Problem 21 - Algebra and Trigonometry

Question

Graph the equation \(4 x^{2}+2 y^{2}=1\) by solving for \(y\) and graphing two equations corresponding to the negative and positive square roots. (This graph is called an ellipse.)

Step-by-Step Solution

Answer

Graph both \(y = \sqrt{\frac{1}{2} - 2x^2}\) and \(y = -\sqrt{\frac{1}{2} - 2x^2}\) to form the ellipse.

Step 1: Rearranging the Equation

The given equation is \(4x^2 + 2y^2 = 1\). Start by isolating \(y^2\) on one side. Divide the entire equation by 2 to simplify it: \(2x^2 + y^2 = \frac{1}{2}\).

Step 2: Solving for y

To solve for \(y\), isolate \(y^2\) by subtracting \(2x^2\) from both sides: \(y^2 = \frac{1}{2} - 2x^2\). Then, take the square root of both sides to solve for \(y\): \(y = \pm \sqrt{\frac{1}{2} - 2x^2}\).

Step 3: Graph the Positive Root

Graph the equation \(y = \sqrt{\frac{1}{2} - 2x^2}\). This represents the upper half of the ellipse.

Step 4: Graph the Negative Root

Graph the equation \(y = -\sqrt{\frac{1}{2} - 2x^2}\). This represents the lower half of the ellipse.

Step 5: Combine the Graphs

The complete graph is formed by combining both halves from Steps 3 and 4. Ensure the values of \(x\) used satisfy \(2x^2 < \frac{1}{2}\), as \(\sqrt{\text{negative number}}\) is not real.

Key Concepts

Solving Quadratic Equations

Quadratic equations are an essential part of algebra and are expressed in the form \(ax^2 + bx + c = 0\). Solving these equations helps you find the values of \(x\) that satisfy the equation.
This exercise involves a slight variation, as it involves \(y^2\) rather than just \(x\). The aim is to express \(y\) in terms of \(x\) by solving the equation \(2x^2 + y^2 = \frac{1}{2}\). Here's how it's done:

• First, rearrange the equation to isolate \(y^2\).
• Then, subtract \(2x^2\) from both sides to have \(y^2 = \frac{1}{2} - 2x^2\).
• Finally, take the square root of both sides to solve for \(y\). This simplifies to \(y = \pm \sqrt{\frac{1}{2} - 2x^2}\).

When working with quadratic equations, always ensure to check your work so that the values you find are correct. Solving these equations sometimes leads to two solutions, as shown here with the positive and negative roots.

Graphing Functions

Graphing functions allows us to visualize equations and understand their behavior. In this case, the function is obtained by solving for \(y\), resulting in two separate equations, \(y = \sqrt{\frac{1}{2} - 2x^2}\) and \(y = -\sqrt{\frac{1}{2} - 2x^2}\).
Here's how we graph them:

• First, understand the domain by noting that \(2x^2 < \frac{1}{2}\), ensuring the expression inside the square root is non-negative.
• The function \(y = \sqrt{\frac{1}{2} - 2x^2}\) represents the upper half of the ellipse.
• The function \(y = -\sqrt{\frac{1}{2} - 2x^2}\) represents the lower half of the ellipse.
• Plot these two functions on the same graph to form the complete ellipse.

By combining both graphs, you obtain a complete picture. Use points within the domain to check your graph, ensuring a smooth, continuous curve.

Conic Sections

Conic sections consist of curves obtained by intersecting a plane with a cone. The ellipse is one type of conic section. It's formed when the plane cuts across the cone at an angle, creating a closed curve.
In this exercise, the graph you obtained from the equation \(4x^2 + 2y^2 = 1\) is an example of an ellipse. Here's why:

• An ellipse's general equation is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Notice the similar structure in our exercise after simplification.
• The coefficients in the equation reflect the shape and orientation of the ellipse.
• In our example, the ellipse is centered at the origin, with axes aligned along the \(x\) and \(y\) axes.

Understanding conic sections, such as ellipses, allows us to apply these principles to various real-world and mathematical scenarios. Recognizing these shapes in equations helps us break down their graphing and characteristics.