Problem 65 - Algebra and Trigonometry

Question

A verbal description of a function is given. Find (a) algebraic, (b) numerical, and (c) graphical representations for the function. To evaluate \(f(x),\) divide the input by 3 and add \(\frac{2}{3}\) to the result.

Step-by-Step Solution

Answer

(a) Algebraic: \(f(x) = \frac{x}{3} + \frac{2}{3}\); (b) Numerical: \(\begin{array}{c|c} x & f(x) \\ \hline 0 & \frac{2}{3} \\ 3 & \frac{5}{3} \\ 6 & \frac{8}{3} \end{array}\); (c) Graphical: A line with slope \(\frac{1}{3}\) passing through \((0, \frac{2}{3})\).

Step 1: Writing the Verbal Description as an Algebraic Formula

To convert the verbal description into an algebraic formula, we follow the described operations. First, we divide the input, which is represented by \(x\), by 3. This is written as \(\frac{x}{3}\). Next, we add \(\frac{2}{3}\) to the result. Thus, the algebraic representation of the function \(f(x)\) is given by \[ f(x) = \frac{x}{3} + \frac{2}{3} \].

Step 2: Constructing the Numerical Representation

To create a numerical representation, we evaluate the function \(f(x)\) for several input values \(x\). For example:- If \(x = 0\), then \(f(0) = \frac{0}{3} + \frac{2}{3} = \frac{2}{3}\).- If \(x = 3\), then \(f(3) = \frac{3}{3} + \frac{2}{3} = 1 + \frac{2}{3} = \frac{5}{3}\).- If \(x = 6\), then \(f(6) = \frac{6}{3} + \frac{2}{3} = 2 + \frac{2}{3} = \frac{8}{3}\).These results can be placed into a table to show the function's values for different inputs.

Step 3: Plotting the Graphical Representation

For the graphical representation, we plot the values obtained in the numerical step on a coordinate plane.1. Plot the point \((0, \frac{2}{3})\).2. Plot the point \((3, \frac{5}{3})\).3. Plot the point \((6, \frac{8}{3})\).4. Draw a line through these points.Since the function \(f(x) = \frac{x}{3} + \frac{2}{3}\) is linear (a line with a slope of \(\frac{1}{3}\)), the graph will be a straight line with this slope.

Key Concepts

Algebraic Representation

When a function is represented algebraically, it means translating a verbal description into a mathematical equation. This is usually the first step in understanding how a function behaves. Let's consider the given problem, where you start by dividing the input, denoted as \(x\), by 3. Mathematically, this operation is expressed as \(\frac{x}{3}\). After performing this division, you then add \(\frac{2}{3}\) to the result. By putting all these pieces together, the algebraic expression for the function \(f(x)\) becomes \(f(x) = \frac{x}{3} + \frac{2}{3}\). This formula succinctly encapsulates the process described in words and allows for easy manipulation and analysis of the function.

Numerical Representation

The numerical representation of a function involves evaluating the function for various input values. This step helps us observe specific outputs and understand how the function behaves at given points. For example, using the function \(f(x) = \frac{x}{3} + \frac{2}{3}\), we can compute:

• \(f(0) = \frac{0}{3} + \frac{2}{3} = \frac{2}{3}\)
• \(f(3) = \frac{3}{3} + \frac{2}{3} = \frac{5}{3}\)
• \(f(6) = \frac{6}{3} + \frac{2}{3} = \frac{8}{3}\)

By listing these calculations, you create a series of input-output pairs that can be organized into a table. This table provides a clear and friendly view of how the function calculates and maps input values to outputs, which is vital for deeper analysis and understanding.

Graphical Representation

Graphical representation transforms the algebraic equation and numerical data into a visual format. Visuals are incredibly helpful for understanding the overall trends and behaviors of a function. For our function \(f(x) = \frac{x}{3} + \frac{2}{3}\), you plot the points obtained from the numerical representation such as:

• (0, \(\frac{2}{3}\))
• (3, \(\frac{5}{3}\))
• (6, \(\frac{8}{3}\))

Once these points are plotted on a graph, draw a line through them. Notice that because the function is linear, this line will follow a straight path with a constant slope of \(\frac{1}{3}\). This slope reflects the rate of change within the function. The graphical method not only shows these individual points but also highlights the continuous nature of the function, providing a comprehensive easy-to-grasp depiction of its behavior.

Linear Function

A linear function is defined by its constant rate of change, which makes it appear as a straight line when graphed. In our function, \(f(x) = \frac{x}{3} + \frac{2}{3}\), we can clearly see this linearity through its components:

• The slope \(\frac{1}{3}\) indicates how much the function's value will increase for every unit increase in \(x\).
• The y-intercept \(\frac{2}{3}\) is the point where the line crosses the y-axis, giving a starting value of the function when \(x = 0\).

Linear functions are foundational in algebra and help provide great insights into real-world situations with constant rates. Their straightforward nature makes them excellent for modeling simple relationships between variables.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. In context, algebra allows us to take a verbal description of a function and express it in an understandable and usable form like \(f(x) = \frac{x}{3} + \frac{2}{3}\). The power of algebra is its ability to give concrete form to abstract ideas, enabling detailed analysis through:

• The formulation and solution of equations, which is vital in interpreting situations theoretically described in words.
• The manipulation of expressions, paving the way for calculations and creating various representations like numerical and graphical forms.

Employing algebra in problems like these helps in transitioning from abstract descriptions to concrete, calculable models. This makes it invaluable in both educational settings and practical applications.