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The image contains two hyperbolic equations, with a task asking to find the coordinates of the center, foci, vertices, and co-vertices of each hyperbola, followed by graphing them.

Here are the equations:

1. \(\frac{(x+1)^2}{4} - \frac{(y-1)^2}{9} = 1\)
2. \(\frac{x^2}{9} - \frac{y^2}{16} = 1\)

Let's go through each equation:

### First Equation:
\[
\frac{(x+1)^2}{4} - \frac{(y-1)^2}{9} = 1
\]

This is the standard form of a horizontal hyperbola:
\[
\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1
\]
- Center: \((h, k) = (-1, 1)\)
- \(a^2 = 4\), so \(a = 2\)
- \(b^2 = 9\), so \(b = 3\)
- Vertices: These lie along the x-axis from the center, at \((-1 - 2, 1)\) and \((-1 + 2, 1)\), so the vertices are \((-3, 1)\) and \((1, 1)\).
- Co-vertices: These lie along the y-axis from the center, at \((-1, 1 - 3)\) and \((-1, 1 + 3)\), so the co-vertices are \((-1, -2)\) and \((-1, 4)\).
- Foci: Use the equation \(c^2 = a^2 + b^2\), so \(c^2 = 4 + 9 = 13\), thus \(c = \sqrt{13} \approx 3.6\). The foci are located at \((-1 - \sqrt{13}, 1)\) and \((-1 + \sqrt{13}, 1)\), which are approximately \((-4.6, 1)\) and \((2.6, 1)\).

### Second Equation:
\[
\frac{x^2}{9} - \frac{y^2}{16} = 1
\]

This is also a horizontal hyperbola:
\[
\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1
\]
- Center: \((0, 0)\)
- \(a^2 = 9\), so \(a = 3\)
- \(b^2 = 16\), so \(b = 4\)
- Vertices: These lie along the x-axis from the center, at \((0 - 3, 0)\) and \((0 + 3, 0)\), so the vertices are \((-3, 0)\) and \((3, 0)\).
- Co-vertices: These lie along the y-axis from the center, at \((0, 0 - 4)\) and \((0, 0 + 4)\), so the co-vertices are \((0, -4)\) and \((0, 4)\).
- Foci: Again, \(c^2 = a^2 + b^2\), so \(c^2 = 9 + 16 = 25\), thus \(c = 5\). The foci are located at \((0 - 5, 0)\) and \((0 + 5, 0)\), which are \((-5, 0)\) and \((5, 0)\).

Would you like further details on graphing or other questions? Here are some related concepts to consider:

1. How do you derive the equations for foci in a hyperbola?
2. What is the difference between horizontal and vertical hyperbolas?
3. How do you calculate the eccentricity of a hyperbola?
4. What is the significance of the transverse and conjugate axes in hyperbolas?
5. Can you compare hyperbolas to ellipses in terms of their geometric properties?

**Tip**: When graphing hyperbolas, always start with the center and plot the vertices, then use the asymptotes to guide the shape of the curves.

Find the coordinates of the center, foci, vertices, and co-vertices for the hyperbola, then graph it.

Learn how to find the coordinates of the center, foci, vertices, and co-vertices of a hyperbola and graph it. This problem explores key concepts in hyperbolas, using formulas for graphing and calculating the foci and vertices. Suitable for high school students in grades 10-12.

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