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To graph the function [tex]\( g(x) = -\frac{3}{2}(x-2)^2 \)[/tex], follow these detailed steps:
1. Understand the Function Format:
The given function is a quadratic function in vertex form: [tex]\( g(x) = a(x-h)^2 + k \)[/tex], where [tex]\( a = -\frac{3}{2} \)[/tex], [tex]\( h = 2 \)[/tex], and [tex]\( k = 0 \)[/tex]. This represents a parabola opening downwards (since [tex]\( a < 0 \)[/tex]), with its vertex at the point [tex]\((h, k) = (2, 0)\)[/tex].
2. Identify Key Points:
- Vertex: The vertex of the parabola is at [tex]\((2,0)\)[/tex].
- Axis of Symmetry: The line [tex]\( x = 2 \)[/tex] is the axis of symmetry.
- Direction: Since the coefficient [tex]\( a \)[/tex] is negative, the parabola opens downwards.
3. Calculate Some Points:
To construct the graph, compute a few values of [tex]\( g(x) \)[/tex] by substituting [tex]\( x \)[/tex] values around the vertex:
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = -\frac{3}{2}(1-2)^2 = -\frac{3}{2}(1) = -\frac{3}{2} = -1.5 \][/tex]
So, the point [tex]\((1, -1.5)\)[/tex] is on the graph.
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ g(3) = -\frac{3}{2}(3-2)^2 = -\frac{3}{2}(1) = -\frac{3}{2} = -1.5 \][/tex]
So, the point [tex]\((3, -1.5)\)[/tex] is on the graph.
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = -\frac{3}{2}(0-2)^2 = -\frac{3}{2}(4) = -6 \][/tex]
So, the point [tex]\((0, -6)\)[/tex] is on the graph.
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ g(4) = -\frac{3}{2}(4-2)^2 = -\frac{3}{2}(2)^2 = -\frac{3}{2}(4) = -6 \][/tex]
So, the point [tex]\((4, -6)\)[/tex] is on the graph.
4. Plot the Points:
- Start with the vertex [tex]\((2, 0)\)[/tex].
- Plot the points [tex]\((1, -1.5)\)[/tex], [tex]\((3, -1.5)\)[/tex], [tex]\((0, -6)\)[/tex], and [tex]\((4, -6)\)[/tex].
5. Draw the Parabola:
- Draw a smooth curve passing through the plotted points.
- Ensure that the curve is symmetric about the line [tex]\( x = 2 \)[/tex].
6. Completing the Graph:
- Label the vertex and the key points.
- Indicate the axis of symmetry [tex]\( x = 2 \)[/tex].
- Optionally, label the function [tex]\( g(x) = -\frac{3}{2}(x-2)^2 \)[/tex].
Here is a rough sketch of the graph:
```
y
|
6-|.................
5-| .................
4-| .................
3-| .................
2-| .................
1-| .................
0-| .................
-1-| ................
-2-| .................
-3-| ...................
-4-| ......................
-5-| .........................
-6-|.................................*
-7-|..................................
+-----------------------------------
-4 -3 -2 -1 0 1 2 3 4 5 6 7
```
- The parabola opens downwards with the vertex at [tex]\( (2, 0) \)[/tex].
- The function value decreases as [tex]\( x \)[/tex] moves away from 2.
This completes the graph of the function [tex]\( g(x) = -\frac{3}{2}(x-2)^2 \)[/tex].
1. Understand the Function Format:
The given function is a quadratic function in vertex form: [tex]\( g(x) = a(x-h)^2 + k \)[/tex], where [tex]\( a = -\frac{3}{2} \)[/tex], [tex]\( h = 2 \)[/tex], and [tex]\( k = 0 \)[/tex]. This represents a parabola opening downwards (since [tex]\( a < 0 \)[/tex]), with its vertex at the point [tex]\((h, k) = (2, 0)\)[/tex].
2. Identify Key Points:
- Vertex: The vertex of the parabola is at [tex]\((2,0)\)[/tex].
- Axis of Symmetry: The line [tex]\( x = 2 \)[/tex] is the axis of symmetry.
- Direction: Since the coefficient [tex]\( a \)[/tex] is negative, the parabola opens downwards.
3. Calculate Some Points:
To construct the graph, compute a few values of [tex]\( g(x) \)[/tex] by substituting [tex]\( x \)[/tex] values around the vertex:
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = -\frac{3}{2}(1-2)^2 = -\frac{3}{2}(1) = -\frac{3}{2} = -1.5 \][/tex]
So, the point [tex]\((1, -1.5)\)[/tex] is on the graph.
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ g(3) = -\frac{3}{2}(3-2)^2 = -\frac{3}{2}(1) = -\frac{3}{2} = -1.5 \][/tex]
So, the point [tex]\((3, -1.5)\)[/tex] is on the graph.
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = -\frac{3}{2}(0-2)^2 = -\frac{3}{2}(4) = -6 \][/tex]
So, the point [tex]\((0, -6)\)[/tex] is on the graph.
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ g(4) = -\frac{3}{2}(4-2)^2 = -\frac{3}{2}(2)^2 = -\frac{3}{2}(4) = -6 \][/tex]
So, the point [tex]\((4, -6)\)[/tex] is on the graph.
4. Plot the Points:
- Start with the vertex [tex]\((2, 0)\)[/tex].
- Plot the points [tex]\((1, -1.5)\)[/tex], [tex]\((3, -1.5)\)[/tex], [tex]\((0, -6)\)[/tex], and [tex]\((4, -6)\)[/tex].
5. Draw the Parabola:
- Draw a smooth curve passing through the plotted points.
- Ensure that the curve is symmetric about the line [tex]\( x = 2 \)[/tex].
6. Completing the Graph:
- Label the vertex and the key points.
- Indicate the axis of symmetry [tex]\( x = 2 \)[/tex].
- Optionally, label the function [tex]\( g(x) = -\frac{3}{2}(x-2)^2 \)[/tex].
Here is a rough sketch of the graph:
```
y
|
6-|.................
5-| .................
4-| .................
3-| .................
2-| .................
1-| .................
0-| .................
-1-| ................
-2-| .................
-3-| ...................
-4-| ......................
-5-| .........................
-6-|.................................*
-7-|..................................
+-----------------------------------
-4 -3 -2 -1 0 1 2 3 4 5 6 7
```
- The parabola opens downwards with the vertex at [tex]\( (2, 0) \)[/tex].
- The function value decreases as [tex]\( x \)[/tex] moves away from 2.
This completes the graph of the function [tex]\( g(x) = -\frac{3}{2}(x-2)^2 \)[/tex].
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