The problem statement is as follows:**8.** One function, $ f(x) $, is defined as $ f(x) = (x+4)^2 - 3 $. A second function, $ g(x) $, is a parabola that passes through the points shown in the table below:\[\begin{array}{|c|c|c|c|c|c|}\hlinex & 0 & 1 & 2 & 3 & 4 & 5 \\\hlineg(x) & 4 & 3 & 4 & 7 & 12 & 19 \\\hline\end{array}\]The question asks: What is the absolute value of the difference of the y-intercepts of $ f(x) $ and $ g(x) $?- 9- 15- 6- 13In order to solve this problem, you need to find the y-intercepts of both functions $ f(x) $ and $ g(x) $.- For $ f(x) = (x+4)^2 - 3 $, plug in $ x = 0 $ to find $ f(0) $: \[ f(0) = (0+4)^2 - 3 = 16 - 3 = 13 \] So the y-intercept of $ f(x) $ is 13.- From the table, the y-intercept of $ g(x) $ (when $ x = 0 $) is 4.The absolute value of the difference between the y-intercepts is:\[|13 - 4| = 9\]The correct answer is **9**. ### Problem 7: Pen Purchase Constraints#### ContextGary owns an office supply store. He needs to order black and red pens for resale. Each box of red pens costs $12, while each box of black pens costs $8. Gary has a budget constraint in which he can spend no more than $250 on the boxes of red and black pens combined. Additionally, he needs to order at least 10 boxes of black pens.#### QuestionWhat are the constraints that Gary must use to determine the possible combinations of red and black pens that he can order?#### Answer Choices1. **Choice A:** \[ 12r + 8b \geq 250 \quad \text{and} \quad r \geq 10 \]2. **Choice B:** \[ 8r + 12b < 250 \quad \text{and} \quad b \geq 10 \]3. **Choice C:** \[ 12r + 8b \leq 250 \quad \text{and} \quad r \leq 10 \]4. **Choice D:** \[ 12r + 8b \leq 250 \quad \text{and} \quad b \geq 10 \]#### AnalysisGary’s constraints can be summarized as follows:- **Budget Constraint:** The total cost of red and black pens must not exceed $250. This is represented by the inequality $12r + 8b \leq 250$.- **Minimum Order Requirement for Black Pens:** Gary needs to order at least 10 boxes of black pens, represented by $b \geq 10$.Thus, the correct set of constraints is represented by **Choice D**:\[12r + 8b \leq 250 \quad \text{and} \quad b \geq 10\]

We have to find As per guideline we have to solve one question

8. One function, f(x), is defined as ƒ(x) = (x +4)' - 3 A second function, g(x), is a parabola that passes through the points shown in the table. 2 3 4 5 g(x) 4 3 12 19 What is the absolute value of the difference of the y-intercepts of fG) and g(x)? O 9 O 15 O 6 O 13

8. One function, f(x), is defined as ƒ(x) = (x +4)' - 3 A second function, g(x), is a parabola that passes through the points shown in the table. 2 3 4 5 g(x) 4 3 12 19 What is the absolute value of the difference of the y-intercepts of fG) and g(x)? O 9 O 15 O 6 O 13

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

The problem statement is as follows:

**8.** One function, $ f(x) $, is defined as $ f(x) = (x+4)^2 - 3 $. A second function, $ g(x) $, is a parabola that passes through the points shown in the table below:

\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
g(x) & 4 & 3 & 4 & 7 & 12 & 19 \\
\hline
\end{array}
\]

The question asks: What is the absolute value of the difference of the y-intercepts of $ f(x) $ and $ g(x) $?

- 9
- 15
- 6
- 13

In order to solve this problem, you need to find the y-intercepts of both functions $ f(x) $ and $ g(x) $.

- For $ f(x) = (x+4)^2 - 3 $, plug in $ x = 0 $ to find $ f(0) $:
\[
f(0) = (0+4)^2 - 3 = 16 - 3 = 13
\]
So the y-intercept of $ f(x) $ is 13.

- From the table, the y-intercept of $ g(x) $ (when $ x = 0 $) is 4.

The absolute value of the difference between the y-intercepts is:
\[
|13 - 4| = 9
\]

The correct answer is **9**.

### Problem 7: Pen Purchase Constraints

#### Context
Gary owns an office supply store. He needs to order black and red pens for resale. Each box of red pens costs $12, while each box of black pens costs $8. Gary has a budget constraint in which he can spend no more than $250 on the boxes of red and black pens combined. Additionally, he needs to order at least 10 boxes of black pens.

#### Question
What are the constraints that Gary must use to determine the possible combinations of red and black pens that he can order?

#### Answer Choices

1. **Choice A:**
\[
12r + 8b \geq 250 \quad \text{and} \quad r \geq 10
\]

2. **Choice B:**
\[
8r + 12b < 250 \quad \text{and} \quad b \geq 10
\]

3. **Choice C:**
\[
12r + 8b \leq 250 \quad \text{and} \quad r \leq 10
\]

4. **Choice D:**
\[
12r + 8b \leq 250 \quad \text{and} \quad b \geq 10
\]

#### Analysis
Gary’s constraints can be summarized as follows:
- **Budget Constraint:** The total cost of red and black pens must not exceed $250. This is represented by the inequality $12r + 8b \leq 250$.
- **Minimum Order Requirement for Black Pens:** Gary needs to order at least 10 boxes of black pens, represented by $b \geq 10$.

Thus, the correct set of constraints is represented by **Choice D**:
\[
12r + 8b \leq 250 \quad \text{and} \quad b \geq 10
\]

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