Pre-Calculus : Hyperbola - Transforming General Form to Standard Form

Summary

In this tutorial from Senior Pablo TV, viewers are guided through the process of converting the general form of a hyperbola to its standard form. The video begins with a refresher on standard hyperbola forms, then tackles a specific problem involving algebraic manipulation to regroup terms and create perfect square trinomials. The final step involves adjusting the equation to match the standard form, showcasing the mathematical steps in a clear and concise manner. The tutorial concludes with the transformed equation and an appreciation for the viewers' attention.

Takeaways

• 📚 The video is a tutorial on converting the general form of a hyperbola to its standard form.
• 📐 The standard form of a hyperbola with the center at the origin is either \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).
• 📍 When the center is not at the origin, the standard form is \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \).
• 🔍 The given problem to solve is \( 4x^2 - 5y^2 + 32x + 30y = 1 \).
• ✂️ The first step is to regroup the terms in the equation to prepare for completing the square.
• 🔢 After regrouping, the equation becomes \( 4x^2 + 32x - 5y^2 - 30y = -13 \).
• 📈 Factor out the coefficients from the x and y terms to simplify the equation.
• 📊 Complete the square for both x and y terms by adding the necessary constants to maintain equality.
• 🔄 Adjust the equation by adding constants to both sides to form perfect square trinomials.
• 📉 The completed square form of the equation is \( (x + 4)^2 - 5(y - 3)^2 = 20 \).
• 🔄 To achieve the standard form, divide the entire equation by 20 to get \( \frac{(x + 4)^2}{5} - \frac{(y - 3)^2}{4} = 1 \).
• 🎯 The final standard form of the hyperbola is \( \frac{(x + 4)^2}{5} - \frac{(y - 3)^2}{4} = 1 \), indicating the center, orientation, and shape of the hyperbola.

Q & A

What is the main topic of the video tutorial?

The main topic of the video tutorial is transforming the general form of a hyperbola to its standard form.

What are the standard forms of a hyperbola when the center is at the origin?

The standard forms of a hyperbola when the center is at the origin are \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).

What is the standard form of a hyperbola when the center is not at the origin?

The standard form of a hyperbola when the center is not at the origin is \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \).

What is the given hyperbola equation in the problem?

The given hyperbola equation in the problem is \( 4x^2 - 5y^2 + 32x + 30y = 1 \).

What is the first step in transforming the given equation to the standard form?

The first step is to regroup the terms of the given equation to form a binomial square on both sides.

How does the script handle the term '32x' in the equation?

The script divides '32x' by the common factor '4' to simplify it to '8x'.

What is the purpose of adding '16' and '9' to the equation in the script?

The purpose of adding '16' and '9' is to complete the square for the 'x' and 'y' terms respectively, to transform the equation into a perfect square trinomial.

What is the final standard form of the hyperbola after the transformation?

The final standard form of the hyperbola is \( \frac{(x + 4)^2}{5} - \frac{(y - 3)^2}{4} = 1 \).

What is the significance of dividing the entire equation by '20' in the script?

Dividing the entire equation by '20' is done to normalize the equation so that the right side equals '1', which is a requirement for the standard form of a hyperbola.

What is the final step in the script to ensure the equation is in the standard form?

The final step is to simplify the coefficients of the squared terms and ensure the equation equals '1', resulting in the standard form of the hyperbola.

Why is it important to transform the general form of a hyperbola to its standard form?

Transforming the general form to the standard form is important because it allows for easier identification of the hyperbola's properties, such as its center, vertices, and asymptotes.