Question

The number of bonus points, \(B(p)\), that a credit card holder receives is given by the function \(B(p)=4 p+7\), where \(p\) represents the number of purchases made. If the number of purchases is increased by 3 , by how much does the number of bonus points increase? A) 3 B) 4 C) 12 D) 19

Short answer

The number of bonus points increases by \(12\) when the number of purchases is increased by 3. The correct answer is option C) 12.

Step-by-step solution

Calculate the initial bonus points B(p)

We are given the function B(p) = 4p + 7, where p represents the number of purchases made.

Calculate the new bonus points B(p+3) when the number of purchases is increased by 3

The new number of purchases can be represented as (p+3). To find the new bonus points, substitute (p+3) into the B(p) function: B(p+3) = 4(p+3) + 7.

Find the difference between the new bonus points B(p+3) and the initial bonus points B(p)

To find the difference between the new bonus points and the initial bonus points, subtract the initial bonus points from the new bonus points: Difference = B(p+3) - B(p)

Match the result of step 3 with the given options to find the correct answer.

From Step 2, we have B(p+3) = 4(p+3) + 7. We already know B(p) = 4p + 7. Difference = (4(p+3) + 7) - (4p + 7) = 4p + 12 + 7 - 4p - 7 \( \) = 12 The number of bonus points increases by 12 when the number of purchases is increased by 3. The correct answer is option C) 12.

Key concepts

Algebraic Expressions

In the world of mathematics, algebraic expressions are the building blocks for conveying relationships between variables and numbers. An algebraic expression consists of numbers, variables, and arithmetic operations. In the example exercise, the expression \( B(p) = 4p + 7 \) is used to describe the number of bonus points earned based on the number of purchases made. Simplifying and manipulating these expressions are crucial skills.

Consider the step-by-step solution to the original problem: The initial bonus points given by \( B(p) \) were calculated, and then the function was updated to find the new bonus points with three additional purchases, yielding \( B(p+3) = 4(p+3) + 7 \). Finally, the increase in bonus points was determined by finding the difference between these two expressions. This method of problem-solving is crucial not only for homework but for any situation where algebra is applied to real-world scenarios.

Function Notation

Function notation is a concise way to define and use functions in mathematics. It involves the use of symbols to represent a relationship between input values and output values. For example, \( B(p) = 4p + 7 \) is a function that gives the number of bonus points \( B \) as a function of the number of purchases \( p \).

In the given exercise, we also see the use of function notation to understand the change in output when the input is increased by 3: \( B(p+3) \). By substituting \( p+3 \) instead of \( p \) into the original function, we analyze how the function changes based on different input values. Function notation is not just a tool for expressing mathematical relationships but also for simplifying and solving real-world problems where variables are interdependent.

SAT Math Preparation

Mastering SAT math involves understanding a wide range of topics, including algebraic expressions and function notation as seen in the given exercise. Preparing for the SAT math section requires practice with these types of problems to develop problem-solving skills and efficient use of time.

When studying for the SAT, focus on breaking down complex problems into smaller, more manageable steps, as seen in the step-by-step solution. Always pay close attention to the function definitions and remember to apply changes systematically to find your solution. Regular practice with textbook exercises, particularly those that challenge your understanding of algebraic expressions and functions, will equip you with the strategies needed to excel during the test. Additionally, leveraging tips like elimination of impossible answers and learning to quickly identify patterns will serve you well on test day.