To find the quadratic regression curve through the points $(1, 4)$, $(3, 7)$, $(4, 5)$, and $(5, 3)$, we need to fit a quadratic equation of the form:

$y = ax^2 + bx + c$

We will use technology (such as a graphing calculator, Excel, or Python) to calculate the coefficients $a$, $b$, and $c$. Here’s the process:

Step-by-Step Explanation:

1. Set up the system of equations using the form $y = ax^2 + bx + c$ for each point:

   • For $(1, 4)$: $4 = a(1^2) + b(1) + c = a + b + c$
   • For $(3, 7)$: $7 = a(3^2) + b(3) + c = 9a + 3b + c$
   • For $(4, 5)$: $5 = a(4^2) + b(4) + c = 16a + 4b + c$
   • For $(5, 3)$: $3 = a(5^2) + b(5) + c = 25a + 5b + c$

2. Solve the system or use technology for quadratic regression to get the best-fit coefficients.

I'll now compute the quadratic regression curve.The quadratic regression curve through the points $(1, 4)$, $(3, 7)$, $(4, 5)$, and $(5, 3)$ is:

$y = -0.7955x^2 + 4.4591x + 0.4000$

The coefficients are:

• $a = -0.7955$
• $b = 4.4591$
• $c = 0.4000$

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How is quadratic regression different from linear regression?
2. Can you verify the curve by plugging in one of the given points?
3. How would adding another point change the regression curve?
4. What is the significance of the coefficients in the quadratic equation?
5. How can you calculate the vertex of the parabola from the quadratic equation?

Tip: Always verify the accuracy of regression curves by checking the fitted values with actual data points.