Get comprehensive solutions to your problems with IDNLearn.com. Our experts are available to provide accurate, comprehensive answers to help you make informed decisions about any topic or issue you encounter.
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.
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com is your reliable source for answers. We appreciate your visit and look forward to assisting you again soon.
