For $$ y^{2}-\frac{(x-4)^{2}}{20}=1 $$, $$ \begin{array}{ll} ext { (a) Identify the center. } & ext { (b) Identify the vertices. } \ ext { (c) Identify the foci. } & ext { (d) Write equations for the asymptotes. } \ ext { (e) Graph the hyperbola. }\end{array} $$ Give exact answers. Express numbers in simplest form using integers, fractions, or radicals.

Question

For $$ y^{2}-\frac{(x-4)^{2}}{20}=1 $$, $$ \begin{array}{ll}	ext { (a) Identify the center. } & 	ext { (b) Identify the vertices. } \ 	ext { (c) Identify the foci. } & 	ext { (d) Write equations for the asymptotes. } \ 	ext { (e) Graph the hyperbola. }\end{array} $$ Give exact answers. Express numbers in simplest form using integers, fractions, or radicals.

UpStudy AI · Accepted Answer

The given equation is

$$
y^{2} - \frac{(x-4)^{2}}{20} = 1
$$

This is the standard form of a hyperbola centered at $$(h, k)$$ with a vertical transverse axis. Let's identify the components step by step.

### (a) Identify the center.
The center of the hyperbola is given by the point $$(h, k)$$. From the equation, we can see that:

- $$h = 4$$
- $$k = 0$$

Thus, the center is $$(4, 0)$$.

### (b) Identify the vertices.
For a hyperbola of the form

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

the vertices are located at $$(h, k \pm a)$$. Here, $$a^{2} = 1$$, so $$a = 1$$. Therefore, the vertices are:

- $$ (4, 0 + 1) = (4, 1) $$
- $$ (4, 0 - 1) = (4, -1) $$

### (c) Identify the foci.
The foci of the hyperbola are located at $$(h, k \pm c)$$, where $$c = \sqrt{a^{2} + b^{2}}$$. We have:

- $$b^{2} = 20$$ so $$b = \sqrt{20} = 2\sqrt{5}$$
- $$c = \sqrt{1 + 20} = \sqrt{21}$$

Thus, the foci are:

- $$ (4, 0 + \sqrt{21}) = (4, \sqrt{21}) $$
- $$ (4, 0 - \sqrt{21}) = (4, -\sqrt{21}) $$

### (d) Write equations for the asymptotes.
The equations for the asymptotes of a hyperbola in this form are given by:

$$
y - k = \pm \frac{a}{b}(x - h)
$$

Substituting the values we have:

- $$a = 1$$
- $$b = \sqrt{20}$$

The equations for the asymptotes are:

$$
y - 0 = \pm \frac{1}{\sqrt{20}}(x - 4)
$$

This simplifies to:

$$
y = \pm \frac{1}{\sqrt{20}}(x - 4)
$$

### (e) Graph the hyperbola.
To graph the hyperbola, we will plot the center, vertices, foci, and asymptotes. The hyperbola opens upwards and downwards, with the vertices at $$(4, 1)$$ and $$(4, -1)$$.

Now, let's summarize the results:

- **Center:** $$(4, 0)$$
- **Vertices:** $$(4, 1)$$ and $$(4, -1)$$
- **Foci:** $$(4, \sqrt{21})$$ and $$(4, -\sqrt{21})$$
- **Asymptotes:** $$y = \pm \frac{1}{\sqrt{20}}(x - 4)$$

Next, I will graph the hyperbola.
It seems there was an issue with graphing the asymptotes directly. However, I can provide the equations for the asymptotes in a clearer format:

The equations for the asymptotes are:

1. $$ y = \frac{1}{\sqrt{20}}(x - 4) $$
2. $$ y = -\frac{1}{\sqrt{20}}(x - 4) $$

These can be simplified further:

1. $$ y = \frac{1}{2\sqrt{5}}(x - 4) $$
2. $$ y = -\frac{1}{2\sqrt{5}}(x - 4) $$

### Summary of Results:
- **Center:** $$(4, 0)$$
- **Vertices:** $$(4, 1)$$ and $$(4, -1)$$
- **Foci:** $$(4, \sqrt{21})$$ and $$(4, -\sqrt{21})$$
- **Asymptotes:** 
  - $$ y = \frac{1}{2\sqrt{5}}(x - 4) $$
  - $$ y = -\frac{1}{2\sqrt{5}}(x - 4) $$

### Graphing the Hyperbola:
To graph the hyperbola, you would plot the center at $$(4, 0)$$, the vertices at $$(4, 1)$$ and $$(4, -1)$$, the foci at $$(4, \sqrt{21})$$ and $$(4, -\sqrt{21})$$, and draw the asymptotes. The hyperbola will open upwards and downwards from the vertices.

If you need a visual representation, please let me know, and I can assist further!
- **Center:** $$(4, 0)$$ - **Vertices:** $$(4, 1)$$ and $$(4, -1)$$ - **Foci:** $$(4, \sqrt{21})$$ and $$(4, -\sqrt{21})$$ - **Asymptotes:** - $$ y = \frac{1}{2\sqrt{5}}(x - 4) $$ - $$ y = -\frac{1}{2\sqrt{5}}(x - 4) $$ To graph the hyperbola, plot the center at $$(4, 0)$$, vertices at $$(4, 1)$$ and $$(4, -1)$$, foci at $$(4, \sqrt{21})$$ and $$(4, -\sqrt{21})$$, and draw the asymptotes $$ y = \frac{1}{2\sqrt{5}}(x - 4) $$ and $$ y = -\frac{1}{2\sqrt{5}}(x - 4) $$. The hyperbola opens upwards and downwards from the vertices.

For ,

Give exact answers. Express numbers in simplest form using integers, fractions, or radicals.

Upstudy AI Solution

Answer

Solution

(a) Identify the center.

(b) Identify the vertices.

© Identify the foci.

(d) Write equations for the asymptotes.

(e) Graph the hyperbola.

Summary of Results:

Graphing the Hyperbola:

Mind Expander

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