Unlocking Quadratics: Solving For Equations With Data Points

Hey everyone! Today, we're diving into the world of quadratic equations. We'll explore how to find the specific quadratic equation that fits a given set of data points. Then, we'll write it in the standard form, which is super important for understanding the equation's properties. So, buckle up because this is going to be a fun ride! This is a core concept in algebra, so understanding it well can unlock many other mathematical concepts. Let's get started, shall we?

Understanding the Quadratic Equation and Standard Form

Alright, before we get our hands dirty with the data points, let's refresh our memories on what a quadratic equation actually is. A quadratic equation is a mathematical expression that looks like this: y = ax² + bx + c. In this equation:

• x is your independent variable (the input).
• y is your dependent variable (the output).
• a, b, and c are coefficients. These are just fancy words for numbers, and they determine the shape and position of the parabola (the U-shaped curve) that the equation represents.

The standard form, as shown above, is a common and super useful way to write a quadratic equation. Each term in the standard form has its own significance. The 'a' value determines if the parabola opens upwards (if a > 0) or downwards (if a < 0). 'b' influences the position of the vertex (the lowest or highest point) and the axis of symmetry. And 'c' is the y-intercept, which is where the parabola crosses the y-axis. Knowing this form, we can then manipulate the data we have, which helps us interpret it more clearly. This knowledge is important for all kinds of applications, and it builds a strong foundation for future mathematical studies. So, you see, it's not just about memorizing the formula; it's about understanding how the parts of the formula influence the overall curve.

So, why is understanding the standard form important? Well, it's because it's the most common way to represent quadratics, which will make it easier to compare them and understand their characteristics. Moreover, the standard form gives us direct access to important characteristics of the parabola represented by the equation. For example, the x-coordinate of the vertex of the parabola can be calculated as -b/2a. The ability to work with this is quite valuable, as we begin to extrapolate data and make predictions.

Setting Up the Equations: Plugging in the Data Points

Now, let's get down to the real fun! We have three data points: (4, -116), (-2, -20), and (1, -5). Each data point gives us an x and a y value that satisfies our quadratic equation. Our job is to use these three points to find the values of a, b, and c. Here's how we'll do it.

Remember our standard form equation: y = ax² + bx + c. We'll plug in the x and y values from each data point to create a system of three equations. Let's start with the point (4, -116):

-116 = a(4)² + b(4) + c -116 = 16a + 4b + c

Next, let's use the point (-2, -20):

-20 = a(-2)² + b(-2) + c -20 = 4a - 2b + c

Finally, the point (1, -5):

-5 = a(1)² + b(1) + c -5 = a + b + c

So, now we have a system of three linear equations with three unknowns (a, b, and c):

1. 16a + 4b + c = -116
2. 4a - 2b + c = -20
3. a + b + c = -5

These three equations will help us solve for 'a', 'b', and 'c'.

Solving the System of Equations: Finding a, b, and c

Okay, now we need to solve this system of equations. There are several ways to do this, such as substitution, elimination, or using matrices. For simplicity's sake, let's use the elimination method here. First, let's subtract equation (3) from equation (2):

(4a - 2b + c) - (a + b + c) = -20 - (-5) 3a - 3b = -15

Now, divide both sides by 3:

a - b = -5. (Equation 4)

Next, subtract equation (3) from equation (1):

(16a + 4b + c) - (a + b + c) = -116 - (-5) 15a + 3b = -111

Divide both sides by 3:

5a + b = -37. (Equation 5)

Now we have two equations with two variables:

4. a - b = -5
5. 5a + b = -37

Add equations (4) and (5) together. The 'b' terms will cancel out:

(a - b) + (5a + b) = -5 + (-37) 6a = -42

Divide by 6:

a = -7

With a = -7, substitute it back into equation (4):

-7 - b = -5 -b = 2

b = -2

Finally, substitute a = -7 and b = -2 into equation (3):

-7 + (-2) + c = -5 -9 + c = -5

c = 4

We have found the values of a, b, and c! We've managed to solve for the values that fit our quadratic equation. We've got a = -7, b = -2, and c = 4. We've overcome the challenge and are one step closer to our final result.

Writing the Quadratic Equation in Standard Form

Great job, guys! We've found the coefficients. Now, let's put it all together. Our quadratic equation in standard form, using a = -7, b = -2, and c = 4, is:

y = -7x² - 2x + 4

And there you have it! The quadratic equation that fits those three data points! We've taken a set of data and found an equation that represents it. Congratulations, we've successfully found the quadratic equation. But our goal is to show the equation in standard form, and that is what we have done! This is something you can build on as you advance through higher-level mathematics. You'll be able to solve more complex problems, and the ability to find a quadratic equation from a data set is a good skill to have. It's a fundamental concept that lays the groundwork for understanding parabolas and other mathematical applications.

Verification and Conclusion

Before we wrap things up, let's quickly verify our equation. We can plug the x-values from our original data points into our equation to see if they yield the correct y-values. We already know the data set: (4, -116), (-2, -20), and (1, -5). Let's start with (4, -116):

y = -7(4)² - 2(4) + 4 y = -7(16) - 8 + 4 y = -112 - 8 + 4 y = -116. This works!

Next, (-2, -20):

y = -7(-2)² - 2(-2) + 4 y = -7(4) + 4 + 4 y = -28 + 8 y = -20. This works too!

Finally, (1, -5):

y = -7(1)² - 2(1) + 4 y = -7 - 2 + 4 y = -5. And this also works!

As you can see, our equation works perfectly! This method of finding a quadratic equation from data points opens doors to modeling real-world phenomena, making it an invaluable tool in various fields. From physics and engineering to economics and data science, quadratic equations play a significant role in understanding and predicting patterns. I hope this helps you understand the concept better. Remember, practice makes perfect! So, try working through more examples, and don't hesitate to ask questions if something isn't clear.

We've covered a lot of ground today, from understanding the basics of quadratic equations to solving for the coefficients using data points, and finally, writing the equation in standard form. It is important to know that mathematics is not just a bunch of formulas to memorize. It's also an exciting exploration that helps us understand the world better. Keep practicing, keep exploring, and keep the passion for learning alive. Thanks for reading, and I'll see you in the next one!