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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=0
x equals 0

x=1
x equals 1

x=−1

3 answers

To complete the input-output table for the function \( f(x) = -(x + 1)^2 + 4 \), we will calculate \( f(x) \) for each given \( x \) value: -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, let's summarize our calculations in the table:

\[ \begin{array}{c|c} x & f(x) \ \hline -1 & 4 \ 0 & 3 \ 1 & 0 \ 2 & -5 \ \end{array} \]

Now, we can determine which \( x \) value produces the largest output for the function \( f(x) \):

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

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

So, the answer is: x = −1

Question
Use input-output tables to determine which graph accurately represents the function f(x)=12x+3
with inputs of −4
, −2
, 0, 2, and 4.(1 point)
Responses

Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from negative 4 to 4 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
Image with alt text: Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from negative 4 to 4 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.

Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from negative 4 to 4 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
Image with alt text: Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from negative 4 to 4 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.

Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from negative 4 to 4 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
Image with alt text: Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from negative 4 to 4 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.

Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from 0 to 7 in increments of 1. The y-axis ranges from negative 5 to 5 in increments of 1.
Image with alt text: Five points are plotted on a coordinate plane, and a line is graphed passing through them. The x-axis ranges from 0 to 7 in increments of 1. The y-axis ranges from negative 5 to 5 in increments of 1.
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A,B,C or D

To determine which graph accurately represents the function \( f(x) = 12x + 3 \) for the input values of \( -4, -2, 0, 2, \) and \( 4 \), we first need to calculate the outputs for each of these inputs.

  1. For \( x = -4 \): \[ f(-4) = 12(-4) + 3 = -48 + 3 = -45 \]

  2. For \( x = -2 \): \[ f(-2) = 12(-2) + 3 = -24 + 3 = -21 \]

  3. For \( x = 0 \): \[ f(0) = 12(0) + 3 = 0 + 3 = 3 \]

  4. For \( x = 2 \): \[ f(2) = 12(2) + 3 = 24 + 3 = 27 \]

  5. For \( x = 4 \): \[ f(4) = 12(4) + 3 = 48 + 3 = 51 \]

Now we can summarize the input-output pairs:

\[ \begin{array}{c|c} x & f(x) \ \hline -4 & -45 \ -2 & -21 \ 0 & 3 \ 2 & 27 \ 4 & 51 \ \end{array} \]

Next, let's analyze the results:

  • The output for \( x = -4 \) is \( -45 \).
  • The output for \( x = -2 \) is \( -21 \).
  • The output for \( x = 0 \) is \( 3 \).
  • The output for \( x = 2 \) is \( 27 \).
  • The output for \( x = 4 \) is \( 51 \).

Now, let's see which points are plotted based on these calculations:

  • \( (-4, -45) \)
  • \( (-2, -21) \)
  • \( (0, 3) \)
  • \( (2, 27) \)
  • \( (4, 51) \)

You should look for a graph that passes through these specific points. The last graph with axises ranging from \( 0 \) to \( 7 \) in increments of \( 1 \) cannot represent these points accurately since the \( x \) values are outside that range.

You should choose the one of the first three options (A, B, or C) that visualizes the points plotted at the outputs calculated above. Since I can't see the options without images, you should select the graph in the images which shows a line passing through the calculated points while maintaining the appropriate scales on both axes.

Thus, without specific visual representations, I’d suggest confirming from your available options which graph correctly includes those points.