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
0
5
negative 4
2
18
1.8
114
4
271
8.2
464
6
625
24.2
1034

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

Answer 1

To determine whether functions \( f(x) \), \( g(x) \), and \( h(x) \) are quadratic, we will calculate the first and second differences of the outputs.

1. Calculate the first differences:

The first difference is calculated as follows: \[ \text{First Difference} = f(x_{i+1}) - f(x_i) \]

Given the data in the table:

• For \( f(x) \):

  • \( f(0) = 0 \)
  • \( f(2) = 18 \)
  • \( f(4) = 271 \)
  • \( f(6) = 625 \)

  First differences:

  • \( 18 - 0 = 18 \)
  • \( 271 - 18 = 253 \)
  • \( 625 - 271 = 354 \)

Thus, the first differences are: \( 18, 253, 354 \).

• For \( g(x) \):

  • \( g(0) = 5 \)
  • \( g(2) = 1.8 \)
  • \( g(4) = 8.2 \)
  • \( g(6) = 24.2 \)

  First differences:

  • \( 1.8 - 5 = -3.2 \)
  • \( 8.2 - 1.8 = 6.4 \)
  • \( 24.2 - 8.2 = 16 \)

Thus, the first differences are: \( -3.2, 6.4, 16 \).

• For \( h(x) \):

  • \( h(0) = -4 \)
  • \( h(2) = 114 \)
  • \( h(4) = 464 \)
  • \( h(6) = 1034 \)

  First differences:

  • \( 114 - (-4) = 118 \)
  • \( 464 - 114 = 350 \)
  • \( 1034 - 464 = 570 \)

Thus, the first differences are: \( 118, 350, 570 \).

2. Calculate the second differences:

The second difference is calculated as follows: \[ \text{Second Difference} = \text{First Difference}_{i+1} - \text{First Difference}_i \]

• For \( f(x) \):
  • Second differences:
  • \( 253 - 18 = 235 \)
  • \( 354 - 253 = 101 \)

Thus, the second differences are: \( 235, 101 \).

• For \( g(x) \):
  • Second differences:
  • \( 6.4 - (-3.2) = 9.6 \)
  • \( 16 - 6.4 = 9.6 \)

Thus, the second differences are: \( 9.6, 9.6 \).

• For \( h(x) \):
  • Second differences:
  • \( 350 - 118 = 232 \)
  • \( 570 - 350 = 220 \)

Thus, the second differences are: \( 232, 220 \).

3. Determine the quadratic nature:

• Function \( f(x) \): The second differences are not constant (235 ≠ 101), thus not quadratic.

• Function \( g(x) \): The second differences are constant (both are 9.6), so \( g(x) \) is exactly quadratic.

• Function \( h(x) \): The second differences are also not constant (232 ≠ 220), thus not quadratic.

Final Conclusion:

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

So the answers are:

• \( f(x) \) is not quadratic,
• \( g(x) \) is exactly quadratic,
• \( h(x) \) is not quadratic.