The table to the right has the​ inputs, x, and the outputs for three​ functions, f,​ g, and h. Use second differences to determine which function is exactly​ quadratic, which is approximately​ quadratic, and which is not quadratic.

Question

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Question content area top right
Part 1
                                                             
x
​f(x)
​g(x)
​h(x)
0
5
negative 2
0
2
3.4
90
8
4
14.6
351
83
6
38.6
774
149

Question content area bottom
Part 1
The function​ f(x) is
▼
approximately
not
exactly
​quadratic, g(x) is
▼
exactly
approximately
not
​quadratic, and​ h(x) is
▼
approximately
not
exactly
quadratic.

Answer 1

To determine which function is exactly quadratic, approximately quadratic, and which is not quadratic using second differences, we will first calculate the first differences of the outputs and then calculate the second differences.

Given Data:

x    |   f(x)   |   g(x)   |   h(x)
---------------------------------------
0    |    5     |   -2     |   0
2    |    3.4   |   90     |   8
4    |   14.6   |   351    |   38.6
6    |   38.6   |   774    |   149

Step 1: Calculate First Differences

The first difference is calculated by subtracting the previous output from the current output.

• For \( f(x) \):

  • \( 3.4 - 5 = -1.6 \)
  • \( 14.6 - 3.4 = 11.2 \)
  • \( 38.6 - 14.6 = 24 \)
  • First differences for f: \( [-1.6, 11.2, 24] \)

• For \( g(x) \):

  • \( 90 - (-2) = 92 \)
  • \( 351 - 90 = 261 \)
  • \( 774 - 351 = 423 \)
  • First differences for g: \( [92, 261, 423] \)

• For \( h(x) \):

  • \( 8 - 0 = 8 \)
  • \( 38.6 - 8 = 30.6 \)
  • \( 149 - 38.6 = 110.4 \)
  • First differences for h: \( [8, 30.6, 110.4] \)

Step 2: Calculate Second Differences

The second difference is calculated by subtracting the previous first difference from the current first difference.

• For \( f(x) \):

  • \( 11.2 - (-1.6) = 12.8 \)
  • \( 24 - 11.2 = 12.8 \)
  • Second differences for f: \( [12.8, 12.8] \)

• For \( g(x) \):

  • \( 261 - 92 = 169 \)
  • \( 423 - 261 = 162 \)
  • Second differences for g: \( [169, 162] \)

• For \( h(x) \):

  • \( 30.6 - 8 = 22.6 \)
  • \( 110.4 - 30.6 = 79.8 \)
  • Second differences for h: \( [22.6, 79.8] \)

Results:

• Function \( f(x) \) has constant second differences of \( 12.8 \), so it is exactly quadratic.
• Function \( g(x) \) has second differences that are close but not constant (values are 169 and 162), which means it is approximately quadratic.
• Function \( h(x) \) has second differences that are not constant (values are 22.6 and 79.8), indicating it is not quadratic.

Final Assignment:

• The function \( f(x) \) is exactly quadratic.
• The function \( g(x) \) is approximately quadratic.
• The function \( h(x) \) is not quadratic.

Conclusion:

The function​ \( f(x) \) is exactly quadratic, \( g(x) \) is approximately quadratic, and \( h(x) \) is not quadratic.