Understanding the Equation for a Vertical Hyperbola: Shape, Orientation, and Graphing

Vertical Hyperbola Formula

The equation for a vertical hyperbola can be written in the form:

(x – h)^2 / a^2 – (y – k)^2 / b^2 = 1

The center of the hyperbola is represented by the point (h, k)

The equation for a vertical hyperbola can be written in the form:

(x – h)^2 / a^2 – (y – k)^2 / b^2 = 1

The center of the hyperbola is represented by the point (h, k). The values a and b determine the shape and orientation of the hyperbola.

Here’s how you can understand these variables:

1. The value of “h” represents the horizontal shift of the center of the hyperbola. If h > 0, the center is shifted to the right. If h < 0, the center is shifted to the left. If h = 0, the center is at the origin. 2. The value of "k" represents the vertical shift of the center of the hyperbola. If k > 0, the center is shifted upwards. If k < 0, the center is shifted downwards. If k = 0, the center is at the origin. 3. "a" determines the distance between the center and the vertices along the x-axis. The value of "a" is always positive. 4. "b" determines the distance between the center and the vertices along the y-axis. The value of "b" is always positive. To graph a vertical hyperbola using the equation above, you can follow these steps: 1. Determine the coordinates of the center (h, k). 2. Determine the distances a and b from the center to the vertices along the x-axis and y-axis, respectively. 3. Plot the center point on the graph. 4. Draw the asymptotes, which are the dashed lines that pass through the center. The equations for the asymptotes are given by y = k ± (b/a)(x - h). 5. Plot the vertices, which are the points (h, k ± a) vertically above and below the center. 6. Sketch the hyperbola, making sure it is vertically stretched by a factor of b/a from the center to the vertices. The curve should intersect the asymptotes at the vertices. Remember, the equation for a vertical hyperbola is just one particular form of the equation. There are other forms, such as the general form and the standard form, which may be more useful or suitable depending on the context or problem you are working on.

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