Question

For the following exercises, sketch the graph of each conic. $$ \frac{x^{2}}{4}+\frac{y^{2}}{16}=1 $$

Short answer

The graph is a vertically oriented ellipse centered at the origin.

Step-by-step solution

Identify the Type of Conic Section

The given equation is \( \frac{x^2}{4} + \frac{y^2}{16} = 1 \). This is the equation of an ellipse in standard form, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), because both terms are positive and equal to 1 on the right side.

Determine the Semi-Axes Lengths

From the standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), we identify that \( a^2 = 4 \) and \( b^2 = 16 \). Thus, \( a = 2 \) and \( b = 4 \). This means the semi-major axis is 4 units and the semi-minor axis is 2 units.

Identify the Orientation of the Ellipse

Since \( b > a \), the semi-major axis is along the y-axis. This tells us that the ellipse is elongated vertically.

Determine the Center of the Ellipse

The given equation is centered at the origin \((0,0)\) since there are no \( (x-h)^2 \) or \( (y-k)^2 \) terms that indicate a shift from the origin.

Sketch the Ellipse

Start by plotting the center at \((0,0)\). From the center, move 4 units up and down along the y-axis and 2 units left and right along the x-axis to plot the vertices and co-vertices. Connect these points in a smooth, oval shape to sketch the ellipse.

Key concepts

Ellipse

An ellipse is a type of conic section that appears as an elongated circle. It can be thought of as a distorted circle, where instead of being perfectly round, it is stretched along one direction. Ellipses are defined by their two semi-axes, which determine their shape and orientation.
The equation of an ellipse can be written in the standard form \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]if the ellipse is centered at the origin. This form reveals crucial information about the ellipse's dimensions and orientation, helping us understand its shape better.

Semi-Major Axis

The semi-major axis is one of the keys to understanding an ellipse's shape. It is the longest radius of the ellipse, extending from the center to the edge along the longest diameter. In an ellipse's equation, the semi-major axis is represented by the larger of the two denominators, compared to the semi-minor axis.

• In the equation \(\frac{x^2}{4} + \frac{y^2}{16} = 1\), the semi-major axis corresponds to \(b^2 = 16\).
• This implies \(b = 4\).

This indicates that the semi-major axis is aligned with the y-axis, making the ellipse taller rather than wider.

Semi-Minor Axis

The semi-minor axis is the shorter radius of the ellipse, aligned perpendicular to the semi-major axis. It extends from the center of the ellipse to the edge along the shortest diameter.

• In the equation \(\frac{x^2}{4} + \frac{y^2}{16} = 1\), the semi-minor axis corresponds to \(a^2 = 4\).
• This implies \(a = 2\).

Thus, the semi-minor axis is aligned with the x-axis, meaning the ellipse is narrower compared to its height.

Graph Sketching

Sketching the graph of an ellipse involves plotting its key components such as the center, vertices, and connecting them smoothly.
First, determine the center of the ellipse, which for the given equation \(\frac{x^2}{4} + \frac{y^2}{16} = 1\), is at the origin \((0,0)\). Next, use the lengths of the semi-major and semi-minor axes to find the vertices:

• Move \(4\) units up and down along the y-axis for the ends of the semi-major axis.
• Move \(2\) units left and right along the x-axis for the ends of the semi-minor axis.

Finally, connect these points with a smooth, oval-like curve to complete the sketch of your ellipse.

Standard Form Equation

Understanding the standard form equation of an ellipse is essential for working with its graph. The equation \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] specifies an ellipse centered at the origin.

• The terms \(x^2/a^2\) and \(y^2/b^2\) determine the orientation and size of the ellipse.
• If \(b > a\), the ellipse is oriented vertically; if \(a > b\), it is oriented horizontally.

Recognizing this equation helps in identifying the ellipse's features and provides a clear framework for sketching its graph accurately.