Quadratic Regression: Find Parabola Equation

Nov 2, 2025 by Admin 45 views

Hey guys! Ever wondered how to find the equation of a parabola that perfectly fits through three given points? It's a common problem in mathematics, especially when dealing with curves and data fitting. In this article, we'll dive into using quadratic regression to solve this, walking through a practical example step-by-step. Let's get started!

Understanding Quadratic Regression

Quadratic regression is a statistical method used to model the relationship between a dependent variable and one independent variable as a quadratic function. Simply put, it helps us find the equation of a parabola that best fits a set of data points. A parabola, as you might remember, is a U-shaped curve, and its standard equation is:

y = ax^2 + bx + c

Where:

y is the dependent variable.
x is the independent variable.
a, b, and c are the coefficients we need to determine.

The goal of quadratic regression is to find the values of a, b, and c that minimize the difference between the actual y values and the y values predicted by the quadratic equation. This 'best fit' curve is what we're after.

When we have three distinct points, we can create a system of three equations, which allows us to uniquely solve for a, b, and c. This is because each point (x, y) gives us an equation when plugged into the quadratic form. So, let's see how it works with an example.

Setting up the Equations

Okay, let's say we want to find the equation of a parabola that passes through three points: (-13, -126), (-7, 24), and (3, -46). Our mission is to find the values for a, b, and c in the quadratic equation y = ax^2 + bx + c.

For each point, we'll plug in the x and y values into the equation. This will give us three equations with three unknowns (a, b, and c). Here’s how it looks:

Point (-13, -126): -126 = a(-13)^2 + b(-13) + c -126 = 169a - 13b + c
Point (-7, 24): 24 = a(-7)^2 + b(-7) + c 24 = 49a - 7b + c
Point (3, -46): -46 = a(3)^2 + b(3) + c -46 = 9a + 3b + c

Now we have a system of three equations:

169a - 13b + c = -126
49a - 7b + c = 24
9a + 3b + c = -46

Next up, we need to solve this system of equations. There are several ways to do it, such as substitution, elimination, or using matrices. We'll go through one of the common methods: elimination.

Solving the System of Equations

Now that we've got our system of equations, let's solve for a, b, and c. We'll use the elimination method, which involves strategically adding or subtracting equations to eliminate variables.

Here's our system again:

169a - 13b + c = -126
49a - 7b + c = 24
9a + 3b + c = -46

Step 1: Eliminate c from equations 1 and 2

Subtract equation 2 from equation 1:

(169a - 13b + c) - (49a - 7b + c) = -126 - 24

This simplifies to:

120a - 6b = -150

We'll call this equation 4.

Step 2: Eliminate c from equations 2 and 3

Subtract equation 3 from equation 2:

(49a - 7b + c) - (9a + 3b + c) = 24 - (-46)

This simplifies to:

40a - 10b = 70

We'll call this equation 5.

Step 3: Simplify equations 4 and 5

We can simplify equation 4 by dividing by 6:

20a - b = -25

And simplify equation 5 by dividing by 10:

4a - b = 7

Step 4: Eliminate b from the simplified equations

Subtract the simplified equation 5 from the simplified equation 4:

(20a - b) - (4a - b) = -25 - 7

This simplifies to:

16a = -32

Divide by 16 to solve for a:

a = -2

Step 5: Solve for b

Plug a = -2 into the simplified equation 5:

4(-2) - b = 7

-8 - b = 7

Add 8 to both sides:

-b = 15

Multiply by -1:

b = -15

Step 6: Solve for c

Plug a = -2 and b = -15 into equation 3:

9(-2) + 3(-15) + c = -46

-18 - 45 + c = -46

-63 + c = -46

Add 63 to both sides:

c = 17

So, we've found our coefficients: a = -2, b = -15, and c = 17.

Constructing the Quadratic Equation

Now that we've found the values of a, b, and c, we can write the quadratic equation for the parabola that passes through the points (-13, -126), (-7, 24), and (3, -46). Remember, the general form of a quadratic equation is:

y = ax^2 + bx + c

We found that a = -2, b = -15, and c = 17. So, we just plug these values into the equation:

y = -2x^2 - 15x + 17

This is the equation of the parabola that passes through the three given points. Pretty cool, right?

To make sure we didn't make any mistakes, it's a good idea to verify our equation. We can do this by plugging the original points back into the equation and checking if the equation holds true.

Let's test it out:

Point (-13, -126): y = -2(-13)^2 - 15(-13) + 17 y = -2(169) + 195 + 17 y = -338 + 195 + 17 y = -126 (Correct!)
Point (-7, 24): y = -2(-7)^2 - 15(-7) + 17 y = -2(49) + 105 + 17 y = -98 + 105 + 17 y = 24 (Correct!)
Point (3, -46): y = -2(3)^2 - 15(3) + 17 y = -2(9) - 45 + 17 y = -18 - 45 + 17 y = -46 (Correct!)

Our equation checks out! Each point satisfies the equation, which confirms that y = -2x^2 - 15x + 17 is indeed the equation of the parabola passing through the given points.

Applications of Quadratic Regression

Understanding and using quadratic regression isn't just a neat math trick; it has practical applications in various fields. Here are a few examples:

Physics: In physics, projectile motion often follows a parabolic path. Quadratic equations can help model the trajectory of objects, like a ball thrown in the air or a rocket launched into space. By knowing the initial conditions and some points along the path, we can predict where the object will land.
Engineering: Engineers use quadratic regression to design curved structures, such as bridges and arches. The parabolic shape provides strength and stability, and the equation helps in calculating the necessary dimensions and forces.
Economics: Economists use quadratic models to analyze trends and predict economic behavior. For example, a company might use quadratic regression to model the relationship between advertising expenditure and sales revenue. This can help them optimize their marketing strategies.
Data Analysis: In data analysis, quadratic regression is used to fit curves to data sets that show a non-linear relationship. This is common in many types of data, such as population growth, chemical reactions, and biological processes.
Computer Graphics: In computer graphics and animation, parabolas are used to create smooth curves and natural-looking motion. Understanding quadratic equations allows developers to create realistic animations and visual effects.

Tips for Accuracy

When working with quadratic regression, accuracy is key. Here are a few tips to help you get the most precise results:

Double-Check Your Equations: Make sure you've correctly substituted the x and y values into the quadratic equation. A small mistake in the setup can lead to a wrong answer.
Be Careful with Signs: Pay close attention to positive and negative signs, especially when dealing with the elimination method. A sign error can throw off the entire solution.
Use a Calculator or Software: Solving systems of equations by hand can be tedious and error-prone. Use a calculator or software that can solve systems of linear equations to minimize mistakes.
Verify Your Solution: Always plug the values of a, b, and c back into the original equations to ensure they hold true. This is a crucial step for catching any errors.
Graph the Parabola: If possible, graph the parabola using the equation you found and plot the original points. This visual check can help you confirm that the parabola passes through the points correctly.

Conclusion

So, there you have it! We've walked through how to use quadratic regression to find the equation of a parabola that passes through three points. By setting up a system of equations, solving for the coefficients, and verifying our solution, we can confidently find the quadratic equation. Whether you're working on a math problem, analyzing data, or designing a structure, quadratic regression is a powerful tool to have in your toolkit. Keep practicing, and you'll become a pro at finding those parabolas! Keep up the great work, guys!