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To graph the equation [tex]\( y = -x^2 + 2x + 3 \)[/tex] and determine its roots and vertex, let's follow these steps:
1. Determine the roots of the equation [tex]\( -x^2 + 2x + 3 = 0 \)[/tex]:
The roots (solutions) of [tex]\( -x^2 + 2x + 3 = 0 \)[/tex] are:
- [tex]\( x = -1.0 \)[/tex]
- [tex]\( x = 3.0 \)[/tex]
Therefore, the points that you will plot for the roots are [tex]\((-1.0, 0)\)[/tex] and [tex]\((3.0, 0)\)[/tex].
2. Calculate the vertex of the parabola [tex]\( y = -x^2 + 2x + 3 \)[/tex]:
The vertex of the parabola, which is the maximum point because the parabola opens downwards, is given by the vertex coordinates:
- [tex]\( (1.0, 4.0) \)[/tex]
Therefore, the vertex point is [tex]\((1.0, 4.0)\)[/tex].
3. Calculate additional points for plotting:
Let's calculate the value of [tex]\(y\)[/tex] for a few selected [tex]\(x\)[/tex] values to get additional points:
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = -(-2)^2 + 2(-2) + 3 = -4 - 4 + 3 = -5 \][/tex]
Point: [tex]\((-2, -5)\)[/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -(-1)^2 + 2(-1) + 3 = -1 - 2 + 3 = 0 \][/tex]
Point: [tex]\((-1, 0)\)[/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -(0)^2 + 2(0) + 3 = 3 \][/tex]
Point: [tex]\((0, 3)\)[/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -(1)^2 + 2(1) + 3 = -1 + 2 + 3 = 4 \][/tex]
Point: [tex]\((1, 4)\)[/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -(2)^2 + 2(2) + 3 = -4 + 4 + 3 = 3 \][/tex]
Point: [tex]\((2, 3)\)[/tex]
4. Plot these points on a set of axes:
- Roots: [tex]\( (-1.0, 0) \)[/tex] and [tex]\( (3.0, 0) \)[/tex]
- Vertex: [tex]\( (1.0, 4.0) \)[/tex]
- Additional points:
- Point: [tex]\( (-2, -5) \)[/tex]
- Point: [tex]\( (0, 3) \)[/tex]
- Point: [tex]\( (2, 3) \)[/tex]
5. Draw the parabola:
Use the plotted points to sketch the parabola. The parabola will be symmetrical around its vertex, which is at [tex]\( (1.0, 4.0) \)[/tex].
By plotting these points, you can clearly see the shape and position of the parabola on the graph. The graph will help you visualize that the roots are [tex]\(x = -1\)[/tex] and [tex]\(x = 3\)[/tex].
1. Determine the roots of the equation [tex]\( -x^2 + 2x + 3 = 0 \)[/tex]:
The roots (solutions) of [tex]\( -x^2 + 2x + 3 = 0 \)[/tex] are:
- [tex]\( x = -1.0 \)[/tex]
- [tex]\( x = 3.0 \)[/tex]
Therefore, the points that you will plot for the roots are [tex]\((-1.0, 0)\)[/tex] and [tex]\((3.0, 0)\)[/tex].
2. Calculate the vertex of the parabola [tex]\( y = -x^2 + 2x + 3 \)[/tex]:
The vertex of the parabola, which is the maximum point because the parabola opens downwards, is given by the vertex coordinates:
- [tex]\( (1.0, 4.0) \)[/tex]
Therefore, the vertex point is [tex]\((1.0, 4.0)\)[/tex].
3. Calculate additional points for plotting:
Let's calculate the value of [tex]\(y\)[/tex] for a few selected [tex]\(x\)[/tex] values to get additional points:
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ y = -(-2)^2 + 2(-2) + 3 = -4 - 4 + 3 = -5 \][/tex]
Point: [tex]\((-2, -5)\)[/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -(-1)^2 + 2(-1) + 3 = -1 - 2 + 3 = 0 \][/tex]
Point: [tex]\((-1, 0)\)[/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -(0)^2 + 2(0) + 3 = 3 \][/tex]
Point: [tex]\((0, 3)\)[/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -(1)^2 + 2(1) + 3 = -1 + 2 + 3 = 4 \][/tex]
Point: [tex]\((1, 4)\)[/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -(2)^2 + 2(2) + 3 = -4 + 4 + 3 = 3 \][/tex]
Point: [tex]\((2, 3)\)[/tex]
4. Plot these points on a set of axes:
- Roots: [tex]\( (-1.0, 0) \)[/tex] and [tex]\( (3.0, 0) \)[/tex]
- Vertex: [tex]\( (1.0, 4.0) \)[/tex]
- Additional points:
- Point: [tex]\( (-2, -5) \)[/tex]
- Point: [tex]\( (0, 3) \)[/tex]
- Point: [tex]\( (2, 3) \)[/tex]
5. Draw the parabola:
Use the plotted points to sketch the parabola. The parabola will be symmetrical around its vertex, which is at [tex]\( (1.0, 4.0) \)[/tex].
By plotting these points, you can clearly see the shape and position of the parabola on the graph. The graph will help you visualize that the roots are [tex]\(x = -1\)[/tex] and [tex]\(x = 3\)[/tex].
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