \( \frac{f(b) - f(a)}{b - a} \), where \( a \) and \( b \) are the given values of \( x \).

Step 2: Identify the given values

In the problem, the given values of \( x \) are \( x = 2 \) and \( x = 3 \). Thus, \( a = 2 \) and \( b = 3 \).

Step 3: Evaluate the function at \( x = 2 \)

Substitute \( x = 2 \) into the function: \[ f(2) = 3(2) - 2 = 6 - 2 = 4 \]

Step 4: Evaluate the function at \( x = 3 \)

Substitute \( x = 3 \) into the function: \[ f(3) = 3(3) - 2 = 9 - 2 = 7 \]

Step 5: Apply the average rate of change formula

Use the values calculated: \[ \frac{f(3) - f(2)}{3 - 2} = \frac{7 - 4}{3 - 2} = \frac{3}{1} = 3 \].

Step 6: Conclude the solution

The average rate of change of the function from \( x = 2 \) to \( x = 3 \) is 3.

Key Concepts

Linear Functions

Linear functions are fundamental concepts in algebra that describe a straight line when graphed on a coordinate plane. A linear function is expressed in the form: \( f(x) = mx + b \) Here, \( m \) represents the slope, and \( b \) is the y-intercept. The slope \( m \) indicates how steep the line is, and the y-intercept \( b \) is the point where the line crosses the y-axis.

• In our function \( f(x) = 3x - 2 \), the slope \( m = 3 \) means the line rises 3 units for every 1 unit increase in \( x \).
• The y-intercept \( b = -2 \) means the line crosses the y-axis at the point (0, -2).

Understanding linear functions helps in predicting how changes in one variable affect another, which is crucial for evaluating the average rate of change.

Evaluating Functions

Evaluating functions involves finding the output value of a function for a particular input value. It is akin to solving for \( y \) in a given function \( f(x) \). To evaluate a function, substitute the given input value into the function and perform the necessary calculations. In our problem, we evaluated the function \( f(x) = 3x - 2 \) at two different points:

• First, for \( x = 2 \): substitute 2 in place of \( x \) to get \( f(2) = 3(2) - 2 = 4 \).
• Next, for \( x = 3 \): substitute 3 in place of \( x \) to get \( f(3) = 3(3) - 2 = 7 \).

By evaluating the function at these specific points, you gain the necessary values to calculate the average rate of change.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation symbols. They are the building blocks of algebra and present various quantities in mathematical statements. Understanding how to manipulate algebraic expressions is key to solving equations and evaluating functions. For our function \( f(x) = 3x - 2 \), substitute different values for \( x \) to calculate the expression's value at those points. This step is crucial for calculating changes between different states, such as finding the average rate of change.

• An algebraic expression like \( 3x - 2 \) incorporates a variable term \( 3x \) and a constant term \(-2\), revealing linearity.
• Understanding how to substitute and simplify algebraic expressions ensures accurate function evaluation.

Practicing the process of substituting and simplifying helps in recognizing patterns and changes in the function's output.