Hyperbolas NOT at the Origin: Videos & Practice Problems

Hyperbolas can be centered at points other than the origin, altering their equations. The standard form for a hyperbola centered at (h, k) is given by $(x-h)^2/a^2-(y-k)^2/b^2=1$. To graph, identify the center, vertices, and asymptotes, and use the relationship $c^2=a^2+b^2$ to find foci. Understanding these transformations is crucial for mastering conic sections.

Graph Hyperbolas NOT at the Origin

Conic sections are a fundamental topic in geometry, encompassing various shapes such as circles, ellipses, parabolas, and hyperbolas. This summary focuses on hyperbolas, particularly those that are not centered at the origin. Understanding how to graph and analyze these hyperbolas involves recognizing the changes in their equations and applying transformation concepts.

The standard equation for a hyperbola centered at the origin is given by:

For a horizontal hyperbola: \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\)

For a vertical hyperbola: \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\)

When the hyperbola is shifted to a new center \((h, k)\), the equations incorporate these values, where \(h\) represents the horizontal shift and \(k\) the vertical shift. This transformation mirrors the adjustments made in the equations of ellipses and other functions.

To graph a hyperbola centered at a point other than the origin, follow these steps:

1. Identify whether the hyperbola is horizontal or vertical based on the arrangement of the terms in the equation. A vertical hyperbola will have the positive term associated with \(y\) first.
2. Determine the center \((h, k)\) from the equation. The values of \(h\) and \(k\) are derived from the constants subtracted from \(x\) and \(y\) respectively.
3. Find the vertices of the hyperbola. For a vertical hyperbola, the vertices are located at \((h, k \pm a)\), where \(a\) is the square root of the first positive term in the equation.
4. Calculate the \(b\) points, which are located horizontally from the center at \((h \pm b, k)\), where \(b\) is the square root of the second term in the equation.
5. Draw a box connecting the vertices and \(b\) points to assist in finding the asymptotes. The asymptotes are represented by lines drawn through the corners of this box.
6. Sketch the branches of the hyperbola, which approach the asymptotes but never intersect them.
7. Finally, determine the foci of the hyperbola using the relationship \(c^2 = a^2 + b^2\). The foci are located at \((h, k \pm c)\), where \(c\) is the square root of the sum of the squares of \(a\) and \(b\).

By following these steps, one can effectively graph a hyperbola that is not centered at the origin, ensuring a clear understanding of its properties and characteristics. This process highlights the interconnectedness of conic sections and the importance of transformations in geometry.

Describe the hyperbola $\frac{\left(x+2\right)^2}{9}-\frac{\left(y-4\right)^2}{16}=1$.

• A. This is a vertical hyperbola centered at $\left(-2,4\right)$ with vertices at $\left(4,2\right),\left(4,-6\right)$ and foci at $\left(4,4\right),\left(4,-8\right)$.
• B. This is a vertical hyperbola centered at $\left(2,-4\right)$ with vertices at $\left(4,1\right),\left(4,-5\right)$ and foci at $\left(4,3\right),\left(4,-7\right)$.
• C. This is a horizontal hyperbola centered at $\left(-2,4\right)$ with vertices at $\left(2,4\right),\left(-6,4\right)$ and foci at $\left(4,4\right),\left(-8,4\right)$.
• D. This is a horizontal hyperbola centered at $\left(-2,4\right)$ with vertices at $\left(1,4\right),\left(-5,4\right)$ and foci at $\left(3,4\right),\left(-7,4\right)$.

Answer: (D)

Describe the hyperbola $y^2-\frac{\left(x-1\right)^2}{4}=1$.

• A. This is a vertical hyperbola centered at $\left(1,0\right)$ with vertices at $\left(1,1\right)$, $\left(1,-1\right)$ and foci at $\left(1,\sqrt5\right)$, $\left(1,-\sqrt5\right)$.
• B. This is a vertical hyperbola centered at $\left(1,0\right)$ with vertices at $\left(1,2\right),\left(1,-2\right)$ and foci at $\left(1,1\right),\left(1,-1\right)$.
• C. This is a horizontal hyperbola centered at $\left(-1,0\right)$ with vertices at $\left(0,0\right),\left(-2,0\right)$ and foci at $\left(\sqrt5-1,0\right),\left(-\sqrt5-1,0\right)$.
• D. This is a horizontal hyperbola centered at $\left(1,0\right)$ with vertices at $\left(0,0\right),\left(-2,0\right)$ and foci at $\left(1,-\sqrt5\right),\left(1,-\sqrt5\right)$.

Answer: (A)

Here’s what students ask on this topic:

What is the standard form equation of a hyperbola not centered at the origin?

The standard form equation of a hyperbola not centered at the origin is given by:

$\frac{{(x-h)}^2}{a^2}-\frac{{(y-k)}^2}{b^2}=1$

Here, (h, k) is the center of the hyperbola, a is the distance from the center to the vertices along the transverse axis, and b is the distance from the center to the co-vertices along the conjugate axis.

How do you determine the vertices of a hyperbola not centered at the origin?

To determine the vertices of a hyperbola not centered at the origin, first identify the center (h, k). For a horizontal hyperbola, the vertices are located at (h ± a, k). For a vertical hyperbola, the vertices are at (h, k ± a). Here, a is the distance from the center to each vertex, which can be found from the equation of the hyperbola where a² is the denominator of the positive term.

How do you find the asymptotes of a hyperbola not centered at the origin?

The asymptotes of a hyperbola not centered at the origin can be found using the slopes derived from the equation. For a hyperbola centered at (h, k), the equations of the asymptotes are:

For a horizontal hyperbola: y = k ± (b/a)(x - h)

For a vertical hyperbola: y = k ± (a/b)(x - h)

Here, a and b are the distances from the center to the vertices and co-vertices, respectively.

How do you find the foci of a hyperbola not centered at the origin?

To find the foci of a hyperbola not centered at the origin, first determine the center (h, k) and the values of a and b from the equation. Calculate c using the relationship c² = a² + b². For a horizontal hyperbola, the foci are at (h ± c, k). For a vertical hyperbola, the foci are at (h, k ± c).

How do you graph a hyperbola not centered at the origin?

To graph a hyperbola not centered at the origin, follow these steps:

1. Identify the center (h, k) from the equation.
2. Determine whether the hyperbola is horizontal or vertical by examining the terms.
3. Find the vertices using the values of a and the center.
4. Calculate the b points to determine the dimensions of the rectangle that helps in drawing the asymptotes.
5. Draw the asymptotes through the corners of the rectangle.
6. Sketch the hyperbola branches approaching the asymptotes from the vertices.