Problem 10 - A Graphical Approach to College Algebra

Question

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$\frac{(x+2)^{2}}{9}+\frac{(y-4)^{2}}{16}=1$$

Step-by-Step Solution

Answer

The equation represents an ellipse.

Step 1: Identify the Standard Form

The equation given is \( \frac{(x+2)^{2}}{9} + \frac{(y-4)^{2}}{16} = 1 \). This form is \( \frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1 \). It resembles the equation of an ellipse centered at \((h, k)\).

Step 2: Determine the Values of h, k, a, and b

In the equation \( \frac{(x+2)^{2}}{9} + \frac{(y-4)^{2}}{16} = 1 \), \(h = -2\), \(k = 4\), \(a^{2} = 9\), and \(b^{2} = 16\). Therefore, \(a = 3\) and \(b = 4\).

Step 3: Classify the Conic Section

Since the equation is of the form \( \frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1 \) with different values for \(a^2\) and \(b^2\), it represents an ellipse.

Key Concepts

Ellipse

An ellipse is a type of conic section that resembles an elongated circle. It is formed by the intersection of a plane with a cone, where the angle of the intersection is less than that required to form a circle. Think of an ellipse like an oval track that people might run around. Ellipses have two key points called foci, and the sum of the distances from any point on the ellipse to these two foci is constant. This property gives the ellipse its distinctive shape.

An ellipse can appear wider or taller, depending on the values of its major and minor axes. If the longer axis is horizontal, the ellipse is wider; if it is vertical, the ellipse is taller. Understanding the structure of an ellipse is crucial for identifying its equation and graphing it correctly.

Standard Form

The standard form of an ellipse's equation helps identify its properties and graph it easily. In general, the standard form of an ellipse is:

• \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]

Here, \( (h, k) \) represents the center of the ellipse, \( a \) is the semi-major axis length, and \( b \) is the semi-minor axis length. The semi-major axis is always the longer one.

To identify the attributes of an ellipse from its equation, we compare the equation to the standard form. Once in standard form, it's easier to determine the ellipse's dimensions and orientation. Identifying which axis is major and minor will help in sketching its graph.

Graphical Equation

Graphical equations allow us to draw and visualize conic sections like ellipses on a coordinate plane. For an ellipse equation in standard form like

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

we can easily plot the center, determine the axes, and sketch the shape using values derived from the equation. By graphing, you can visually inspect features such as orientation and symmetry.

The graphical representation of an ellipse involves identifying key points: the center, the vertices (endpoints of the major axis), and the co-vertices (endpoints of the minor axis). Incorporating these elements into a sketch helps build a clear and accurate visual model of the ellipse, aiding in the understanding of its geometry.

Center of Ellipse

The center of an ellipse is a crucial point that determines its position on the coordinate plane. It is denoted by \( (h, k) \) in the standard form of the equation:

• \[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \].

The point \( (h, k) \) acts as the center around which the entire ellipse is drawn. This is the midpoint of both the major and minor axes, providing symmetry.

Knowing the center enables you to position the ellipse correctly when graphing. If you shift the coordinates \( h \) units horizontally and \( k \) units vertically, the center of the ellipse moves from the origin to your new point. Hence, the values \( h \) and \( k \) allow precise control over the ellipse's location, which simplifies many real-world applications and problems in geometry.