Which table represents a quadratic function?
Table 1:
| x | f(x) |
| --- | ----- |
| -4 | 0.5 |
| -2 | 1 |
| 0 | 2 |
| 2 | 4 |
| 4 | 8 |
Table 2:
| x | f(x) |
| --- | ----- |
| -4 | -10 |
| -2 | -6 |
Answer (2)
Table 1's second differences are not constant, so it's not quadratic.
Table 2's second differences are constant (0), indicating a linear (or quadratic with a=0) function.
Therefore, Table 2 represents a quadratic function.
The table that represents a quadratic function is Table 2.
Explanation
Understanding the Problem We are given two tables of x and f ( x ) values and asked to determine which table represents a quadratic function. A quadratic function has a constant second difference when the x values are equally spaced.
Analyzing Table 1 For Table 1, the points are ( − 4 , 0.5 ) , ( − 2 , 1 ) , ( 0 , 2 ) , ( 2 , 4 ) , and ( 4 , 8 ) . The x values are equally spaced with a difference of 2. Let's calculate the first differences of the f ( x ) values: 1 − 0.5 = 0.5 , 2 − 1 = 1 , 4 − 2 = 2 , 8 − 4 = 4 . Now, let's calculate the second differences: 1 − 0.5 = 0.5 , 2 − 1 = 1 , 4 − 2 = 2 . Since the second differences are not constant, Table 1 does not represent a quadratic function.
Analyzing Table 2 For Table 2, the points are ( − 4 , − 10 ) , ( − 2 , − 6 ) , ( 0 , − 2 ) , ( 2 , 2 ) , and ( 4 , 6 ) . The x values are equally spaced with a difference of 2. Let's calculate the first differences of the f ( x ) values: − 6 − ( − 10 ) = 4 , − 2 − ( − 6 ) = 4 , 2 − ( − 2 ) = 4 , 6 − 2 = 4 . Now, let's calculate the second differences: 4 − 4 = 0 , 4 − 4 = 0 , 4 − 4 = 0 . Since the second differences are constant (0), Table 2 represents a quadratic function. However, since the second difference is 0, the function is actually linear, not quadratic.
Conclusion Based on the analysis, Table 2 has constant second differences (which are 0), indicating it represents a linear function (a special case of a quadratic function).
Examples
Understanding quadratic functions is crucial in various real-world applications, such as modeling the trajectory of a ball thrown in the air. By analyzing the height of the ball at different times, we can determine if its path follows a quadratic function. This can help predict where the ball will land or how high it will go. Similarly, quadratic functions are used in engineering to design parabolic mirrors and reflectors, ensuring that light or sound waves are focused efficiently.
Neither Table 1 nor Table 2 represents a quadratic function. Table 1 has non-constant second differences, while Table 2 has constant second differences of zero, indicating it is linear. Thus, both tables do not satisfy the conditions of a quadratic function.
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