To sketch the graph of the function \( y = 2(x-1)^2 - 5 \), let's break it down step by step:

### 1. **Identify the Base Function:**
The base function is \( y = (x-1)^2 \), which is a parabola. The parabola opens upwards because the coefficient of \( x^2 \) is positive.

### 2. **Vertical Stretch:**
The coefficient 2 outside the squared term stretches the parabola vertically. This means the parabola will be narrower than the standard parabola \( y = x^2 \).

### 3. **Horizontal Shift:**
The term \( (x-1)^2 \) represents a shift of the parabola 1 unit to the right. This moves the vertex of the parabola from \( (0, 0) \) to \( (1, 0) \).

### 4. **Vertical Shift:**
The term \( -5 \) translates the entire graph 5 units downwards. So the new vertex will be at \( (1, -5) \).

### 5. **Graph Characteristics:**
- **Vertex:** The vertex of the parabola is at \( (1, -5) \).
- **Axis of symmetry:** The axis of symmetry is the vertical line \( x = 1 \).
- **Direction:** The parabola opens upwards because the leading coefficient of the squared term is positive.
- **Width:** The parabola is vertically stretched (narrower) compared to the standard parabola \( y = x^2 \).

### Sketch Description:
1. Start by plotting the vertex at \( (1, -5) \).
2. The axis of symmetry is a vertical line through \( x = 1 \).
3. The parabola opens upwards and is narrower due to the stretch factor of 2.
4. Plot a few points on either side of the vertex for accuracy, for example at \( x = 0 \) and \( x = 2 \), where \( y = 2(0-1)^2 - 5 = -3 \), so points \( (0, -3) \) and \( (2, -3) \) will be on the graph.

Would you like to see a visual sketch of this graph?

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### 5 Related Questions:
1. How do you find the vertex of any parabola in vertex form?
2. What does a vertical stretch do to the shape of a graph?
3. How does shifting a parabola left or right affect its graph?
4. How can you determine the direction a parabola opens?
5. What is the significance of the axis of symmetry in parabolas?

**Tip:** The vertex form of a parabola, \( y = a(x-h)^2 + k \), gives you the vertex \( (h, k) \) directly!

Learn how to graph the quadratic function y=2(x-1)^2-5, with detailed steps on identifying vertical stretches, shifts, and vertex placement. This problem explores algebraic concepts like vertex form and graph transformations, ideal for students in Grades 9-10.

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