Solving a System of Equations for a Horizontally Opening Parabola

Understanding and solving the system of equations to find a parabola that opens horizontally and passes through specific points is a fundamental concept in algebra and calculus. This article will guide you through the process step-by-step, including how to solve the system, interpret the results, and verify the findings using both traditional methods and modern calculators like the TI-84. Let's dive into the details!

Equation Form and Given Points

The equation of a horizontally opening parabola in the standard form is given by:

ay - k^2j x

We are given the points (20, -3), (12, 1), and (33, -2) which the parabola should pass through. These points provide the necessary constraints to solve for the unknowns a, j, and k.

System of Equations

We now establish the system of equations based on the given points:

a(-3) - k^2j 20 a(1) - k^2j 12 a(-2) - k^2j 33

Step-by-Step Solution

Let's denote the equations for convenience:

A: a(-3) - k^2j 20 B: a(1) - k^2j 12 C: a(-2) - k^2j 33

Subtract equation B from equation A:

A - B: a(-3) - k^2j - a(1) k^2j 20 - 12

Simplify to:

a(-3 1) - 8k^2j 8

8(-2k^2j) 8

Solve for k^2:

-2k^2 1

k^2 -frac{1}{2}

Since k^2 must be non-negative for real solutions, there seems to be an error in the simplification process. Let's revisit the system:

A - B: a(-3) - a(1) 20 - 12

-4a 8

a -2

C - A: a(-2) - a(-3) 33 - 20

-a 3a 13

2a 13

a frac{13}{2}

Now solving for j and k using a frac{13}{2}:

A - B: frac{13}{2}(-3 - 1) - 8k^2j 8

-13 - 8k^2j 8

-8k^2j 21

8k^2j -21

C - B: frac{13}{2}(-2 - 1) - 21 12 - 12

-13 - 21 0

-34 0

This simplification confirms the original system's complexity and the need for reevaluation.

Verification Using Calculators

Using a TI-84 calculator, we can simplify this process:

Enter the y-values of the points (-3, 1, -2) into list L1. Clear and enter the corresponding x-values (20, 12, 33) into list L2. Select STAT CALC 5:QuadReg to get the quadratic regression formula.

The result from the calculator might give you the quadratic function in the form:

y -5x^2 - 12x 29

To solve for x, interchange the x and y in the quadratic formula:

x -5y^2 - 12y 29

Conclusion

Solving the system of equations for the horizontally opening parabola involves careful algebraic manipulation and the use of technological tools. The final equation in the desired form is:

x -5y^2 - 12y 29

By understanding and applying these steps, you can accurately solve similar problems involving parabolas in various forms.