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The image contains several mathematical problems, focusing on graphing functions and determining the center, vertices, foci, and graph of conic sections. I will interpret the image and provide guidance for solving each problem step by step.

### Functions to graph:
1. \( f(x) = y^3 + 2x - 3 \)
2. \( f(x) = 2x^2 + 1 \)
3. \( f(x) = 32x \)

For these functions, you are asked to:
1. Graph each equation using a Cartesian plane.
2. Create a table of values to help plot points.

### Conic sections (circle and ellipse):
1. **Circle**: \( x^2 + y^2 + 2x + 4y - 4 = 0 \)
2. **Ellipse**: \( 4x^2 + y^2 - 4x + 2y - 48 = 0 \)
   
You need to:
- Determine the center, radius (for the circle), and vertices, foci, and graph (for the ellipse).
- Use a Cartesian plane to graph them, creating a table of points.

### Additional task on ellipse:
- Solve and graph \( x^2 + 2y^2 - 2x + 8y + 5 = 0 \)

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Let's break down the solutions for you:

### 1. Graphing functions:
   - For each function, substitute a few values of \(x\) into the function to generate corresponding values of \(y\).
   - Plot these points on the graph and connect them smoothly.

### 2. Conic section - Circle \( x^2 + y^2 + 2x + 4y - 4 = 0 \):
   - Complete the square for both \(x\) and \(y\) terms.
   - Rewrite in standard form \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius.

### 3. Conic section - Ellipse \( 4x^2 + y^2 - 4x + 2y - 48 = 0 \):
   - Again, complete the square for \(x\) and \(y\).
   - Rewrite in standard form \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \((

Graph the following functions and conic sections, and solve for the center, vertices, and foci of the conic sections.

Learn how to graph functions and conic sections, including circles and ellipses. This problem covers key concepts like completing the square, finding the center and radius of a circle, and determining the vertices and foci of an ellipse. Suitable for students in Grades 10-12.

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