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

Steps to Solve

1. 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.

2. 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.

3. 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.

4. 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