Problem 67 - Algebra and Trigonometry

Question

A verbal description of a function is given. Find (a) algebraic, (b) numerical, and (c) graphical representations for the function. Let \(T(x)\) be the amount of sales tax charged in Lemon County on a purchase of \(x\) dollars. To find the tax, take 8\(\%\) of the purchase price.

Step-by-Step Solution

Answer

The function is \( T(x) = 0.08x \) with a linear graph and values like 8 for $100.

Step 1: Identify the Given Function Description

The problem describes a function where sales tax is calculated as 8% of the purchase amount. This means for any purchase of amount \( x \), the tax \( T(x) \) is 8% of \( x \).

Step 2: Formulate the Algebraic Representation

To find the algebraic representation, express the tax as a function of the purchase price. Since the tax is 8% of the price \( x \), the function is \( T(x) = 0.08x \).

Step 3: Create the Numerical Representation

For the numerical representation, compute the tax for various purchase values. For example, for \( x = 100 \), \( T(100) = 0.08 \times 100 = 8 \). For \( x = 200 \), \( T(200) = 0.08 \times 200 = 16 \). Continue for more values like 50, 150, etc.

Step 4: Develop the Graphical Representation

Graph the function \( T(x) = 0.08x \). Plot several points such as \((50, 4)\), \((100, 8)\), and \((150, 12)\), and draw a line through these points. The line represents the linear relationship between \( x \) and \( T(x) \), indicating that as the purchase amount increases, the tax increases linearly.

Key Concepts

Algebraic Representation

An algebraic representation uses an equation to describe a function mathematically based on given conditions. In this exercise, we’re tasked with finding a way to express the sales tax charged in Lemon County. The tax is calculated as 8% of any purchase amount, meaning for a purchase amount of \( x \) dollars, the tax charged, \( T(x) \), can be represented by an equation. By converting 8% into its decimal form, we use 0.08. Thus, algebraically, the sales tax function is expressed as \( T(x) = 0.08x \). This equation succinctly represents how the tax depends directly on the purchase price.

Numerical Representation

A numerical representation shows various specific examples of how the function works. It transforms the abstract equation into relatable numbers. Here, we compute the tax for several different purchase values. This allows us to observe the function's behavior:

• For a purchase of \( x = 50 \) dollars, the tax is \( T(50) = 0.08 \times 50 = 4 \) dollars.
• If \( x = 100 \) dollars, then \( T(100) = 0.08 \times 100 = 8 \) dollars.
• With a larger purchase of \( x = 200 \) dollars, \( T(200) = 0.08 \times 200 = 16 \) dollars.

These examples highlight how the sales tax increases as the purchase price increases, providing tangible data that supports our algebraic expression.

Graphical Representation

A graphical representation gives a visual understanding of the relationship described by a function. We graph the sales tax function, \( T(x) = 0.08x \). To do this, it's helpful to plot points we computed in the numerical representation, such as

• \((50, 4)\)
• \((100, 8)\)
• \((150, 12)\)

By connecting these points, we form a straight line. This line graphically illustrates that the sales tax increases linearly with the purchase amount. It reinforces the idea that for each dollar increase in the purchase amount, the tax rises by a constant rate of 0.08. Such visual aids are valuable in understanding the direct relationship between the variables.

Sales Tax Function

The sales tax function is a practical example of a linear function. In this scenario, the function represents the method by which Lemon County calculates sales tax. It is crucial because it forms a basis to understand how taxes impact purchases. Knowing that the function is \( T(x) = 0.08x \) allows consumers to predict their total costs more accurately.

• The key components include the purchase price, \( x \), and the sales tax rate of 8%.
• Understanding this function helps individuals budget more effectively by foreseeing tax expenses on their purchases.
• It also illustrates a foundational concept in algebra where functions describe relationships between variables.

Through this function, the abstract concept of "8% of a price" is translated into a workable model that aids in financial planning and decision-making.