If The Points In The Table Lie On A Parabola, Write The Equation Whose Graph Is The Parabola.$\[ \begin{array}{|c|c|c|c|c|} \hline x & -1 & 1 & 3 & 5 \\ \hline y & -13 & 15 & -13 & -97 \\ \hline \end{array} \\]$\[ Y = \square
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
The problem provides a table with four points: (-1, -13), (1, 15), (3, -13), and (5, -97). We are asked to write the equation of the parabola that passes through these points. To solve this problem, we will use the concept of quadratic equations and the properties of parabolas.
Using the Points to Determine the Equation
To determine the equation of the parabola, we can use the fact that the points lie on the parabola. This means that each point must satisfy the equation of the parabola. We can use this information to set up a system of equations and solve for the constants a, b, and c.
Let's start by assuming that the equation of the parabola is in the form y = ax^2 + bx + c. We can plug in each point into this equation and set up a system of equations.
For the point (-1, -13), we have:
-13 = a(-1)^2 + b(-1) + c
Simplifying this equation, we get:
-13 = a - b + c
For the point (1, 15), we have:
15 = a(1)^2 + b(1) + c
Simplifying this equation, we get:
15 = a + b + c
For the point (3, -13), we have:
-13 = a(3)^2 + b(3) + c
Simplifying this equation, we get:
-13 = 9a + 3b + c
For the point (5, -97), we have:
-97 = a(5)^2 + b(5) + c
Simplifying this 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 various methods, such as substitution or elimination.
Let's use the elimination method to solve the system of equations. We can start by subtracting the first equation from the second equation to eliminate the constant c.
(15) - (-13) = (a + b + c) - (a - b + c)
Simplifying this equation, we get:
28 = 2b
Dividing both sides by 2, we get:
b = 14
Now that we have found the value of b, we can substitute this value into one of the original equations to solve for a. Let's use the first equation:
-13 = a - b + c
Substituting b = 14, we get:
-13 = a - 14 + c
Simplifying this equation, we get:
a - c = -1
Now we have two equations with two unknowns (a and c). We can solve this system of equations using substitution or elimination.
Let's use substitution to solve the system of equations. We can solve the equation a - c = -1 for a:
a = c - 1
Now we can substitute this expression for a into one of the original equations. Let's use the third equation:
-13 = 9a + 3b + c
Substituting a = c - 1 and b = 14, we get:
-13 = 9(c - 1) + 3(14) + c
Simplifying this equation, we get:
-13 = 9c - 9 + 42 + c
Combine like terms:
-13 = 10c + 33
Subtract 33 from both sides:
-46 = 10c
Divide both sides by 10:
c = -4.6
Now that we have found the value of c, we can substitute this value into the expression a = c - 1 to solve for a:
a = -4.6 - 1
a = -5.6
Writing the Equation of the Parabola
We have now found the values of a, b, and c. We can substitute these values into the equation y = ax^2 + bx + c to write the equation of the parabola.
y = -5.6x^2 + 14x - 4.6
This is the equation of the parabola that passes through the points (-1, -13), (1, 15), (3, -13), and (5, -97).
Conclusion
Introduction
In our previous article, we showed how to write the equation of a parabola given a set of points. We used the concept of quadratic equations and the properties of parabolas to determine the equation of the parabola. In this article, we will answer some common questions related to writing the equation of a parabola.
Q: What is a parabola?
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.
Q: How do I know if a set of points lie on a parabola?
You can use the concept of quadratic equations and the properties of parabolas to determine if a set of points lie on a parabola. If the points satisfy the equation of a parabola, then they lie on the parabola.
Q: What are the steps to write the equation of a parabola?
The steps to write the equation of a parabola are:
- Determine the form of the equation: The equation of a parabola is in the form of y = ax^2 + bx + c, where a, b, and c are constants.
- Plug in the points: Plug in the given points into the equation to set up a system of equations.
- Solve the system of equations: Solve the system of equations using various methods, such as substitution or elimination.
- Write the equation of the parabola: Substitute the values of a, b, and c into the equation y = ax^2 + bx + c to write the equation of the parabola.
Q: What are some common mistakes to avoid when writing the equation of a parabola?
Some common mistakes to avoid when writing the equation of a parabola are:
- Not using the correct form of the equation: Make sure to use the correct form of the equation, which is y = ax^2 + bx + c.
- Not plugging in the points correctly: Make sure to plug in the points correctly into the equation to set up a system of equations.
- Not solving the system of equations correctly: Make sure to solve the system of equations correctly using various methods, such as substitution or elimination.
- Not writing the equation of the parabola correctly: Make sure to substitute the values of a, b, and c into the equation y = ax^2 + bx + c to write the equation of the parabola.
Q: Can I use a graphing calculator to write the equation of a parabola?
Yes, you can use a graphing calculator to write the equation of a parabola. A graphing calculator can help you to visualize the parabola and determine the equation of the parabola.
Q: What are some real-world applications of writing the equation of a parabola?
Some real-world applications of writing the equation of a parabola are:
- Projectile motion: The equation of a parabola can be used to model the trajectory of a projectile, such as a thrown ball or a rocket.
- Optics: The equation of a parabola can be used to model the shape of a mirror or a lens.
- Engineering: The equation of a parabola can be used to design and optimize the shape of a structure, such as a bridge or a building.
Conclusion
In this article, we have answered some common questions related to writing the equation of a parabola. We have discussed the steps to write the equation of a parabola, common mistakes to avoid, and real-world applications of writing the equation of a parabola. We hope that this article has been helpful in understanding the concept of writing the equation of a parabola.