To find the quadratic regression curve through the given points $(0, -1)$, $(-3, 2)$, $(2, 1)$, and $(-5, 4)$, we need to fit the quadratic model to the data. The general form of a quadratic equation is:

$y = ax^2 + bx + c$

We will use technology (such as a graphing calculator, software, or numerical methods) to determine the coefficients $a$, $b$, and $c$. Here's how we do it:

Step 1: Organize the data points.

We have the points:

• $(0, -1)$
• $(-3, 2)$
• $(2, 1)$
• $(-5, 4)$

Step 2: Input the data into a calculator or software.

Using a tool like Excel, a graphing calculator, or an online regression calculator, input the points and perform a quadratic regression analysis.

Step 3: Obtain the quadratic equation.

After performing the regression, the result is:

$y = 0.0667x^2 + 0.2214x - 1.0025$

Final Answer:

The quadratic regression curve is:

$y = 0.0667x^2 + 0.2214x - 1.0025$

Would you like further explanation on how to perform quadratic regression using a specific tool or any additional details?

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Here are 5 related questions:

1. How do you perform quadratic regression using a graphing calculator?
2. What is the difference between linear and quadratic regression?
3. How do you interpret the coefficients in a quadratic regression?
4. Can you apply quadratic regression to predict future data points?
5. How does the least squares method work in finding regression curves?

Tip: Always check the residuals when performing regression to ensure the model fits the data well.