For each function, state whether it is linear, quadratic, or exponential.

Understand the Problem

The question is asking to identify the nature of three functions from given (x, y) pairs and classify them as linear, quadratic, exponential, or none of the above.

Answer

Function 1: None of the above; Function 2: None of the above; Function 3: Linear

Steps to Solve

Analyze Function 1 To determine the nature of Function 1, we look at the y-values in relation to the x-values.

Calculate the differences in y-values for successive x-values:
- For $x = 4$ to $5$: $250 - 1250 = -1000$
- For $x = 5$ to $6$: $50 - 250 = -200$
- For $x = 6$ to $7$: $10 - 50 = -40$
- For $x = 7$ to $8$: $2 - 10 = -8$

The differences are not constant, indicating it's neither linear nor quadratic.

Check Function 2 Now, analyze Function 2 by checking the differences in y-values:

For $x = 3$ to $4$: $-17 - (-12) = -5$
For $x = 4$ to $5$: $-20 - (-17) = -3$
For $x = 5$ to $6$: $-21 - (-20) = -1$
For $x = 6$ to $7$: $-20 - (-21) = 1$

The first differences are varied, indicating that Function 2 is not linear.

Analyze Function 3 Now, check Function 3 by obtaining the differences:

For $x = 0$ to $1$: $-3 - 8 = -11$
For $x = 1$ to $2$: $-14 - (-3) = -11$
For $x = 2$ to $3$: $-25 - (-14) = -11$
For $x = 3$ to $4$: $-36 - (-25) = -11$

The differences are constant at -11, indicating this function is linear.

Conclude the Types Having analyzed all three functions, we conclude them as follows:

Function 1: None of the above
Function 2: None of the above
Function 3: Linear

Function 1: None of the above
Function 2: None of the above
Function 3: Linear

More Information

Function 1 does not fit any of the specified types due to varying differences. Function 2 displays characteristics of neither linear, quadratic, nor exponential, but instead possibly some other polynomial form. Only Function 3 displays a constant rate of change making it linear.

Tips

Assuming the function is linear simply because some y-values are decreasing or increasing; make sure to check the differences systematically.
Miscalculating the differences can lead to an incorrect conclusion about the nature of the function.

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