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!