Use the table to answer the question.

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

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

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

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

Now we summarize the results:

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

From these results:

• \( f(-1) = 4 \)
• \( f(0) = 3 \)
• \( f(1) = 0 \)
• \( f(2) = -5 \)

The largest output value is \( f(-1) = 4 \).

Thus, the x value that produces the largest output for the function is:

x = -1.

Answer 2

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

(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

(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

(1,25)
, (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)
left parenthesis 1 comma 5 right parenthesis , left parenthesis 1.5 comma 7.5 right parenthesis , left parenthesis 2 comma 10 right parenthesis , left parenthesis 2.5 comma 12.5 right parenthesis , left parenthesis 3 comma 15 right parenthesis

Answer 3

To create a function for Ari's yard service business, we first need to determine how much he earns based on the number of hours worked. The total earnings can be expressed as:

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

Where:

• \( f(h) \) = total earnings
• \( h \) = number of hours worked

Now, let's calculate the earnings for the specified hours: 1, 1.5, 2, 2.5, and 3 hours.

1. For \( h = 1 \): \[ f(1) = 10 + 15 \times 1 = 10 + 15 = 25 \]

2. For \( h = 1.5 \): \[ f(1.5) = 10 + 15 \times 1.5 = 10 + 22.5 = 32.5 \]

3. For \( h = 2 \): \[ f(2) = 10 + 15 \times 2 = 10 + 30 = 40 \]

4. For \( h = 2.5 \): \[ f(2.5) = 10 + 15 \times 2.5 = 10 + 37.5 = 47.5 \]

5. For \( h = 3 \): \[ f(3) = 10 + 15 \times 3 = 10 + 45 = 55 \]

Now, creating a table of values:

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

The responses that match our calculations are:

(1, 25), (1.5, 32.5), (2, 40), (2.5, 47.5), (3, 55)

So the correct option is:

(1, 25), (1.5, 32.5), (2, 40), (2.5, 47.5), (3, 55).

Answer 4

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=105x+32
y equals 105 x plus 32

y=32x−105
y equals 32 x minus 105

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

y=32x

Answer 5

To determine the function that correctly relates Noah's reading progress, we can define the variables as follows:

• Let \( y \) be the total number of pages read.
• Let \( x \) be the number of days Noah reads.

1. Noah has already read 105 pages.
2. Noah can read 32 pages per day.

The total number of pages read after \( x \) days can be represented by adding the pages already read (105 pages) to the number of pages read in \( x \) days (32 pages per day). Therefore, the algebraic equation can be structured as:

\[ y = 32x + 105 \]

This equation indicates that the total pages read \( y \) is a function of the number of days \( x \), where 32 pages are read each day, and an initial 105 pages have already been read.

So, the correct response is:

y = 32x + 105.