Question

A craft store sells specialty beads for $$\$ 1.00$$ for a single bead but will give a discount if a customer buys more than one bead. For each bead after the first, the price per bead goes down until it reaches 75 cents per bead, the lowest possible price, once 5 or more beads are purchased. Which of the following graphs represents the cost per bead in cents, \(y\), of buying \(x\) beads in a single visit?

Short answer

The short answer based on the given step-by-step solution is: We calculated the cost per bead, \(y\), for different number of beads purchased, \(x\), and found the following data points: (1, \$1.00) (2, \$0.875) (3, \$0.833) (4, \$0.8125) (\(x \geq 5\), \$0.75) Choose the graph that has these data points, a continuously decreasing cost per bead, and a constant price of 75 cents for \(x \geq 5\).

Step-by-step solution

List given information

The price per bead is given as follows: 1. The first bead costs $1.00. 2. The price goes down for each bead after the first. 3. The lowest possible price is 75 cents per bead. 4. The lowest price is reached when 5 or more beads are purchased. We have to find a graph representing the cost per bead in cents, \(y\), of buying \(x\) beads in a single visit.

Calculate the price per bead for different quantities of beads

We will calculate the cost per bead for different values of \(x\) (number of beads purchased) to understand how the price changes. For \(x = 1\), the cost per bead \(y = \$1.00\). For \(x = 2\), the bead costs are \((\$1.00 + \$0.25)\) off. Thus, the cost per bead is \(\frac{\$1.00 + \$0.75}{2} = \$0.875\). For \(x = 3\), the bead costs are \((\$1.00 + \$0.50)\) off. Thus, the cost per bead is \(\frac{\$1.00 + \$0.50}{3} = \$0.833\). For \(x = 4\), the bead costs are \((\$1.00 + \$0.75)\) off. Thus, the cost per bead is \(\frac{\$1.00 + \$0.25}{4} = \$0.8125\). For \(x \geq 5\), the cost per bead \(y = \$0.75\).

Determine the graph that represents this relationship

We now have the following data points for the cost per bead, \(y\), based on the number of beads, \(x\): (1, \$1.00) (2, \$0.875) (3, \$0.833) (4, \$0.8125) (\(x \geq 5\), \$0.75) Inspect each of the provided graphs and choose the one with data points that match these calculations. Make sure that the graph has a continuously decreasing cost per bead, reaching the lowest price at \(x = 5\) and staying constant at 75 cents thereafter.

Key concepts

SAT Math Preparation

Preparing for the math section of the SAT can seem daunting, but understanding different types of functions, including piecewise functions, is a key stepping stone. These exams often include questions about the cost of items or services, which requires you to not only do basic calculations but also understand more complex relationships as depicted in graphs.

When you're faced with word problems or graph interpretations, start by carefully reading the questions to identify key pieces of information. Practice by listing down given values and operations, much like you would in the step-by-step example provided. Determine the function that represents the problem, and remember to double-check your work against potential answer choices. This will improve your problem-solving speed and accuracy, which are crucial for test-taking.

Piecewise Functions

Piecewise functions are mathematical expressions that have different values based on the input. In real-world terms, this could represent something like a store discount, where the price of a product changes depending on how many units you purchase.

These functions are defined using multiple sub-functions, each applying to a certain interval of the domain. In our example, the price per bead is a function of the quantity of beads purchased and changes at different intervals. For SAT preparation, it's vital to understand how to interpret such functions and their graphs, as they reflect a variety of real-world scenarios.

Price Discount Problems

Price discount problems are a classic example of real-world math and are frequently featured in standardized tests. To solve these problems, identify how discounts apply to the given scenario. For instance, the craft store bead problem shows a price per unit that decreases incrementally with each additional bead purchased.

To tackle such problems, list the price changes and calculate the new price at each interval. Then, determine the overall cost pattern, which in this case is piecewise, since the price changes after the purchase of each bead until a certain point. Understanding how to calculate discounts in a piecewise function format is an essential skill for students.

Interpreting Graphs

Interpreting graphs involves more than recognizing the shape of a line; it requires a thorough understanding of what the axes represent and how the variables are related. For instance, in the piecewise function that represents the bead pricing structure, the graph illustrates the cost adjustments at different intervals of the number of beads purchased.

Practice by plotting points given by the function and then connecting them to showcase how the function behaves in different segments. Remember that the graph for our bead example should feature a decreasing line that eventually plateaus, indicating the minimum price reached. This is a common representation for discounts over quantities and a valuable skill to have when dissecting such problems.