How do you graph a semi ellipse?

To graph a semi-ellipse, you typically start with the equation of a full ellipse and then restrict its domain or range to display only a specific half. This involves solving for one variable (usually y) and selecting either the positive or negative root, or by restricting x values.

Understanding the Semi-Ellipse

A semi-ellipse is half of a complete ellipse, meaning it could be the top, bottom, left, or right half. For instance, as demonstrated in educational resources like a video titled "Graphing a Half-Ellipse," a semi-ellipse can appear as "just the bottom half" of a full ellipse, forming a distinct curved shape.

Steps to Graph a Semi-Ellipse

Graphing a semi-ellipse involves a few key steps, primarily centered on modifying the standard ellipse equation.

1. Identify the Standard Ellipse Equation:
   The general equation for an ellipse centered at (h, k) is:

   • Horizontal Major Axis: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$
   • Vertical Major Axis: $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$

   Where:

   • (h, k) is the center of the ellipse.
   • a is the distance from the center to the ellipse along the major axis.
   • b is the distance from the center to the ellipse along the minor axis.

2. Determine Which Half You Need:
   Decide whether you want to graph the top, bottom, left, or right half of the ellipse. This decision will dictate how you modify the equation.

3. Solve for y (for Top/Bottom Halves) or x (for Left/Right Halves):

   For Top or Bottom Halves:

   Start with the general ellipse equation and solve for y:
   $\frac{(y-k)^2}{b^2} = 1 - \frac{(x-h)^2}{a^2}$
   $(y-k)^2 = b^2 \left(1 - \frac{(x-h)^2}{a^2}\right)$
   $y-k = \pm \sqrt{b^2 \left(1 - \frac{(x-h)^2}{a^2}\right)}$
   $y = k \pm b \sqrt{1 - \frac{(x-h)^2}{a^2}}$

   • To graph the Top Half: Use the positive square root:
     $y = k + b \sqrt{1 - \frac{(x-h)^2}{a^2}}$
   • To graph the Bottom Half: Use the negative square root. As illustrated in college algebra discussions on graphing a half-ellipse, focusing on "just the bottom half" involves this form:
     $y = k - b \sqrt{1 - \frac{(x-h)^2}{a^2}}$

   For Left or Right Halves:

   Start with the general ellipse equation and solve for x:
   $\frac{(x-h)^2}{a^2} = 1 - \frac{(y-k)^2}{b^2}$
   $(x-h)^2 = a^2 \left(1 - \frac{(y-k)^2}{b^2}\right)$
   $x-h = \pm \sqrt{a^2 \left(1 - \frac{(y-k)^2}{b^2}\right)}$
   $x = h \pm a \sqrt{1 - \frac{(y-k)^2}{b^2}}$

   • To graph the Right Half: Use the positive square root:
     $x = h + a \sqrt{1 - \frac{(y-k)^2}{b^2}}$
   • To graph the Left Half: Use the negative square root:
     $x = h - a \sqrt{1 - \frac{(y-k)^2}{b^2}}$

Practical Application

Here's a table summarizing the equations for different semi-ellipse orientations:

Graphing Process:

1. Plot the Center: Mark the point (h, k).
2. Determine Endpoints:
   • For horizontal major axis (a): The full ellipse extends a units horizontally from the center.
   • For vertical major axis (b): The full ellipse extends b units vertically from the center.
   • For a semi-ellipse, these define the boundaries of your half. For instance, for a top/bottom semi-ellipse, your x-values will range from h-a to h+a.
3. Plot Key Points: Use the specific semi-ellipse equation to calculate several points within the appropriate range of x or y values. For example, for a top semi-ellipse, calculate y values for various x values between h-a and h+a.
4. Draw the Curve: Connect the plotted points with a smooth curve to form the semi-ellipse. Ensure the curve starts and ends at the major axis points (for top/bottom halves) or minor axis points (for left/right halves), and respects the boundary defined by the center.

By following these steps, you can accurately graph any semi-ellipse based on its desired orientation and the parameters of its corresponding full ellipse.