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Sagot :
To determine which table represents a quadratic function, we need to check the values of [tex]\(f(x)\)[/tex] and see if they can be expressed in a quadratic form, which is generally given by:
[tex]\[ f(x) = ax^2 + bx + c \][/tex]
Let's examine each table one by one:
### Table 1
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -4 & -5 \\ \hline -2 & -2 \\ \hline 0 & 1 \\ \hline 2 & 4 \\ \hline 4 & 7 \\ \hline \end{array} \][/tex]
We attempt to fit these points to a quadratic function. If we look at the differences of [tex]\(f(x)\)[/tex]:
- The first differences (change in [tex]\(f(x)\)[/tex]) are: 3, 3, 3, and 3.
- The second differences (change in the first differences) are: 0, 0, and 0.
The constant second differences suggest that this is a quadratic function. Therefore, Table 1 represents a quadratic function.
### Table 2
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -4 & -10 \\ \hline -2 & -6 \\ \hline 0 & 2 \\ \hline 2 & 6 \\ \hline 4 & 10 \\ \hline \end{array} \][/tex]
Examining the first and second differences:
- The first differences are: 4, 8, 4, and 4.
- The second differences are: 4, -4, and 4.
Since the second differences are not constant, Table 2 does not represent a quadratic function.
### Table 3
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -4 & 2 \\ \hline -2 & -7 \\ \hline 0 & -10 \\ \hline 2 & -7 \\ \hline 4 & 2 \\ \hline \end{array} \][/tex]
Examining the first and second differences:
- The first differences are: -9, -3, 3, and 9.
- The second differences are: 6, 6, and 6.
Although the second differences are not all 6, they suggest a symmetry around [tex]\( x = 0 \)[/tex]. This is characteristic of some quadratic functions but not definitive with these numbers.
### Table 4
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -4 & 0.5 \\ \hline -2 & 1 \\ \hline 0 & 2 \\ \hline 2 & 4 \\ \hline 4 & 8 \\ \hline \end{array} \][/tex]
Examining the first and second differences:
- The first differences are: 0.5, 1, 2, and 4.
- The second differences are: 0.5, 1, and 2.
The second differences are not constant, indicating that Table 4 likely does not represent a quadratic function.
Upon analyzing each table, we observe that Table 1 has constant second differences, a clear indication that it represents a quadratic function. Therefore, the answer is:
Table 1 represents a quadratic function.
[tex]\[ f(x) = ax^2 + bx + c \][/tex]
Let's examine each table one by one:
### Table 1
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -4 & -5 \\ \hline -2 & -2 \\ \hline 0 & 1 \\ \hline 2 & 4 \\ \hline 4 & 7 \\ \hline \end{array} \][/tex]
We attempt to fit these points to a quadratic function. If we look at the differences of [tex]\(f(x)\)[/tex]:
- The first differences (change in [tex]\(f(x)\)[/tex]) are: 3, 3, 3, and 3.
- The second differences (change in the first differences) are: 0, 0, and 0.
The constant second differences suggest that this is a quadratic function. Therefore, Table 1 represents a quadratic function.
### Table 2
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -4 & -10 \\ \hline -2 & -6 \\ \hline 0 & 2 \\ \hline 2 & 6 \\ \hline 4 & 10 \\ \hline \end{array} \][/tex]
Examining the first and second differences:
- The first differences are: 4, 8, 4, and 4.
- The second differences are: 4, -4, and 4.
Since the second differences are not constant, Table 2 does not represent a quadratic function.
### Table 3
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -4 & 2 \\ \hline -2 & -7 \\ \hline 0 & -10 \\ \hline 2 & -7 \\ \hline 4 & 2 \\ \hline \end{array} \][/tex]
Examining the first and second differences:
- The first differences are: -9, -3, 3, and 9.
- The second differences are: 6, 6, and 6.
Although the second differences are not all 6, they suggest a symmetry around [tex]\( x = 0 \)[/tex]. This is characteristic of some quadratic functions but not definitive with these numbers.
### Table 4
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -4 & 0.5 \\ \hline -2 & 1 \\ \hline 0 & 2 \\ \hline 2 & 4 \\ \hline 4 & 8 \\ \hline \end{array} \][/tex]
Examining the first and second differences:
- The first differences are: 0.5, 1, 2, and 4.
- The second differences are: 0.5, 1, and 2.
The second differences are not constant, indicating that Table 4 likely does not represent a quadratic function.
Upon analyzing each table, we observe that Table 1 has constant second differences, a clear indication that it represents a quadratic function. Therefore, the answer is:
Table 1 represents a quadratic function.
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