Problem 40 - Algebra A Combined Function

Question

Identify whether each equation, when graphed, will be a parabola, circle,ellipse, or hyperbola. Sketch the graph of each equation. If a parabola, label the vertex. If a circle, label the center and note the radius. If an ellipse, label the center. If a hyperbola, label the \(x\) - or \(y\) -intercepts. $$ 9 x^{2}+4 y^{2}=36 $$

Step-by-Step Solution

Answer

The equation represents an ellipse centered at (0,0) with a vertical major axis.

Step 1: Identify the General Form of the Equation

The given equation is \(9x^2 + 4y^2 = 36\). Compare this with the standard forms of conic sections. The general forms are: \(Ax^2 + By^2 = C\) for ellipses, \(x^2 = 4py\) for parabolas, \(x^2 + y^2 = r^2\) for circles, and \(Ax^2 - By^2 = C\) for hyperbolas. Since both \(x^2\) and \(y^2\) terms are positive and their coefficients are different, the equation represents an ellipse.

Step 2: Rewrite the Equation in Standard Form

Divide each term by 36 to simplify the equation: \(\frac{9x^2}{36} + \frac{4y^2}{36} = 1\). This simplifies to \(\frac{x^2}{4} + \frac{y^2}{9} = 1\). This is now in the standard form of an ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a^2 = 4\) and \(b^2 = 9\).

Step 3: Identify the Features of the Ellipse

From the standard form \(\frac{x^2}{4} + \frac{y^2}{9} = 1\), we identify that the ellipse is centered at the origin \((0,0)\). Here, \(a^2 = 4\) gives \(a = 2\), and \(b^2 = 9\) gives \(b = 3\). Since \(b > a\), the major axis is vertical.

Step 4: Sketch the Graph of the Ellipse

Draw the ellipse centered at the origin. Mark the lengths of the semi-major and semi-minor axes. The semi-major axis is 3 units long (vertical direction along the y-axis), and the semi-minor axis is 2 units long (horizontal direction along the x-axis).

Key Concepts

Ellipse

An ellipse is a type of conic section that appears as an elongated circle. Think of it as a circle that got stretched either vertically or horizontally. Key features of an ellipse include:

• Two axes: the major and minor axes.
• A center point where these axes intersect.
• The major axis is the longer one, passing through the foci of the ellipse.
• The minor axis is the shorter one.

Ellipses appear frequently in real life, such as in planetary orbits, which are typically elliptical. They are defined mathematically with certain properties allowing us to manipulate and analyze them effectively.

Standard Form

The standard form of an ellipse’s equation is highly informative. It reveals important characteristics about the ellipse by organizing the equation as: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] Here, \(a^2\) and \(b^2\) represent values related to the lengths of the semi-major and semi-minor axes. Depending on whether \(a > b\) or \(b > a\), you can determine if the ellipse is stretched more along the x-axis or y-axis. If \(b > a\), as in our example \(\frac{x^2}{4} + \frac{y^2}{9} = 1\), it indicates the major axis is vertical. When you have an ellipse centered at the origin, this form quickly tells you about the ellipse's shape and size without requiring additional complex calculations. It simplifies the process of graphing and understanding ellipses.

General Form

Equations of conic sections often start in what is known as the general form. For ellipses, this can be seen as: \[Ax^2 + By^2 = C\] By examining coefficients \(A\) and \(B\), where both are positive but not equal, you can recognize the structure of an ellipse. The goal often involves transforming this general form into the more user-friendly standard form through algebraic manipulation. Beginning with the general form allows you to utilize patterns to quickly identify different conic sections without initially needing a graph. Through the conversion process, you gain insights into dimensions and orientation, which shapes your understanding of the ellipse's features before graphing it.

Graphing

Graphing an ellipse from its equation involves careful steps to align perfectly with its mathematical description. Here's how:

• Identify the ellipse's center point, often at the origin or mathematically shifted.
• Determine lengths of the semi-major and semi-minor axes using \(a\) and \(b\).
• Note the orientation: is the major axis horizontal or vertical?
• Plot the center, marking intersections at distances \(a\) and \(b\) from the center in their respective directions.
• Draw a smooth curve connecting these points to form the ellipse.

Graphing provides a visual context, making it easier to grasp the spatial relationships represented algebraically. It solidifies understanding by offering tangible geometric insight into how calculated values translate into the physical shape of an ellipse.