Question

Use the table to answer the question.

x f(x)
−1

0
1
2
Finish creating the input-output table for the function f(x)=−(x+1)2+4
. Which of the given x values produces the largest output for the function?

(1 point)
Responses

x=2
x equals 2

x=−1
x equals negative 1

x=0
x equals 0

x=1

Answer 1

Ari is starting a yard service business. He charges a flat rate of $10 plus an additional $15 per hour, regardless of the service. Create a function and use a table to determine how much Ari will make if he works for 1, 1.5, 2, 2.5, and 3 hours.(1 point)
Responses

(15,1)
, (17.5,1.5)
, (20,2)
, (22.5,2.5)
, (25,3)
left parenthesis 15 comma 1 right parenthesis , left parenthesis 17.5 comma 1.5 right parenthesis , left parenthesis 20 comma 2 right parenthesis , left parenthesis 22.5 comma 2.5 right parenthesis , left parenthesis 25 comma 3 right parenthesis

(1,16)
, (1.5,16.5)
, (2,17)
, (2.5,17.5)
, (3,18)
left parenthesis 1 comma 16 right parenthesis , left parenthesis 1.5 comma 16.5 right parenthesis , left parenthesis 2 comma 17 right parenthesis , left parenthesis 2.5 comma 17.5 right parenthesis , left parenthesis 3 comma 18 right parenthesis

(1,25)
, (1.5,32.5)
, (2,40)
, (2.5,47.5)
, (3,55)
left parenthesis 1 comma 25 right parenthesis , left parenthesis 1.5 comma 32.5 right parenthesis , left parenthesis 2 comma 40 right parenthesis , left parenthesis 2.5 comma 47.5 right parenthesis , left parenthesis 3 comma 55 right parenthesis

(1,5)
, (1.5,7.5)
, (2,10)
, (2.5,12.5)
, (3,15)

Answer 2

Noah has to read a book for a project. They have already read 105 pages, and they can read about 32 pages per day. Which of the following functions correctly relates the verbal information about the function to an algebraic equation?(1 point)
Responses

y=32x+105
y equals 32 x plus 105

y=105x+32
y equals 105 x plus 32

y=32x−105
y equals 32 x minus 105

y=32x

Answer 3

To complete the input-output table for the function \( f(x) = - (x + 1)^2 + 4 \), we will calculate the output values for each of the given \( x \) values: -1, 0, 1, and 2.

**Calculations:**

1. **For \( x = -1 \)**:
\[
f(-1) = -((-1) + 1)^2 + 4 = -0 + 4 = 4
\]

2. **For \( x = 0 \)**:
\[
f(0) = -((0) + 1)^2 + 4 = -1^2 + 4 = -1 + 4 = 3
\]

3. **For \( x = 1 \)**:
\[
f(1) = -((1) + 1)^2 + 4 = -2^2 + 4 = -4 + 4 = 0
\]

4. **For \( x = 2 \)**:
\[
f(2) = -((2) + 1)^2 + 4 = -3^2 + 4 = -9 + 4 = -5
\]

**Summary of the Outputs:**

| x | f(x) |
|-----|------|
| -1 | 4 |
| 0 | 3 |
| 1 | 0 |
| 2 | -5 |

Now we can compare the output values:
- \( f(-1) = 4 \)
- \( f(0) = 3 \)
- \( f(1) = 0 \)
- \( f(2) = -5 \)

The largest output value occurs at \( x = -1 \), which gives \( f(-1) = 4 \).

**Conclusion:**
The value of \( x \) that produces the largest output for the function is:

**x = -1**.

Answer 4

To create a function that represents how much Ari will make based on the number of hours worked, we can define the function as follows:

Let \( h \) be the number of hours worked. The total earnings \( E(h) \) can be calculated with the formula:

\[
E(h) = 10 + 15h
\]

where $10 is the flat rate and $15 is the hourly charge.

Now, let's compute the total earnings for the given hours: 1, 1.5, 2, 2.5, and 3.

### Calculations:
1. **For 1 hour (h = 1)**:
\[
E(1) = 10 + 15(1) = 10 + 15 = 25
\]

2. **For 1.5 hours (h = 1.5)**:
\[
E(1.5) = 10 + 15(1.5) = 10 + 22.5 = 32.5
\]

3. **For 2 hours (h = 2)**:
\[
E(2) = 10 + 15(2) = 10 + 30 = 40
\]

4. **For 2.5 hours (h = 2.5)**:
\[
E(2.5) = 10 + 15(2.5) = 10 + 37.5 = 47.5
\]

5. **For 3 hours (h = 3)**:
\[
E(3) = 10 + 15(3) = 10 + 45 = 55
\]

### Summary of Earnings:

| Hours Worked (h) | Earnings (E(h)) |
|-------------------|------------------|
| 1 | 25 |
| 1.5 | 32.5 |
| 2 | 40 |
| 2.5 | 47.5 |
| 3 | 55 |

### Conclusion:
The pairs (h, E(h)) representing the earnings based on hours worked are:
- \( (1, 25) \)
- \( (1.5, 32.5) \)
- \( (2, 40) \)
- \( (2.5, 47.5) \)
- \( (3, 55) \)

Therefore, the correct response is:
**(1, 25), (1.5, 32.5), (2, 40), (2.5, 47.5), (3, 55)**.

Answer 5

To determine which function correctly relates the verbal information to an algebraic equation, we need to interpret the details given:

- Noah has already read **105 pages**.
- They can read **32 pages per day**.
- Let \( y \) represent the total number of pages read after \( x \) days.

From this information, we can build the function that describes the total number of pages read over time:

1. Noah starts with **105 pages** already read.
2. For each day \( x \), Noah reads an additional **32 pages**.

Thus, the total number of pages read after \( x \) days can be expressed as:

\[
y = 32x + 105
\]

where:
- \( 32x \) represents the pages read after \( x \) days,
- \( 105 \) represents the pages already read.

Now, let's review the given options:

1. **\( y = 32x + 105 \)** **(Correct)**
2. \( y = 105x + 32 \)
3. \( y = 32x - 105 \)
4. \( y = 32x \)

The correct function that relates the verbal information is:

**\( y = 32x + 105 \)**.