The number of bonus points, B(p), that a credit card holder receives is given by the function `B(p)=4p+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?

Options

A. 3
B. 4
C. 12
D. 19

Answer

To solve the problem step by step, we will analyze the function for bonus points and how it changes when the number of purchases increases.

### Step 1: Understand the function for bonus points
The function for the number of bonus points \( B(p) \) is given by:
\[ B(p) = 4p + 7 \]
where \( p \) is the number of purchases made.

**Hint:** Recognize that the function describes a linear relationship between the number of purchases and the bonus points.

### Step 2: Calculate the initial bonus points
Let’s denote the initial number of purchases as \( p \). The initial bonus points can be calculated as:
\[ B(p) = 4p + 7 \]

**Hint:** This is the starting point before any changes are made to the number of purchases.

### Step 3: Determine the new number of purchases
If the number of purchases is increased by 3, the new number of purchases becomes:
\[ p + 3 \]

**Hint:** This step involves simple addition to the variable representing purchases.

### Step 4: Calculate the new bonus points
Now, we need to find the bonus points for the new number of purchases:
\[ B(p + 3) = 4(p + 3) + 7 \]
Expanding this expression gives:
\[ B(p + 3) = 4p + 12 + 7 = 4p + 19 \]

**Hint:** Remember to distribute the 4 across the terms in the parentheses.

### Step 5: Find the increase in bonus points
To find out how much the bonus points have increased, we subtract the initial bonus points from the new bonus points:
\[ \text{Increase in bonus points} = B(p + 3) - B(p) \]
Substituting the expressions we found:
\[ \text{Increase in bonus points} = (4p + 19) - (4p + 7) \]
Simplifying this gives:
\[ \text{Increase in bonus points} = 4p + 19 - 4p - 7 = 12 \]

**Hint:** This step involves basic subtraction of like terms.

### Conclusion
Thus, when the number of purchases is increased by 3, the number of bonus points increases by **12**.

**Final Answer:** The increase in bonus points is **12**.