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Sagot :
To sketch the graph of the function [tex]\(y = 2(x - 2)^2\)[/tex] and identify its axis of symmetry, let's go through the following detailed steps.
### Step 1: Understand the Equation
The given function is in the form [tex]\(y = a(x - h)^2 + k\)[/tex], which represents a parabola. For the equation [tex]\(y = 2(x - 2)^2\)[/tex]:
- [tex]\(a = 2\)[/tex]
- [tex]\(h = 2\)[/tex]
- [tex]\(k = 0\)[/tex] (since there is no constant term outside the squared term)
### Step 2: Identify the Vertex
The vertex of the parabola given by [tex]\(y = a(x - h)^2 + k\)[/tex] is the point [tex]\((h, k)\)[/tex]. For our equation, the vertex is:
- [tex]\(h = 2\)[/tex]
- [tex]\(k = 0\)[/tex]
So the vertex is [tex]\((2, 0)\)[/tex].
### Step 3: Axis of Symmetry
The axis of symmetry of a parabola in the form [tex]\(y = a(x - h)^2 + k\)[/tex] is the vertical line that passes through the vertex. This line is given by:
[tex]\[ x = h \][/tex]
For our equation [tex]\(y = 2(x - 2)^2\)[/tex], the axis of symmetry is at [tex]\(x = 2\)[/tex].
### Step 4: Plot the Vertex and Axis of Symmetry
On a coordinate plane:
1. Plot the vertex at [tex]\((2, 0)\)[/tex].
2. Draw a dotted vertical line through [tex]\(x = 2\)[/tex], representing the axis of symmetry.
### Step 5: Plot Additional Points
To get a more accurate graph, plot additional points on either side of the vertex and reflect them across the axis of symmetry.
For example:
- Choose [tex]\(x = 1\)[/tex]:
[tex]\[ y = 2(1 - 2)^2 = 2(1)^2 = 2 \][/tex]
So the point [tex]\((1, 2)\)[/tex] is on the graph.
- Reflect [tex]\((1, 2)\)[/tex] across the axis of symmetry ([tex]\(x = 2\)[/tex]):
The reflected point is [tex]\((3, 2)\)[/tex].
Similarly, for another point:
- Choose [tex]\(x = 3\)[/tex]:
[tex]\[ y = 2(3 - 2)^2 = 2(1)^2 = 2 \][/tex]
So the point [tex]\((3, 2)\)[/tex] is on the graph.
- Reflect [tex]\((3, 2)\)[/tex] across the axis of symmetry ([tex]\(x = 2\)[/tex]):
Since this point is already on the other side of the axis, it stays [tex]\((1, 2)\)[/tex].
### Step 6: Sketch the Graph
Draw a smooth curve through these points, making sure the parabola opens upwards (since [tex]\(a > 0\)[/tex]).
### Conclusion
The graph of [tex]\(y = 2(x - 2)^2\)[/tex] is a parabola with its vertex at [tex]\((2, 0)\)[/tex] and opens upwards. The axis of symmetry for this parabola is the vertical line [tex]\(x = 2\)[/tex]. This aligns with the provided numerical answer.
So, the correct answer to the question is:
[tex]\[ x = 2 \][/tex]
### Step 1: Understand the Equation
The given function is in the form [tex]\(y = a(x - h)^2 + k\)[/tex], which represents a parabola. For the equation [tex]\(y = 2(x - 2)^2\)[/tex]:
- [tex]\(a = 2\)[/tex]
- [tex]\(h = 2\)[/tex]
- [tex]\(k = 0\)[/tex] (since there is no constant term outside the squared term)
### Step 2: Identify the Vertex
The vertex of the parabola given by [tex]\(y = a(x - h)^2 + k\)[/tex] is the point [tex]\((h, k)\)[/tex]. For our equation, the vertex is:
- [tex]\(h = 2\)[/tex]
- [tex]\(k = 0\)[/tex]
So the vertex is [tex]\((2, 0)\)[/tex].
### Step 3: Axis of Symmetry
The axis of symmetry of a parabola in the form [tex]\(y = a(x - h)^2 + k\)[/tex] is the vertical line that passes through the vertex. This line is given by:
[tex]\[ x = h \][/tex]
For our equation [tex]\(y = 2(x - 2)^2\)[/tex], the axis of symmetry is at [tex]\(x = 2\)[/tex].
### Step 4: Plot the Vertex and Axis of Symmetry
On a coordinate plane:
1. Plot the vertex at [tex]\((2, 0)\)[/tex].
2. Draw a dotted vertical line through [tex]\(x = 2\)[/tex], representing the axis of symmetry.
### Step 5: Plot Additional Points
To get a more accurate graph, plot additional points on either side of the vertex and reflect them across the axis of symmetry.
For example:
- Choose [tex]\(x = 1\)[/tex]:
[tex]\[ y = 2(1 - 2)^2 = 2(1)^2 = 2 \][/tex]
So the point [tex]\((1, 2)\)[/tex] is on the graph.
- Reflect [tex]\((1, 2)\)[/tex] across the axis of symmetry ([tex]\(x = 2\)[/tex]):
The reflected point is [tex]\((3, 2)\)[/tex].
Similarly, for another point:
- Choose [tex]\(x = 3\)[/tex]:
[tex]\[ y = 2(3 - 2)^2 = 2(1)^2 = 2 \][/tex]
So the point [tex]\((3, 2)\)[/tex] is on the graph.
- Reflect [tex]\((3, 2)\)[/tex] across the axis of symmetry ([tex]\(x = 2\)[/tex]):
Since this point is already on the other side of the axis, it stays [tex]\((1, 2)\)[/tex].
### Step 6: Sketch the Graph
Draw a smooth curve through these points, making sure the parabola opens upwards (since [tex]\(a > 0\)[/tex]).
### Conclusion
The graph of [tex]\(y = 2(x - 2)^2\)[/tex] is a parabola with its vertex at [tex]\((2, 0)\)[/tex] and opens upwards. The axis of symmetry for this parabola is the vertical line [tex]\(x = 2\)[/tex]. This aligns with the provided numerical answer.
So, the correct answer to the question is:
[tex]\[ x = 2 \][/tex]
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