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x f(x)
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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=1

x=0

x=−1

5 answers

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

Let's calculate:

  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 can summarize the output values:

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

The values of \( f(x) \) are:

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

The largest output is \( 4 \), which occurs when \( x = -1 \).

Thus, the correct response is: \( x = -1 \) produces the largest output for the function.

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.
Responses:

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

(1,16), (1.5,16.5), (2,17), (2.5,17.5), (3,18)

(15,1), (17.5,1.5), (20,2), (22.5,2.5), (25,3)

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

To create a function that calculates 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. Ari charges a flat rate of $10 plus $15 for each hour worked.

The function can be defined as: \[ f(h) = 10 + 15h \]

Now, let's calculate \( f(h) \) for the specified hours: 1, 1.5, 2, 2.5, and 3.

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

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

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

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

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

Now, let's summarize the results in a table:

| \( h \) (hours) | \( f(h) \) (dollars) | |:----------------:|:--------------------:| | 1 | 25 | | 1.5 | 32.5 | | 2 | 40 | | 2.5 | 47.5 | | 3 | 55 |

Now we can match the calculated results with the given options:

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

This matches with the calculated values.

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

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
Responses:

y=32x

y=32x+105

y=105x+32

y=32x−105

To relate the verbal information to an algebraic equation, let's break down the problem:

  • Noah has already read 105 pages, which is a constant.
  • Noah can read 32 pages each day.

Let:

  • \( y \) represent the total number of pages read after \( x \) days.
  • \( x \) represent the number of days Noah continues to read.

Given this information, after \( x \) days, Noah's total number of pages read can be modeled as:

\[ y = 32x + 105 \]

This equation shows that for each day \( x \), Noah reads an additional 32 pages, starting from the 105 pages already read.

Now let's examine the options:

  • \( y = 32x \) (This does not account for the 105 pages already read).
  • \( y = 32x + 105 \) (This correctly accounts for both the pages read each day and the pages already read).
  • \( y = 105x + 32 \) (This incorrectly suggests that Noah reads 105 pages each day, which is not the case).
  • \( y = 32x - 105 \) (This incorrectly implies a negative total before subtracting the 105 pages).

The correct function that relates the verbal information to an algebraic equation is:

\( y = 32x + 105 \).