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The image contains several quadratic equations that need to be rewritten in standard form, and you are asked to identify the graph of each.

Standard form for a quadratic equation is typically written as:

\[
Ax^2 + Bx + C = 0
\]

Let's rewrite each equation in standard form and analyze them:

1. **Original equation**: 
   \[
   -x^2 + 10x + y = 32 - y
   \]
   Rearranging:
   \[
   -x^2 + 10x + y + y = 32 \quad \Rightarrow \quad -x^2 + 10x + 2y = 32
   \]
   Now, subtract 32 from both sides to get it in standard form:
   \[
   -x^2 + 10x + 2y - 32 = 0
   \]

2. **Original equation**:
   \[
   x^2 + 4x - 2x - y = 6
   \]
   Combine like terms:
   \[
   x^2 + 2x - y = 6
   \]
   Now, subtract 6 from both sides:
   \[
   x^2 + 2x - y - 6 = 0
   \]

3. **Original equation**:
   \[
   x^2 - x - 2x - 10y + 4 = 6
   \]
   Combine like terms:
   \[
   x^2 - 3x - 10y + 4 = 6
   \]
   Subtract 6:
   \[
   x^2 - 3x - 10y + 4 - 6 = 0 \quad \Rightarrow \quad x^2 - 3x - 10y - 2 = 0
   \]

4. **Original equation**:
   \[
   x^2 - 8x + y + 2 = 77
   \]
   Subtract 77 from both sides:
   \[
   x^2 - 8x + y + 2 - 77 = 0 \quad \Rightarrow \quad x^2 - 8x + y - 75 = 0
   \]

5. **Original equation**:
   \[
   3x^2 + 7x + 4x + 3 = 60
   \]
   Combine like terms:
   \[
   3x^2 + 11x + 3 = 60
   \]
   Subtract 60:
   \[
   3x^2 + 11x + 3 - 60 = 0 \quad \Rightarrow \quad 3x^2 + 11x - 57 = 0
   \]

These are the quadratic equations in standard form. To identify their graphs, we'd need to explore the characteristics of each, such as the shape of the parabola (if applicable) and the vertex.

Would you like more details on solving any specific part of this, or help with graphing these equations?

Here are some related questions:
1. How do you determine the direction a parabola opens?
2. What is the significance of the vertex in a quadratic equation?
3. How do you find the axis of symmetry of a parabola?
4. What method can you use to solve quadratic equations?
5. How do you convert a quadratic equation into vertex form?

**Tip**: For graphing parabolas, the coefficient of \(x^2\) determines whether the parabola opens upwards (\(A > 0\)) or downwards (\(A < 0\)).

Rewrite the given equations in standard form and identify the graphs.

Learn how to rewrite quadratic equations in standard form and identify their graphs. This exercise covers important algebraic concepts, including transforming equations into the form Ax^2 + Bx + C = 0, and determining the corresponding graph characteristics. Suitable for students in Grades 9-12.

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