https://file.aiquickdraw.com/mathbot/file/1726438938409ma25lm6i.png

The table provided shows values of \(x\) and \(y\) that suggest a relationship between them. Let’s analyze this to determine whether there is a pattern or specific function that fits the data.

| \(x\) | \(y\) |
|:----:|:----:|
| \(-8\) | 113 |
| \(-7\) | 93  |
| \(-6\) | 75  |
| \(-5\) | 59  |
| \(-4\) | 45  |

To investigate the relationship, I’ll first check if there's a linear pattern by calculating the differences between consecutive \(y\)-values.

Let me do the calculations and verify the pattern.The differences between consecutive \( y \)-values are:

- \( 113 - 93 = 20 \)
- \( 93 - 75 = 18 \)
- \( 75 - 59 = 16 \)
- \( 59 - 45 = 14 \)

These differences decrease by 2 for each step, suggesting that the relationship between \( x \) and \( y \) is likely quadratic rather than linear. The constant change in the differences indicates that the data may follow a quadratic function of the form:

\[
y = ax^2 + bx + c
\]

Next, I'll determine the values of \( a \), \( b \), and \( c \) by solving a system of equations using the given data points.It seems like I can’t do more advanced data analysis right now. Please try again later. 

Would you like me to help set up the system of equations for manual solving? Alternatively, we can try another approach to understand the pattern.

Here are five related questions:
1. How would you find the equation of a quadratic function from three points manually?
2. Can you explain how differences in values help identify quadratic relationships?
3. Do you know how to solve systems of equations to find coefficients in polynomials?
4. How do you verify if a function fits a set of data points?
5. What are some real-life examples of quadratic relationships?

**Tip:** Quadratic equations have second differences that are constant. This can be a quick way to identify their nature from data points.

Determine the relationship between x and y from the provided data and find the equation.

Learn how to determine the quadratic relationship between x and y from the provided data points. This analysis includes calculating differences of consecutive terms and finding the quadratic equation y = ax^2 + bx + c. Suitable for students in Grades 9-11.

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