Problem 35 - 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. $$ \frac{x^{2}}{16}+\frac{y^{2}}{4}=1 $$

Step-by-Step Solution

Answer

The equation represents an ellipse centered at (0,0).

Step 1: Recognizing the Equation Form

The given equation is \( \frac{x^2}{16} + \frac{y^2}{4} = 1 \). Notice that both \(x^2\) and \(y^2\) terms are positive and there is a 1 on the other side of the equation. This is the standard form of an ellipse equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

Step 2: Identifying the Ellipse Components

In the equation \( \frac{x^2}{16} + \frac{y^2}{4} = 1 \), compare it with \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a^2 = 16 \) and \( b^2 = 4 \). Hence, \( a = 4 \) and \( b = 2 \). The greater value is \(a = 4\), which indicates that the major axis is along the x-axis.

Step 3: Finding the Center of the Ellipse

The equation is in standard form and centered at the origin \((0,0)\) because there is no \( (x - h)^2 \) or \( (y - k)^2 \) term.

Step 4: Sketching the Graph of the Ellipse

To sketch the ellipse, plot the center at \((0,0)\). The x-axis extends 4 units in both directions (left and right) from the center, and the y-axis extends 2 units in both directions (up and down) from the center. Connect these points in an oval shape to represent the ellipse.

Key Concepts

Ellipse Equation

Understanding the rectangle parameters in conic sections is crucial. For an ellipse, its equation is typically in the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This is called the "ellipse equation in standard form". The terms \( a^2 \) and \( b^2 \) are squared values of the semi-major and semi-minor axes, respectively. In context, the given equation \( \frac{x^2}{16} + \frac{y^2}{4} = 1 \) clearly follows this pattern. Subsequently, by matching it to the general form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), you can determine the parameters of the ellipse. Simply replace \( a^2 \) with 16 and \( b^2 \) with 4 to find \( a = 4 \) and \( b = 2 \). This knowledge helps you recognize key aspects of graphing ellipses.

Ellipse Graphing

Graphing an ellipse involves understanding its geometric properties. The equation \( \frac{x^2}{16} + \frac{y^2}{4} = 1 \) tells us that the ellipse is centered at the origin \((0,0)\) because there are no terms like \((x - h)^2\) or \((y - k)^2\), which would indicate shifts away from the origin.

To graph the ellipse:

• First, plot the center at the origin \((0,0)\).
• The semi-major axis (the longer one) is along the x-axis. From the center, extend 4 units left and 4 units right.
• The semi-minor axis (the shorter one) is along the y-axis. From the center, extend 2 units up and 2 units down.
• Connect these points smoothly to form an oval, which is the shape of the ellipse.

These steps illustrate why the components \( a \) and \( b \) derived from the equation are essential for constructing the precise graph of the ellipse.

Standard Forms

Recognizing standard forms is essential in understanding conic sections. Each type of conic section has its own distinct form. For ellipses, the standard form can be either \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\) or \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\). The first form is used when the major axis is horizontal, and the second when it is vertical.

In our case, \( \frac{x^2}{16} + \frac{y^2}{4} = 1 \) is a horizontal ellipse because \( a^2 = 16 \) (greater than \( b^2 = 4 \)), indicating that the major axis is horizontal.

By comparing a specific ellipse equation to its standard form, you can immediately identify its orientation and the lengths of its axes, ensuring accurate graphing.