Problem 32 - Intermediate Algebra

Question

Graph each equation. \(4(x-2)^{2}+9(y-4)^{2}=144\)

Step-by-Step Solution

Answer

Graph the ellipse centered at (2, 4) with vertices at (8, 4) and (-4, 4), and co-vertices at (2, 8) and (2, 0).

Step 1: Identify the Equation Type

The given equation \(4(x - 2)^2 + 9(y - 4)^2 = 144\) is in the form of the general equation of an ellipse, which is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\). Identify the type by comparing the equation format.

Step 2: Convert to Standard Form

To graph the ellipse, first convert the given equation into the standard form by dividing all terms by 144, the right side value. \[\frac{4(x-2)^2}{144} + \frac{9(y-4)^2}{144} = 1\]. Simplify this to get: \[\frac{(x-2)^2}{36} + \frac{(y-4)^2}{16} = 1\].

Step 3: Identify Center and Axes

From the standard form \(\frac{(x-2)^2}{36} + \frac{(y-4)^2}{16} = 1\), identify the center (h, k) = (2, 4). The term \((x-2)^2/36\) indicates the major axis along x (since 36 > 16), so \(a^2 = 36\) (a = 6 and b = 4). The ellipse is wider than it is tall.

Step 4: Determine the Vertices

The vertices of the ellipse along the major axis (x-direction) are obtained by moving 6 units from the center (2, 4) in both directions. The vertices are at \((2+6, 4) = (8, 4)\) and \((2-6, 4) = (-4, 4)\).

Step 5: Determine the Co-Vertices

The co-vertices along the minor axis (y-direction) are found by moving 4 units from the center (2, 4) up and down. The co-vertices are at \((2, 4+4) = (2, 8)\) and \((2, 4-4) = (2, 0)\).

Step 6: Graph the Ellipse

Plot the center (2, 4), the vertices (8, 4) and (-4, 4), and the co-vertices (2, 8) and (2, 0) on a coordinate plane. Draw the ellipse by connecting these points smoothly, ensuring that it is elongated along the x-axis.

Key Concepts

Ellipse Standard Form

The Standard Form for an ellipse is a straightforward method to easily graph ellipses. An ellipse equation in standard form looks like this: \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]This form helps define the shape and position of the ellipse. In this equation:

• \((h, k)\) is the center of the ellipse.
• \(a^2\) and \(b^2\) are the denominators that represent the squares of the lengths of the semi-major and semi-minor axes respectively.

Depending on whether \(a > b\) or \(b > a\), the ellipse will be wider in the x-direction or taller in the y-direction. For graphing, transforming the equation into this standard form simplifies the process of identifying key features like the center and axes of the ellipse.

Vertices and Co-Vertices

The vertices and co-vertices of an ellipse are critical components in understanding its geometry. Based on the standard form:- The vertices lie along the major axis.- The co-vertices lie along the minor axis.For our example, the ellipse has a center at \((2, 4)\). The vertices, located on the major axis (in this case x-axis because 36 is greater than 16), are 6 units away from the center. Thus they are at \((8, 4)\) and \((-4, 4)\).
The co-vertices are positioned on the minor axis (y-axis) and are 4 units away from the center. They are found at \((2, 8)\) and \((2, 0)\). These points help complete the geometric outline when graphing the ellipse.

Coordinate Geometry

Coordinate Geometry, often called analytic geometry, is the study of geometric figures through the coordinate plane. This approach allows us to analyze shapes like ellipses using algebraic techniques. The basic idea is to use coordinates (like \((x, y)\)) to explore spatial relationships between points.In the context of ellipses, coordinate geometry helps in determining specific locations such as:

• The center – where the axes intersect, defining the symmetrical middle of the ellipse.
• The vertices and co-vertices as previously discussed – these define the farthest and closest points from the center along each axis.

Graphing an ellipse using coordinate geometry involves plotting key points and smoothly connecting them. This alignment of algebra and geometry makes graphing ellipses not only precise but also a bit easier to visualize on the coordinate plane.