Problem 41 - 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. $$ \frac{(x-1)^{2}}{49}+\frac{(y+2)^{2}}{25}=1 $$

Step-by-Step Solution

Answer

It's an ellipse centered at (1, -2).

Step 1: Identify the Type of Conic Section

The given equation is \( \frac{(x-1)^{2}}{49} + \frac{(y+2)^{2}}{25} = 1 \). This equation is of the form \( \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \), which is the standard form of an ellipse.

Step 2: Identify the Center of the Ellipse

From the equation \( \frac{(x-1)^{2}}{49} + \frac{(y+2)^{2}}{25} = 1 \), we can identify the center of the ellipse as \((h, k) = (1, -2)\).

Step 3: Determine the Length of the Major and Minor Axes

Since \(49 > 25\), the major axis is along the \(x\)-axis. The semi-major axis length \(a\) is \(\sqrt{49} = 7\) and the semi-minor axis length \(b\) is \(\sqrt{25} = 5\).

Step 4: Sketch the Graph

The ellipse is centered at \((1, -2)\), with a major axis of length 14 (extending 7 units in both directions from the center along the \(x\)-axis) and a minor axis of length 10 (extending 5 units in both directions from the center along the \(y\)-axis). Plotting these points will help in sketching the ellipse.

Key Concepts

Understanding the Ellipse

An ellipse is a fascinating shape that looks like a flattened circle. It’s one of the four types of conic sections, which also include parabolas, circles, and hyperbolas. Understanding ellipses is crucial in fields like astronomy and physics, where they describe planetary orbits and electromagnetic waves. In mathematics, an ellipse can be expressed in a standard form equation: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). Here, \((h,k)\) indicates the center, \(a\) is half the length of the major axis, and \(b\) is half the length of the minor axis.

• If \(a^2 > b^2\), the major axis lies on the x-axis.
• If \(b^2 > a^2\), the major axis lies on the y-axis.

By understanding the equation structure, you can easily identify and graph an ellipse.

Graphing Ellipse Equations

Graphing an ellipse begins with writing its equation in standard form, which is essential in determining its properties. This involves identifying the center and lengths of the axes. In the given example, we have \( \frac{(x-1)^{2}}{49} + \frac{(y+2)^{2}}{25} = 1 \). By comparing this with the standard form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), it's clear that:
- The center of the ellipse is \((h,k)\), or \((1,-2)\).
- The semi-major axis, \(a\), is \(\sqrt{49} = 7\).
- The semi-minor axis, \(b\), is \(\sqrt{25} = 5\).
Once these values are identified, plot/graph the ellipse by marking the center and drawing the axes. The major and minor axes help shape the boundary of the ellipse, as you extend half the lengths on both sides of the center respectively along the x and y axes.

Finding the Ellipse Center

The center of an ellipse is a key component in its graphing process. You find the center point directly from the equation in standard form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). The terms \((x-h)^2\) and \((y-k)^2\) revert to their vertex forms, revealing the center of the ellipse as \((h, k)\).
To derive the center in the given problem, look closely at the terms in the equation \( \frac{(x-1)^{2}}{49} + \frac{(y+2)^{2}}{25} = 1 \). It tells us that:
- The \((x-1)\) part indicates a horizontal shift, and \((y+2)\) signifies a vertical shift.
- Therefore, the center of the ellipse is \((1, -2)\), derived from the numbers reducing \((x-h)\) and \((y-k)\).
Understanding how to interpret these shifts is fundamental to accurately pinpointing the ellipse's center and drawing it correctly on a graph.