To determine if this function is linear, quadratic, or exponential, we need to analyze the relationship between the $x$ and $y$ values.

Step 1: Check for a linear relationship

A function is linear if the change in $y$ is constant for each unit change in $x$. Let's check the differences between consecutive $y$-values.

For $x = -6$ to $x = -5$: $y(-5) - y(-6) = -122.5 - (-176.4) = 53.9$

For $x = -5$ to $x = -4$: $y(-4) - y(-5) = -78.4 - (-122.5) = 44.1$

For $x = -4$ to $x = -3$: $y(-3) - y(-4) = -44.1 - (-78.4) = 34.3$

For $x = -3$ to $x = -2$: $y(-2) - y(-3) = -19.6 - (-44.1) = 24.5$

Since the differences are not constant, the function is not linear.

Step 2: Check for a quadratic relationship

A quadratic function has the form $y = ax^2 + bx + c$, and the second differences in the $y$-values should be constant.

Let's check the second differences:

First differences:

• $53.9, 44.1, 34.3, 24.5$

Second differences:

• $44.1 - 53.9 = -9.8$
• $34.3 - 44.1 = -9.8$
• $24.5 - 34.3 = -9.8$

Since the second differences are constant (-9.8), the function is quadratic.

Step 3: Check for an exponential relationship

An exponential function has the form $y = ab^x$. To check for this, the ratio of consecutive $y$-values should be constant.

Let's compute the ratios:

• $\frac{-122.5}{-176.4} \approx 0.694$
• $\frac{-78.4}{-122.5} \approx 0.640$
• $\frac{-44.1}{-78.4} \approx 0.563$
• $\frac{-19.6}{-44.1} \approx 0.444$

Since the ratios are not constant, the function is not exponential.

Conclusion:

This function is quadratic.

Would you like more details on quadratic functions or further questions on this topic?

Here are 5 related questions for deeper understanding:

1. How can we find the equation of a quadratic function from data points?
2. How can we verify if a function is quadratic by plotting it?
3. What is the formula for calculating second differences?
4. How can we tell if a quadratic function opens upwards or downwards?
5. What are the real-life applications of quadratic functions?

Tip: To identify whether a function is quadratic, always check the second differences between the $y$-values. If they are constant, the function is quadratic!