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To graph the parabola [tex]\( y = 3x^2 - 12x + 9 \)[/tex] and plot the five specific points, follow these steps:
1. Identify the Vertex:
The formula for the x-coordinate of the vertex in a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] is given by:
[tex]\[ x_{\text{vertex}} = \frac{-b}{2a} \][/tex]
Here, [tex]\( a = 3 \)[/tex], [tex]\( b = -12 \)[/tex], and [tex]\( c = 9 \)[/tex]. Plugging in these values:
[tex]\[ x_{\text{vertex}} = \frac{-(-12)}{2 \cdot 3} = \frac{12}{6} = 2 \][/tex]
To find the y-coordinate of the vertex, substitute [tex]\( x = 2 \)[/tex] back into the equation:
[tex]\[ y_{\text{vertex}} = 3(2)^2 - 12(2) + 9 = 12 - 24 + 9 = -3 \][/tex]
Thus, the vertex is [tex]\( (2, -3) \)[/tex].
2. Calculate Points to the Left of the Vertex:
Choose [tex]\( x \)[/tex] values 1 and 0, which are to the left of [tex]\( x = 2 \)[/tex].
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 3(1)^2 - 12(1) + 9 = 3 - 12 + 9 = 0 \][/tex]
So, the point is [tex]\( (1, 0) \)[/tex].
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 3(0)^2 - 12(0) + 9 = 9 \][/tex]
So, the point is [tex]\( (0, 9) \)[/tex].
3. Calculate Points to the Right of the Vertex:
Choose [tex]\( x \)[/tex] values 3 and 4, which are to the right of [tex]\( x = 2 \)[/tex].
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = 3(3)^2 - 12(3) + 9 = 27 - 36 + 9 = 0 \][/tex]
So, the point is [tex]\( (3, 0) \)[/tex].
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 3(4)^2 - 12(4) + 9 = 48 - 48 + 9 = 9 \][/tex]
So, the point is [tex]\( (4, 9) \)[/tex].
4. Plot the Points:
The five points to be plotted on the graph are:
- Vertex: [tex]\( (2, -3) \)[/tex]
- Points to the left of the vertex: [tex]\( (1, 0) \)[/tex], [tex]\( (0, 9) \)[/tex]
- Points to the right of the vertex: [tex]\( (3, 0) \)[/tex], [tex]\( (4, 9) \)[/tex]
These points can be plotted on a coordinate plane, and then draw a smooth curve through these points to form the graph of the parabola.
Remember, a parabola is symmetric about its vertex. Thus, points on both sides of the vertex should mirror each other. This symmetry is reflected in the calculated points here.
1. Identify the Vertex:
The formula for the x-coordinate of the vertex in a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] is given by:
[tex]\[ x_{\text{vertex}} = \frac{-b}{2a} \][/tex]
Here, [tex]\( a = 3 \)[/tex], [tex]\( b = -12 \)[/tex], and [tex]\( c = 9 \)[/tex]. Plugging in these values:
[tex]\[ x_{\text{vertex}} = \frac{-(-12)}{2 \cdot 3} = \frac{12}{6} = 2 \][/tex]
To find the y-coordinate of the vertex, substitute [tex]\( x = 2 \)[/tex] back into the equation:
[tex]\[ y_{\text{vertex}} = 3(2)^2 - 12(2) + 9 = 12 - 24 + 9 = -3 \][/tex]
Thus, the vertex is [tex]\( (2, -3) \)[/tex].
2. Calculate Points to the Left of the Vertex:
Choose [tex]\( x \)[/tex] values 1 and 0, which are to the left of [tex]\( x = 2 \)[/tex].
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 3(1)^2 - 12(1) + 9 = 3 - 12 + 9 = 0 \][/tex]
So, the point is [tex]\( (1, 0) \)[/tex].
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 3(0)^2 - 12(0) + 9 = 9 \][/tex]
So, the point is [tex]\( (0, 9) \)[/tex].
3. Calculate Points to the Right of the Vertex:
Choose [tex]\( x \)[/tex] values 3 and 4, which are to the right of [tex]\( x = 2 \)[/tex].
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = 3(3)^2 - 12(3) + 9 = 27 - 36 + 9 = 0 \][/tex]
So, the point is [tex]\( (3, 0) \)[/tex].
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 3(4)^2 - 12(4) + 9 = 48 - 48 + 9 = 9 \][/tex]
So, the point is [tex]\( (4, 9) \)[/tex].
4. Plot the Points:
The five points to be plotted on the graph are:
- Vertex: [tex]\( (2, -3) \)[/tex]
- Points to the left of the vertex: [tex]\( (1, 0) \)[/tex], [tex]\( (0, 9) \)[/tex]
- Points to the right of the vertex: [tex]\( (3, 0) \)[/tex], [tex]\( (4, 9) \)[/tex]
These points can be plotted on a coordinate plane, and then draw a smooth curve through these points to form the graph of the parabola.
Remember, a parabola is symmetric about its vertex. Thus, points on both sides of the vertex should mirror each other. This symmetry is reflected in the calculated points here.
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