Problem 58 - Precalculus with Calculus Previews

Question

The graph of the ellipse \((x-1)^{2} / 9+(y-4)^{2}=1\) is shifted 5 units to the left and 3 units up. What are the center, foci, vertices, and endpoints of the minor axis for the shifted graph?

Step-by-Step Solution

Answer

New center: (-4, 7), Foci: (-4±2√2, 7), Vertices: (-7, 7) and (-1, 7), Minor axis endpoints: (-4, 8) and (-4, 6).

Step 1: Identify the original ellipse components

The given ellipse equation is \(\frac{(x-1)^2}{9} + (y-4)^2 = 1\). It is in the standard form of an ellipse centered at \((h, k)\), which is \((1, 4)\). Here, \(a^2 = 9\), so \(a = 3\), indicating the semi-major axis length lies along the x-axis. The semi-minor axis has \(b = 1\).

Step 2: Determine the new center after shifting

The ellipse is shifted 5 units to the left and 3 units up from its original center \((1, 4)\), leading to a new center of \((1-5, 4+3) = (-4, 7)\).

Step 3: Calculate the foci of the shifted ellipse

The distance of the foci from the center is calculated as \(c = \sqrt{a^2 - b^2} = \sqrt{9 - 1} = 2\sqrt{2}\). The foci, originally at \((h \pm c, k)\) or \((1 \pm 2\sqrt{2}, 4)\), are shifted along with the ellipse to \((-4 \pm 2\sqrt{2}, 7)\).

Step 4: Identify the vertices of the shifted ellipse

The vertices are located \(a\) units from the center along the x-axis. Originally, they are at \((1 \pm 3, 4)\). After shifting, the vertices are at \((-4 \pm 3, 7)\), resulting in the points \((-1, 7)\) and \((-7, 7)\).

Step 5: Determine the endpoints of the minor axis

For the minor axis, the endpoints are \(b\) units up and down from the center along the y-axis. Initially at \((1, 4 \pm 1)\), after shifting, these endpoints are \((-4, 7 \pm 1)\), resulting in \((-4, 8)\) and \((-4, 6)\).

Key Concepts

Ellipse Equation

An ellipse is a geometric shape that looks like a stretched circle. It can be described using a mathematical expression known as the ellipse equation. In standard form, the equation for an ellipse centered at

• a point \(h, k\)

is given by:\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]Here,

• \(a\) is the semi-major axis length
• \(b\) is the semi-minor axis length

These axes represent the distances from the center to the farthest and shortest points on the ellipse, respectively. If \(a > b\), the major axis is horizontal, while \(a < b\) signifies a vertical major axis. To change the position of an ellipse, you can shift its center by adding values to \(h\) and \(k\). For instance, shifting left decreases \(h\), and shifting up increases \(k\).

Center of Ellipse

The center of an ellipse is a critical element as it defines the position of the overall shape in the coordinate plane. The center is typically found at the point \(h, k\) in the ellipse's standard equation.

• For example, in the equation \(\frac{(x-1)^2}{9} + \frac{(y-4)^2}{1} = 1\), the center is at \( (1, 4) \).
• This point acts as a reference from which other crucial features of the ellipse are determined.

When an ellipse is shifted, like in our exercise, the center moves accordingly. Shifting the ellipse left by 5 units changes \(h\) to \(-4\), and moving it up by 3 units alters \(k\) to \(7\). Thus, the new center becomes \(-4, 7\). Knowing how to adjust the center is key to translating the entire ellipse and maintaining its shape orientation.

Foci of Ellipse

The foci, pronounced as "foe-sigh," are two specific points inside an ellipse. These points are crucial because they define the ellipse's shape and eccentricity. The sum of the distances from any point on the ellipse to these two foci remains constant.

• The formula to find the distance \(c\) from the center to each focus is: \[ c = \sqrt{a^2 - b^2} \]

For our initial ellipse, this results in \(c = \sqrt{9 - 1} = \sqrt{8} = 2\sqrt{2}\). The foci are located horizontally at \(h \pm c, k\). Shifting the entire ellipse moves the foci as well. So the coordinates change from \(1 \pm 2\sqrt{2}, 4\) to \(-4 \pm 2\sqrt{2}, 7\). Understanding how to calculate and shift foci is essential for accurately mapping the interior attributes of an ellipse.

Vertices of Ellipse

The vertices of an ellipse are found along its major axis and play a vital role in unveiling its geometry. These points are the furthest from the center and indicate the ellipse's length across the major axis.

• Vertices can be found \(a\) units away from the center in the equation's direction of the larger denominator (either \(a^2\) or \(b^2\)).

In our problem, \(a = 3\), directing vertices along the x-axis initially at coordinates \(1 \pm 3, 4\).

• Once shifted, they settle at \(-4 \pm 3, 7\), forming \(-1, 7\) and \(-7, 7\).

This transformation influences the span and location of the ellipse in a coordinate plane. Recognizing how to spot and adjust the vertices ensures that students can accurately portray the ellipse's extent and orientation.