How to Find the Parabolic Equation y ax^2 bx c Given Two Points and a Y-Intercept

Introduction:

Understanding the equation of a parabola is a fundamental concept in algebra and graphing. Often, you might find yourself in a situation where you need to determine the equation of a parabola given two specific points it passes through and the y-intercept value. This guide will walk you through the process on how to achieve this using step-by-step methods.

Understanding the Parabolic Equation y ax^2 bx c

The general form of a parabolic equation is y ax^2 bx c. Here, a, b, and c are constants, and a ≠ 0. The y-intercept c is the point at which the parabola crosses the y-axis. The values of a and b determine the shape and orientation of the parabola. Given that you know the y-intercept c and two points on the parabola, you can determine the exact values of a and b.

Steps to Find the Parabolic Equation

Here are the step-by-step instructions to find the equation of the parabola given two points and the y-intercept:

1. Identify the Points

Let’s assume the two given points are (x1, y1) and (x2, y2), and the y-intercept is c.

2. Substitute the Known Values into the Equation

Given the equation of the parabola y ax^2 bx c and the known points, you can write two equations:

For the first point (x1, y1): y1 ax1^2 bx1 c For the second point (x2, y2): y2 ax2^2 bx2 c

3. Rearrange the Equations

Rearrange the equations to isolate a and b:

a x1^2 b x1 y1 - c a x2^2 b x2 y2 - c

4. Set Up a System of Equations

These two rearranged equations form a system of linear equations with two unknowns, a and b:

ax1^2 bx1 y1 - c ax2^2 bx2 y2 - c

5. Solve the System of Equations

To solve for a and b, you can use various methods including substitution, elimination, or matrix methods. One common method is to subtract the first equation from the second:

a(x2^2 - x1^2) b(x2 - x1) (y2 - c) - (y1 - c)

a(x2 - x1)^2 b(x2 - x1) y2 - y1

Express one variable in terms of the other:

b (y2 - y1 - a(x2 - x1)^2) / (x2 - x1)

Substitute this expression back into one of the original equations, and solve for a and b. Once you find these values, you can substitute them back into the general equation y ax^2 bx c to get the final equation of the parabola.

Example Problem

Suppose you are given the following points and y-intercept:

Points: (1, 3) and (2, 5) Value of c 1

Let's walk through the process:

1. For point (1, 3):

3 a(1^2) b(1) 1

3 a b 1

a b 2 - equation 1

2. For point (2, 5):

5 a(2^2) b(2) 1

5 4a 2b 1

4a 2b 4 - equation 2

3. Solve the system:

From equation 1:

b 2 - a

4. Substitute into equation 2:

4a 2(2 - a) 4

4a 4 - 2a 4

2a 0

a 0

5. Substitute a 0 into equation 1:

0 b 2

b 2

6. The equation of the parabola is:

y ^2 2x 1

y 2x 1

This indicates that the points lie on a line rather than a parabola. This is an important reminder that not all sets of points will form a parabola.

By understanding these steps, you can confidently solve similar problems dealing with parabolic equations and points, ensuring your process is both accurate and efficient.

Conclusion

Finding the equation of a parabola given two points and the y-intercept is a useful skill in algebra and graphing. By following the steps outlined in this guide, you can determine the values of a and b in the parabolic equation, ensuring that you have a solid understanding of the relationship between the given points and the y-intercept. This knowledge can be applied in a variety of real-world scenarios, from physics and engineering to data analysis and statistics.