Problem 3 - Precalculus: Functions and Graphs

Question

Exer. 1-14: Find the vertices and foci of the ellipse. Sketch its graph, showing the foci. $$\frac{x^{2}}{15}+\frac{y^{2}}{16}=1$$

Step-by-Step Solution

Answer

Vertices: (0, ±4); Foci: (0, ±1)

Step 1: Identify the values of a² and b²

The equation of the ellipse is \( \frac{x^2}{15} + \frac{y^2}{16} = 1 \). Compare this with the standard form of an ellipse \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \). Here \( a^2 = 16 \) and \( b^2 = 15 \).

Step 2: Calculate a and b

Since \( a^2 = 16 \), take the square root to find \( a = 4 \). Similarly, since \( b^2 = 15 \), take the square root to find \( b = \sqrt{15} \).

Step 3: Find the vertices

For a vertical ellipse, the vertices are at \((0, \pm a)\). Thus, the vertices are at \((0, \pm 4)\).

Step 4: Calculate c

For an ellipse, \( c^2 = a^2 - b^2 \). Here, \( c^2 = 16 - 15 = 1 \). Thus, \( c = 1 \).

Step 5: Find the foci

For a vertical ellipse, the foci are located at \((0, \pm c)\). Therefore, the foci are at \((0, \pm 1)\).

Step 6: Sketch the graph

Plot the ellipse with the center at the origin \((0,0)\). The vertices are at \((0, 4)\) and \((0, -4)\). The foci are at \((0, 1)\) and \((0, -1)\). Draw the ellipse, ensuring it is taller along the y-axis.

Key Concepts

Ellipse Equation

Let's delve into the concept of the ellipse equation and how it helps us understand ellipses better. The standard form for an ellipse equation is \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \]This equation represents an ellipse centered at the origin \((0,0)\). If \(a > b\), it means the major axis is vertical, making the ellipse taller. Otherwise, if \(b > a\), the major axis is horizontal, resulting in a wider ellipse.

In our exercise, the equation given is \[ \frac{x^2}{15} + \frac{y^2}{16} = 1 \]Comparing this with the standard equation, we identify that \(a^2 = 16\) and \(b^2 = 15\), leading us to conclude the major axis is vertical since \(a > b\). Understanding this form helps us describe the direction and dimensions of the ellipse properly.

Ellipse Vertices

Vertices are crucial points that show us how far the ellipse stretches along its major axis. For ellipses centered at the origin, finding vertices is straightforward. Since the equation indicated our ellipse has a vertical major axis \((a > b)\), the vertices are located at \((0, \pm a)\).

To find the exact vertices, calculate \(a\) by taking the square root of \(a^2\). With \(a^2 = 16\), we get \(a = 4\). Therefore, the vertices are \((0, 4)\) and \((0, -4)\).

• The major axis of the ellipse is always twice the distance from the center to a vertex, here, it equals 8, running along the y-axis.

This places the vertices at the northernmost and southernmost points of the ellipse.

Ellipse Foci

While the vertices mark the tips of the ellipse, the foci (plural of focus) are distances that help complete its unique shape. The foci lie along the major axis, inside the ellipse. To find them, use the formula \( c^2 = a^2 - b^2 \), where \(a\), \(b\), and \(c\) describe the ellipse's major axis, semi-minor axis, and distance to each focus from the center, respectively.

Using \(c^2 = 16 - 15 = 1\), we find \(c = 1\). Therefore, the foci are located at \((0, \pm 1)\), closer to the center than the vertices.

• Foci positions show us how much the ellipse deviates from being a circle. If foci coincide with the center, the ellipse becomes a circle.

It's this deviation that characterizes the ellipse's shape.

Precalculus

Understanding ellipses and their components is a vital topic in precalculus, the mathematics course focusing on functions and their graphs. This course prepares students for calculus by introducing concepts like conic sections, of which ellipses are a part.

Working through ellipses involves:

• Understanding their standard equations.
• Learning to find vertices and foci.
• Graphing these shapes accurately based on algebraic data.

This foundation in precalculus is crucial because it teaches students to visualize and graph mathematical concepts, paving the way for the more intricate studies of calculus.
By mastering precalculus concepts, students enhance their logical thinking skills, enabling them to tackle a wide array of mathematical challenges in their higher education journey.