Problem 29 - 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 the ellipse by plotting \(y = \pm \sqrt{\frac{1 - 4x^2}{2}}\) within the domain \(-\frac{1}{2} \leq x \leq \frac{1}{2}\).

Step 1: Understand the Equation

The given equation is \(4x^2 + 2y^2 = 1\). This is an equation of an ellipse, which resembles a squished or stretched circle.

Step 2: Solve for y^2

To solve for \(y^2\), move the \(4x^2\) term to the other side of the equation: \(2y^2 = 1 - 4x^2\). Divide both sides by 2 to isolate \(y^2\): \(y^2 = \frac{1 - 4x^2}{2}\).

Step 3: Solve for y

Now, take the square root of both sides to solve for \(y\): \(y = \pm \sqrt{\frac{1 - 4x^2}{2}}\). This means there are two equations to graph: \(y = \sqrt{\frac{1 - 4x^2}{2}}\) and \(y = -\sqrt{\frac{1 - 4x^2}{2}}\).

Step 4: Set Domain Restrictions

Because the expression under the square root must be non-negative, we solve \(\frac{1 - 4x^2}{2} \geq 0\). Simplifying, we find \(-\frac{1}{2} \leq x^2 \leq \frac{1}{4}\), so \(-\frac{1}{2} \leq x \leq \frac{1}{2}\).

Step 5: Graph the Equations

Using a graphing tool or graph paper, plot the two equations. The graphs will be symmetrical across the x-axis and will form an ellipse centered at the origin (0, 0). The x-intercepts occur at the extremes of the domain \(-\frac{1}{2}\) and \(\frac{1}{2}\).

Key Concepts

Graphing Equations

Graphing an equation means visually representing it on a coordinate plane. An ellipse is a specific type of conic section, appearing as an elongated circle. To graph the ellipse equation given, we start by transforming the original equation, \( 4x^2 + 2y^2 = 1 \), into a more workable form by solving for \( y \). We will graph two separate equations: \( y = \sqrt{\frac{1 - 4x^2}{2}} \) and \( y = -\sqrt{\frac{1 - 4x^2}{2}} \). Both of these equations represent the upper and lower halves of the ellipse.
Using either graph paper or a graphing tool, you plot values for \( x \) within the found domain, and calculate \( y \) for each \( x \).
This process will outline the shape of the ellipse centered at the origin with its longest axis horizontally due to how the terms are balanced in the equation. It's important to ensure symmetry is maintained when plotting these equations, reflecting the same above and below the x-axis, capturing the true elliptical shape.

Solving for Variables

Solving for a variable involves manipulating the equation to express one variable in terms of another. Here, with \( 4x^2 + 2y^2 = 1 \), we want to isolate \( y \). First, you'll move the \( 4x^2 \) term to the other side. This makes it \( 2y^2 = 1 - 4x^2 \). Then, you divide by 2, resulting in \( y^2 = \frac{1 - 4x^2}{2} \).
To fully isolate \( y \), take the square root of both sides, adding both a positive and negative case, as square roots can yield two results. Therefore, you obtain the expression \( y = \pm \sqrt{\frac{1 - 4x^2}{2}} \).

• The positive equation captures the top half of the ellipse.
• The negative equation captures the bottom half of the ellipse.

This method helps break down more complex elliptical equations into manageable parts, giving us straightforward equations that we can use for graphing.

Domain Restrictions

Before graphing, it's critical to consider domain restrictions, which define the values that \( x \) can take in an equation. For our ellipse equation, this means ensuring the expression under the square root remains non-negative.
Setting the inequality \( \frac{1 - 4x^2}{2} \geq 0 \) and solving it gives the domain of \( x \). Simplify it to find \(-\frac{1}{2} \leq x^2 \leq \frac{1}{4} \). As we are dealing with real numbers, the variables \( x \) can also turn out to be \(-\frac{1}{2} \leq x \leq \frac{1}{2} \).
This means \( x \) can only range between these values, otherwise, the square root would involve a negative number, which is not defined in the set of real numbers.

• This effectively shapes the width of the ellipse.
• Any \( x \) value outside this range leads to undefined \( y \) values.

Understanding domain restrictions is crucial for accurately graphing conic sections like ellipses.