Problem 49 - Algebra 2

Question

Graph each pair of functions. Identify the conic section represented by the graph and write each equation in standard form. \(y=\sqrt{36-4 x^{2}}\) \(y=-\sqrt{36-4 x^{2}}\)

Step-by-Step Solution

Answer

The graph of the functions forms an ellipse. The standard form of the equation is \((\frac{x}{3})^{2} + (\frac{y}{3})^{2} = 1\).

Step 1: Graphing the Functions

Use a graphing tool to plot the equations. For \(y=\sqrt{36-4 x^{2}}\), plot the points that satisfy the function when values are substituted for \(x\). Similarly, for \(y=-\sqrt{36-4 x^{2}}\), plot the corresponding points. Since the range of the square root function is non-negative, the upper part of the graph lies above the x-axis and the lower part lies below the x-axis.

Step 2: Identify the Conic Section

Examine the graph of the two functions together. The combination of the two graphs appears to form a single, continuous shape, which is reflected on both sides of the x-axis. Since there are no cross sections or breaks in the curve, the shape appears to be an ellipse.

Step 3: Write in Standard Form

To reveal the conic section represented, rewrite each equation in standard form. For an ellipse, this form is \((\frac{x}{a})^{2} + (\frac{y}{b})^{2} = 1\). Isolate the \(x^{2}\) term in the functions: \(x^{2} = 9 - y^{2} / 4\). By adding the equations of the two functions, we get \(x^{2} + y^{2} = 9\), or \((\frac{x}{3})^{2} + (\frac{y}{3})^{2} = 1\). This confirms that the graph represents an ellipse.

Key Concepts

Ellipse

An ellipse is a smooth, closed curve that looks like a stretched circle. It's one of the fundamental shapes in conic sections. You encounter it when cutting through a cone at an angle. In this exercise, you have an ellipse that forms when both functions, \( y = \sqrt{36 - 4x^2} \) and \( y = -\sqrt{36 - 4x^2} \), are graphed together. These functions represent the upper and lower halves of the ellipse. The ellipse's symmetry and smoothness make it easy to recognize.

Standard Form of Conic Sections

The standard form of an ellipse is instrumental in understanding its shape and properties. It's given by the equation:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

Here, \(a\) and \(b\) represent the ellipse's horizontal and vertical axes, respectively. For the given functions, after algebraic manipulation, we get \( \left(\frac{x}{3}\right)^2 + \left(\frac{y}{3}\right)^2 = 1 \). This tells us both axes have equal lengths, indicating that our ellipse is actually a circle when drawn.
In general:

• If \(a = b\), it's a circle.
• If \(a > b\), the ellipse is stretched along the x-axis.
• If \(b > a\), the ellipse is stretched along the y-axis.

Graphing Functions

To grasp the concept of graphing, start by plotting points for each function. With functions like \( y = \sqrt{36 - 4x^2} \), you deal only with non-negative outputs because square roots of negative numbers aren't real. Plot points for the positive and negative roots separately to see how they form an ellipse above and below the x-axis.
Tips for graphing:

• Choose x-values within the domain. Here, it means values that don't make \(36 - 4x^2\) negative.
• Plot the positive root for the top half and the negative root for the bottom half.

Square Root Functions

Square root functions can be tricky, but they're a core part of these equations. The function \( y = \sqrt{36 - 4x^2} \) restricts values to non-negative because square roots involve only real numbers.
Breaking it down:

• The domain is where \(36 - 4x^2 \geq 0\), giving you \(-3 \leq x \leq 3\).
• The range is \(0 \leq y \leq 6\), as we take the square root values.

The negative root, \( y = -\sqrt{36 - 4x^2} \), simply mirrors the positive function below the x-axis, creating the ellipse's bottom half.