Problem 11 - Algebra A Combined Function

Question

Graph each ellipse. $$ \frac{(x-1)^{2}}{4}+\frac{(y-1)^{2}}{25}=1 $$

Step-by-Step Solution

Answer

The ellipse is centered at (1, 1) with semi-major axis 5 and semi-minor axis 2.

Step 1: Identify the center of the ellipse

The equation of the ellipse is \( \frac{(x-1)^{2}}{4} + \frac{(y-1)^{2}}{25} = 1 \). The standard form of an ellipse centered at \((h, k)\) is \( \frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1 \). From the given equation, \(h = 1\) and \(k = 1\), so the center of the ellipse is \((1, 1)\).

Step 2: Determine the lengths of the semi-axes

The equation \( \frac{(x-1)^{2}}{4} + \frac{(y-1)^{2}}{25} = 1 \) reveals that \(a^{2} = 4\) and \(b^{2} = 25\). The values of \(a\) and \(b\) are the lengths of the semi-axes. Thus, \(a = 2\) and \(b = 5\).

Step 3: Plot the center and axes lengths on the coordinate plane

Start by plotting the center \((1, 1)\) on the graph. From the center, move 2 units along the x-axis in both directions to plot the vertices \((3, 1)\) and \((-1, 1)\). Move 5 units along the y-axis in both directions to plot the vertices \((1, 6)\) and \((1, -4)\).

Step 4: Draw the ellipse

Connect the vertices smoothly, keeping in mind the oval shape of an ellipse. Ensure the ellipse passes through the four vertices, forming a symmetric shape about the center point \((1, 1)\).

Key Concepts

Standard Form of Ellipse

The standard form of an ellipse's equation helps us to easily identify its properties. An ellipse on a coordinate plane can be expressed in the form \( \frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1 \). Here, \((h, k)\) denotes the center of the ellipse.
This equation structure is significant because it sets up a comparison between horizontal and vertical distances relative to the center point.
If \(a\) and \(b\) are the semi-axes, then \(a^2\) and \(b^2\) give insights into how far the ellipse stretches along the x-axis and y-axis, respectively.
For our specific exercise, substituting in the values, we have \(h=1\), \(k=1\), \(a^2=4\), and \(b^2=25\).
This form is crucial for graphing and understanding the properties of an ellipse.

Center of Ellipse

Finding the center of an ellipse is the first step in graphing it correctly. In the standard form equation \( \frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1 \), the center is simply \((h, k)\).
For our equation \( \frac{(x-1)^{2}}{4} + \frac{(y-1)^{2}}{25} = 1 \), it's clear that the center is at the point \((1, 1)\).
This central point is essential because all the features of the ellipse—its vertices, foci, and axes—are symmetric around it.
To plot the ellipse, you start from this center and use it as a reference point for all other measurements.

Semi-axes of Ellipse

The semi-axes define the size and orientation of the ellipse. These are the half-lengths of the ellipse's major and minor axes.
In the equation \( \frac{(x-1)^{2}}{4} + \frac{(y-1)^{2}}{25} = 1 \), we extract the semi-axis lengths from \(a^2=4\) and \(b^2=25\).
The values of \(a\) and \(b\) turn out to be 2 and 5 respectively, since \( a = \sqrt{4} \) and \( b = \sqrt{25} \).
The larger value, here \(b=5\), indicates the y-axis is the major axis, making the ellipse vertically oriented.
Understanding these axes is vital as they guide the plotting of the ellipse's shape on the coordinate plane.

Coordinate Plane Plotting

Plotting the ellipse on the coordinate plane involves moving from the center to establish the ellipse's borders.
Given the center at \((1, 1)\), move 2 units left and right along the x-axis to mark the ellipse's width. These points are \((-1, 1)\) and \((3, 1)\).
Next, plot 5 units up and down along the y-axis from the center to determine the height. Mark these at \((1, 6)\) and \((1, -4)\).
Now, connect these points smoothly in an oval shape.
This plotting verifies the ellipse's symmetry and orientation, making sure it passes through all critical points for accuracy.