Question
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 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.
.
.
.
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.
Answers
GPT-4o mini
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.
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.