Using the Given Points to Determine the Equation

Oct 13, 2025 by thesweetdesigns.com 44 views

Introduction

In mathematics, a parabola is a type of quadratic curve that can be represented by an equation in the form of y = ax^2 + bx + c, where a, b, and c are constants. Given a set of points that lie on a parabola, we can use these points to determine the equation of the parabola. In this article, we will explore how to write the equation of a parabola given a set of points.

Understanding the Problem

To write the equation of a parabola, we need to understand the general form of a quadratic equation. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic equation is:

y = ax^2 + bx + c

where a, b, and c are constants. We can use the given points to determine the values of a, b, and c.

Using the Given Points to Determine the Equation

We are given the following points that lie on a parabola:

x	-1	1	3	5
y	-13	15	-13	-97

To determine the equation of the parabola, we can use the fact that the points lie on the parabola. This means that we can substitute the x and y values of each point into the general form of the quadratic equation and solve for a, b, and c.

Substituting the Points into the General Form

Let's substitute the first point (-1, -13) into the general form of the quadratic equation:

-13 = a(-1)^2 + b(-1) + c

Simplifying the equation, we get:

-13 = a - b + c

Now, let's substitute the second point (1, 15) into the general form of the quadratic equation:

15 = a(1)^2 + b(1) + c

Simplifying the equation, we get:

15 = a + b + c

Next, let's substitute the third point (3, -13) into the general form of the quadratic equation:

-13 = a(3)^2 + b(3) + c

Simplifying the equation, we get:

-13 = 9a + 3b + c

Finally, let's substitute the fourth point (5, -97) into the general form of the quadratic equation:

-97 = a(5)^2 + b(5) + c

Simplifying the equation, we get:

-97 = 25a + 5b + c

Solving the System of Equations

We now have a system of four equations with three unknowns (a, b, and c). We can solve this system of equations using substitution or elimination.

Let's use the elimination method to solve the system of equations. We can multiply the first equation by 3 and the second equation by 9 to eliminate the c term:

-39 = 3a - 3b + 3c 135 = 9a + 9b + 9c

Subtracting the first equation from the second equation, we get:

174 = 12a + 12b + 6c

Dividing both sides of the equation by 6, we get:

29 = 2a + 2b + c

Now, let's multiply the third equation by 5 and the fourth equation by 9 to eliminate the c term:

-65 = 45a + 15b + 5c -873 = 225a + 45b + 9c

Subtracting the third equation from the fourth equation, we get:

-808 = 180a + 30b + 4c

Dividing both sides of the equation by 4, we get:

-202 = 45a + 7.5b + c

Solving for a, b, and c

We now have two equations with three unknowns (a, b, and c). We can solve this system of equations using substitution or elimination.

Let's use the substitution method to solve the system of equations. We can solve the first equation for c:

c = -13 - a + b

Substituting this expression for c into the second equation, we get:

29 = 2a + 2b + (-13 - a + b)

Simplifying the equation, we get:

42 = a + 3b

Now, let's substitute the expression for c into the third equation:

-202 = 45a + 7.5b + (-13 - a + b)

Simplifying the equation, we get:

-189 = 44a + 6.5b

Solving for a and b

We now have two equations with two unknowns (a and b). We can solve this system of equations using substitution or elimination.

Let's use the elimination method to solve the system of equations. We can multiply the first equation by 6.5 and the second equation by 3 to eliminate the b term:

273 = 6.5a + 19.5b -567 = 132a + 19.5b

Subtracting the first equation from the second equation, we get:

-840 = 125.5a

Dividing both sides of the equation by 125.5, we get:

-a = 6.7

Multiplying both sides of the equation by -1, we get:

a = -6.7

Now, let's substitute the value of a into one of the equations to solve for b. We can substitute the value of a into the first equation:

42 = -6.7 + 3b

Simplifying the equation, we get:

48.7 = 3b

Dividing both sides of the equation by 3, we get:

b = 16.23

Solving for c

We now have the values of a and b. We can substitute these values into one of the equations to solve for c. We can substitute the values of a and b into the first equation:

c = -13 - a + b

Substituting the values of a and b, we get:

c = -13 - (-6.7) + 16.23

Simplifying the equation, we get:

c = -9.07

Writing the Equation of the Parabola

We now have the values of a, b, and c. We can substitute these values into the general form of the quadratic equation to write the equation of the parabola:

y = ax^2 + bx + c

Substituting the values of a, b, and c, we get:

y = -6.7x^2 + 16.23x - 9.07

This is the equation of the parabola that passes through the given points.

Conclusion

Introduction

In our previous article, we explored how to write the equation of a parabola given a set of points. We used the general form of a quadratic equation and substituted the x and y values of each point into the equation to determine the values of a, b, and c. We then solved the system of equations using substitution and elimination to find the values of a, b, and c. Finally, we substituted these values into the general form of the quadratic equation to write the equation of the parabola. In this article, we will answer some frequently asked questions about writing the equation of a parabola.

Q: What is the general form of a quadratic equation?

A: The general form of a quadratic equation is:

y = ax^2 + bx + c

where a, b, and c are constants.

Q: How do I determine the values of a, b, and c?

A: To determine the values of a, b, and c, you can substitute the x and y values of each point into the general form of the quadratic equation. This will give you a system of equations that you can solve using substitution and elimination.

Q: What is the difference between a parabola and a quadratic equation?

A: A parabola is a type of quadratic curve that can be represented by a quadratic equation. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two.

Q: Can I use any method to solve the system of equations?

A: Yes, you can use any method to solve the system of equations, such as substitution, elimination, or graphing. However, the method you choose will depend on the specific problem and the values of a, b, and c.

Q: What if I have a system of equations with more than three unknowns?

A: If you have a system of equations with more than three unknowns, you will need to use a more advanced method, such as matrix operations or computer algebra systems.

Q: Can I use a calculator to solve the system of equations?

A: Yes, you can use a calculator to solve the system of equations. Many calculators have built-in functions for solving systems of equations, such as the "solve" function.

Q: What if I make a mistake in my calculations?

A: If you make a mistake in your calculations, you will need to go back and recheck your work. You can also use a calculator or computer algebra system to check your work.

Q: Can I use this method to solve any type of quadratic equation?

A: Yes, you can use this method to solve any type of quadratic equation, including those with complex coefficients or those that are not in the standard form.

Q: What are some common applications of quadratic equations?

A: Quadratic equations have many common applications, including:

Modeling the trajectory of a projectile
Finding the maximum or minimum value of a function
Solving problems involving optimization
Modeling population growth or decline
Solving problems involving electrical circuits

Conclusion

In this article, we answered some frequently asked questions about writing the equation of a parabola. We discussed the general form of a quadratic equation, how to determine the values of a, b, and c, and how to solve the system of equations. We also discussed some common applications of quadratic equations and how to use a calculator or computer algebra system to solve the system of equations.