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Let's go through each quadratic function, finding the axis of symmetry, vertex, y-intercept, maximum or minimum point, and the domain and range.

### Function 1: \( f(x) = 3x^2 - 6x + 1 \)

1. **Axis of Symmetry**: The axis of symmetry for a quadratic function \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \).
   - Here, \( a = 3 \) and \( b = -6 \).
   - So, \( x = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1 \).

2. **Vertex**: Substitute \( x = 1 \) into the function to find the y-coordinate of the vertex.
   - \( f(1) = 3(1)^2 - 6(1) + 1 = 3 - 6 + 1 = -2 \).
   - Vertex: \( (1, -2) \).

3. **Y-intercept**: The y-intercept is the value of \( f(x) \) when \( x = 0 \).
   - \( f(0) = 3(0)^2 - 6(0) + 1 = 1 \).
   - Y-intercept: \( (0, 1) \).

4. **Maximum or Minimum**: Since \( a = 3 > 0 \), this parabola opens upwards, so it has a **minimum** at the vertex.
   - Minimum point: \( (1, -2) \).

5. **Domain and Range**:
   - Domain: All real numbers, \( (-\infty, \infty) \).
   - Range: \( [-2, \infty) \), since the minimum value is -2.

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### Function 2: \( f(x) = -x^2 - 2x + 3 \)

1. **Axis of Symmetry**:
   - \( a = -1 \) and \( b = -2 \).
   - \( x = -\frac{-2}{2 \cdot -1} = \frac{2}{-2} = -1 \).

2. **Vertex**:
   - Substitute \( x = -1 \) into the function.
   - \( f(-1) = -(-1)^2 - 2(-1) + 3 = -1 + 2 + 3 = 4 \).
   - Vertex: \( (-1, 4) \).

3. **Y-intercept**:
   - \( f(0) = -0^2 - 2(0) + 3 = 3 \).
   - Y-intercept: \( (0, 3) \).

4. **Maximum or Minimum**: Since \( a = -1 < 0 \), this parabola opens downwards, so it has a **maximum** at the vertex.
   - Maximum point: \( (-1, 4) \).

5. **Domain and Range**:
   - Domain: All real numbers, \( (-\infty, \infty) \).
   - Range: \( (-\infty, 4] \), since the maximum value is 4.

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### Function 3: \( f(x) = x^2 - 4 \)

1. **Axis of Symmetry**:
   - \( a = 1 \) and \( b = 0 \).
   - \( x = -\frac{0}{2 \cdot 1} = 0 \).

2. **Vertex**:
   - Substitute \( x = 0 \) into the function.
   - \( f(0) = (0)^2 - 4 = -4 \).
   - Vertex: \( (0, -4) \).

3. **Y-intercept**:
   - \( f(0) = -4 \).
   - Y-intercept: \( (0, -4) \).

4. **Maximum or Minimum**: Since \( a = 1 > 0 \), this parabola opens upwards, so it has a **minimum** at the vertex.
   - Minimum point: \( (0, -4) \).

5. **Domain and Range**:
   - Domain: All real numbers, \( (-\infty, \infty) \).
   - Range: \( [-4, \infty) \), since the minimum value is -4.

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Would you like more detailed explanations for any of these steps, or do you have further questions?

### Additional Related Questions

1. How do you determine if a quadratic function has a maximum or minimum point?
2. How can you find the x-intercepts of a quadratic function?
3. What is the significance of the axis of symmetry in a quadratic function?
4. How can transformations affect the graph of a quadratic function?
5. How do the values of \( a \), \( b \), and \( c \) in \( ax^2 + bx + c \) affect the shape and position of the graph?

### Tip
Remember that the vertex of a quadratic function \( ax^2 + bx + c \) provides the point where the function reaches its maximum or minimum, depending on whether the parabola opens up or down.

Graph each quadratic function, find the axis of symmetry, vertex, y-intercept, maximum or minimum point, and then state the domain and range.

Learn how to graph quadratic functions, identify the axis of symmetry, vertex, y-intercept, and find maximum or minimum points. This includes functions like 3x^2 - 6x + 1, -x^2 - 2x + 3, and x^2 - 4, suitable for students in Grades 9-11.

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