graph each hyberbola by hand give the domain and range  (x^(2))/(16 ) -  (y^(2))/(9) =1

To graph the hyperbola given by the equation:

\[ \frac{x^2}{16} - \frac{y^2}{9} = 1 \]

we need to follow these steps:

1. **Identify the center, vertices, and asymptotes:**
   - This hyperbola is centered at the origin \((0,0)\) because there are no \(h\) and \(k\) values in the equation.
   - The equation is in the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), where \(a^2 = 16\) and \(b^2 = 9\). Thus, \(a = 4\) and \(b = 3\).

2. **Vertices:**
   - For hyperbolas of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the vertices are at \((\pm a, 0)\).
   - Therefore, the vertices are \((\pm 4, 0)\), which are \((4, 0)\) and \((-4, 0)\).

3. **Asymptotes:**
   - The equations of the asymptotes for hyperbolas in this form are \(y = \pm \frac{b}{a} x\).
   - Substituting \(a = 4\) and \(b = 3\), the asymptote equations are \(y = \pm \frac{3}{4} x\).

4. **Graphing the Hyperbola:**
   - Draw the center at the origin \((0,0)\).
   - Plot the vertices at \((4,0)\) and \((-4,0)\).
   - Draw the asymptotes \(y = \frac{3}{4}x\) and \(y = -\frac{3}{4}x\). These lines pass through the origin and have slopes of \(\pm \frac{3}{4}\).
   - Sketch the hyperbola opening left and right, approaching but never touching the asymptotes.

5. **Domain and Range:**
   - For the hyperbola \(\frac{x^2}{16} - \frac{y^2}{9} = 1\):
     - The domain (set of all possible \(x\)-values) is \(x \in (-\infty, -4] \cup [4, \infty)\).
     - The range (set of all possible \(y\)-values) is \(y \in (-\infty, \infty)\).

Here's the graph of the hyperbola:

1. Draw a set of coordinate axes.
2. Mark the center at \((0,0)\).
3. Plot the vertices at \((4,0)\) and \((-4,0)\).
4. Draw the asymptotes: \(y = \frac{3}{4}x\) and \(y = -\frac{3}{4}x\).
5. Sketch the branches of the hyperbola, opening to the left and right, approaching the asymptotes.

Would you like more details or have any questions?

Here are some related questions:

1. How do you derive the standard form of a hyperbola from its general equation?
2. What are the differences between the equations of horizontal and vertical hyperbolas?
3. How do you determine the foci of a hyperbola?
4. How do you find the lengths of the transverse and conjugate axes of a hyperbola?
5. How do you determine the eccentricity of a hyperbola?
6. How do the asymptotes help in sketching the graph of a hyperbola?
7. How can you verify if a given equation represents a hyperbola?
8. What are the properties of hyperbolas that differentiate them from other conic sections?

**Tip:** When graphing a hyperbola, always start by identifying the center, vertices, and asymptotes. These key elements guide the accurate sketching of the hyperbola's shape.

Learn how to graph the hyperbola \(\frac{x^2}{16} - \frac{y^2}{9} = 1\) with a detailed step-by-step solution. Includes identifying the center, vertices, asymptotes, domain, and range. Suitable for advanced high school students.

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