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Sagot :
Certainly! Let's plot five key points on the parabola given by the equation [tex]\( y = 2x^2 - 16x + 34 \)[/tex].
### Step-by-Step Solution:
1. Find the vertex of the parabola:
The vertex formula for a quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex] is given by [tex]\( x = \frac{-b}{2a} \)[/tex].
Here, [tex]\( a = 2 \)[/tex] and [tex]\( b = -16 \)[/tex].
- Calculate the x-coordinate of the vertex:
[tex]\[ x = \frac{-(-16)}{2 \times 2} = \frac{16}{4} = 4 \][/tex]
- Plug [tex]\( x = 4 \)[/tex] back into the equation to find the y-coordinate of the vertex:
[tex]\[ y = 2(4)^2 - 16(4) + 34 = 2 \times 16 - 64 + 34 = 32 - 64 + 34 = 2 \][/tex]
- The vertex is at the point [tex]\( (4, 2) \)[/tex].
2. Calculate two points to the left of the vertex:
- First point to the left:
[tex]\[ \text{Let } x = 3. \][/tex]
[tex]\[ y = 2(3)^2 - 16(3) + 34 = 2 \times 9 - 48 + 34 = 18 - 48 + 34 = 4 \][/tex]
[tex]\[ \text{Point: } (3, 4) \][/tex]
- Second point to the left:
[tex]\[ \text{Let } x = 2. \][/tex]
[tex]\[ y = 2(2)^2 - 16(2) + 34 = 2 \times 4 - 32 + 34 = 8 - 32 + 34 = 10 \][/tex]
[tex]\[ \text{Point: } (2, 10) \][/tex]
3. Calculate two points to the right of the vertex:
- First point to the right:
[tex]\[ \text{Let } x = 5. \][/tex]
[tex]\[ y = 2(5)^2 - 16(5) + 34 = 2 \times 25 - 80 + 34 = 50 - 80 + 34 = 4 \][/tex]
[tex]\[ \text{Point: } (5, 4) \][/tex]
- Second point to the right:
[tex]\[ \text{Let } x = 6. \][/tex]
[tex]\[ y = 2(6)^2 - 16(6) + 34 = 2 \times 36 - 96 + 34 = 72 - 96 + 34 = 10 \][/tex]
[tex]\[ \text{Point: } (6, 10) \][/tex]
### Summary of the Points:
- Vertex: [tex]\( (4, 2) \)[/tex]
- Two points to the left of the vertex: [tex]\( (3, 4) \)[/tex] and [tex]\( (2, 10) \)[/tex]
- Two points to the right of the vertex: [tex]\( (5, 4) \)[/tex] and [tex]\( (6, 10) \)[/tex]
These points are:
- [tex]\((4.0, 2.0)\)[/tex]
- [tex]\((3.0, 4.0)\)[/tex]
- [tex]\((2.0, 10.0)\)[/tex]
- [tex]\((5.0, 4.0)\)[/tex]
- [tex]\((6.0, 10.0)\)[/tex]
Using these five points, you can plot the parabola [tex]\( y = 2x^2 - 16x + 34 \)[/tex] on a graph.
### Step-by-Step Solution:
1. Find the vertex of the parabola:
The vertex formula for a quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex] is given by [tex]\( x = \frac{-b}{2a} \)[/tex].
Here, [tex]\( a = 2 \)[/tex] and [tex]\( b = -16 \)[/tex].
- Calculate the x-coordinate of the vertex:
[tex]\[ x = \frac{-(-16)}{2 \times 2} = \frac{16}{4} = 4 \][/tex]
- Plug [tex]\( x = 4 \)[/tex] back into the equation to find the y-coordinate of the vertex:
[tex]\[ y = 2(4)^2 - 16(4) + 34 = 2 \times 16 - 64 + 34 = 32 - 64 + 34 = 2 \][/tex]
- The vertex is at the point [tex]\( (4, 2) \)[/tex].
2. Calculate two points to the left of the vertex:
- First point to the left:
[tex]\[ \text{Let } x = 3. \][/tex]
[tex]\[ y = 2(3)^2 - 16(3) + 34 = 2 \times 9 - 48 + 34 = 18 - 48 + 34 = 4 \][/tex]
[tex]\[ \text{Point: } (3, 4) \][/tex]
- Second point to the left:
[tex]\[ \text{Let } x = 2. \][/tex]
[tex]\[ y = 2(2)^2 - 16(2) + 34 = 2 \times 4 - 32 + 34 = 8 - 32 + 34 = 10 \][/tex]
[tex]\[ \text{Point: } (2, 10) \][/tex]
3. Calculate two points to the right of the vertex:
- First point to the right:
[tex]\[ \text{Let } x = 5. \][/tex]
[tex]\[ y = 2(5)^2 - 16(5) + 34 = 2 \times 25 - 80 + 34 = 50 - 80 + 34 = 4 \][/tex]
[tex]\[ \text{Point: } (5, 4) \][/tex]
- Second point to the right:
[tex]\[ \text{Let } x = 6. \][/tex]
[tex]\[ y = 2(6)^2 - 16(6) + 34 = 2 \times 36 - 96 + 34 = 72 - 96 + 34 = 10 \][/tex]
[tex]\[ \text{Point: } (6, 10) \][/tex]
### Summary of the Points:
- Vertex: [tex]\( (4, 2) \)[/tex]
- Two points to the left of the vertex: [tex]\( (3, 4) \)[/tex] and [tex]\( (2, 10) \)[/tex]
- Two points to the right of the vertex: [tex]\( (5, 4) \)[/tex] and [tex]\( (6, 10) \)[/tex]
These points are:
- [tex]\((4.0, 2.0)\)[/tex]
- [tex]\((3.0, 4.0)\)[/tex]
- [tex]\((2.0, 10.0)\)[/tex]
- [tex]\((5.0, 4.0)\)[/tex]
- [tex]\((6.0, 10.0)\)[/tex]
Using these five points, you can plot the parabola [tex]\( y = 2x^2 - 16x + 34 \)[/tex] on a graph.
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