IDNLearn.com: Your trusted source for accurate and reliable answers. Find the answers you need quickly and accurately with help from our knowledgeable and experienced experts.
Sagot :
Sure! Let's graph the given parabola [tex]\( y = (x-4)^2 - 1 \)[/tex] and identify five key points including the vertex and four additional points.
### Step-by-Step Solution:
1. Understanding the Equation:
- The given equation is [tex]\( y = (x-4)^2 - 1 \)[/tex].
- This is a standard form of a parabola [tex]\( y = a(x-h)^2 + k \)[/tex] where the vertex is at the point [tex]\((h, k)\)[/tex].
2. Identify the Vertex:
- By comparing [tex]\( y = (x-4)^2 - 1 \)[/tex] with [tex]\( y = a(x-h)^2 + k \)[/tex]:
- [tex]\( h = 4 \)[/tex]
- [tex]\( k = -1 \)[/tex]
- So, the vertex of the parabola is at [tex]\((4, -1)\)[/tex].
3. Choose Additional Points:
- We will pick points around the vertex to keep it simple, say [tex]\( x = 3, 5, 2, \)[/tex] and [tex]\( 6 \)[/tex].
4. Calculate y-values for Chosen x-values:
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = (4-4)^2 - 1 = 0 - 1 = -1 \][/tex]
[tex]\((4, -1)\)[/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = (3-4)^2 - 1 = 1^2 - 1 = 1 - 1 = 0 \][/tex]
[tex]\((3, 0)\)[/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ y = (5-4)^2 - 1 = 1^2 - 1 = 1 - 1 = 0 \][/tex]
[tex]\((5, 0)\)[/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = (2-4)^2 - 1 = 2^2 - 1 = 4 - 1 = 3 \][/tex]
[tex]\((2, 3)\)[/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[ y = (6-4)^2 - 1 = 2^2 - 1 = 4 - 1 = 3 \][/tex]
[tex]\((6, 3)\)[/tex]
5. Plot the Points and the Parabola:
- Now we plot the points [tex]\((4, -1)\)[/tex], [tex]\((3, 0)\)[/tex], [tex]\((5, 0)\)[/tex], [tex]\((2, 3)\)[/tex], and [tex]\((6, 3)\)[/tex].
- Also, we draw the curve of the parabola [tex]\( y = (x-4)^2 - 1 \)[/tex].
### Graphical Representation:
- Vertex: The parabola’s vertex is [tex]\( (4, -1) \)[/tex].
- Points:
- [tex]\( (4, -1) \)[/tex]
- [tex]\( (3, 0) \)[/tex]
- [tex]\( (5, 0) \)[/tex]
- [tex]\( (2, 3) \)[/tex]
- [tex]\( (6, 3) \)[/tex]
Here's how you'd sketch the graph on Cartesian coordinates:
- Plot the vertex at [tex]\((4, -1)\)[/tex].
- Plot the other points: [tex]\((3, 0)\)[/tex], [tex]\((5, 0)\)[/tex], [tex]\((2, 3)\)[/tex], and [tex]\((6, 3)\)[/tex].
- The parabola opens upwards because the coefficient of [tex]\((x-4)^2\)[/tex] is positive.
- Draw a smooth curve through these points to complete the parabola.
### Summary
- Vertex of the parabola: [tex]\((4, -1)\)[/tex]
- Additional Points: [tex]\((3, 0)\)[/tex], [tex]\((5, 0)\)[/tex], [tex]\((2, 3)\)[/tex], and [tex]\((6, 3)\)[/tex]
- The equation of the parabola is [tex]\( y = (x-4)^2 - 1 \)[/tex].
This should guide you in graphing the parabola with the specified points.
### Step-by-Step Solution:
1. Understanding the Equation:
- The given equation is [tex]\( y = (x-4)^2 - 1 \)[/tex].
- This is a standard form of a parabola [tex]\( y = a(x-h)^2 + k \)[/tex] where the vertex is at the point [tex]\((h, k)\)[/tex].
2. Identify the Vertex:
- By comparing [tex]\( y = (x-4)^2 - 1 \)[/tex] with [tex]\( y = a(x-h)^2 + k \)[/tex]:
- [tex]\( h = 4 \)[/tex]
- [tex]\( k = -1 \)[/tex]
- So, the vertex of the parabola is at [tex]\((4, -1)\)[/tex].
3. Choose Additional Points:
- We will pick points around the vertex to keep it simple, say [tex]\( x = 3, 5, 2, \)[/tex] and [tex]\( 6 \)[/tex].
4. Calculate y-values for Chosen x-values:
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = (4-4)^2 - 1 = 0 - 1 = -1 \][/tex]
[tex]\((4, -1)\)[/tex]
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ y = (3-4)^2 - 1 = 1^2 - 1 = 1 - 1 = 0 \][/tex]
[tex]\((3, 0)\)[/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ y = (5-4)^2 - 1 = 1^2 - 1 = 1 - 1 = 0 \][/tex]
[tex]\((5, 0)\)[/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = (2-4)^2 - 1 = 2^2 - 1 = 4 - 1 = 3 \][/tex]
[tex]\((2, 3)\)[/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[ y = (6-4)^2 - 1 = 2^2 - 1 = 4 - 1 = 3 \][/tex]
[tex]\((6, 3)\)[/tex]
5. Plot the Points and the Parabola:
- Now we plot the points [tex]\((4, -1)\)[/tex], [tex]\((3, 0)\)[/tex], [tex]\((5, 0)\)[/tex], [tex]\((2, 3)\)[/tex], and [tex]\((6, 3)\)[/tex].
- Also, we draw the curve of the parabola [tex]\( y = (x-4)^2 - 1 \)[/tex].
### Graphical Representation:
- Vertex: The parabola’s vertex is [tex]\( (4, -1) \)[/tex].
- Points:
- [tex]\( (4, -1) \)[/tex]
- [tex]\( (3, 0) \)[/tex]
- [tex]\( (5, 0) \)[/tex]
- [tex]\( (2, 3) \)[/tex]
- [tex]\( (6, 3) \)[/tex]
Here's how you'd sketch the graph on Cartesian coordinates:
- Plot the vertex at [tex]\((4, -1)\)[/tex].
- Plot the other points: [tex]\((3, 0)\)[/tex], [tex]\((5, 0)\)[/tex], [tex]\((2, 3)\)[/tex], and [tex]\((6, 3)\)[/tex].
- The parabola opens upwards because the coefficient of [tex]\((x-4)^2\)[/tex] is positive.
- Draw a smooth curve through these points to complete the parabola.
### Summary
- Vertex of the parabola: [tex]\((4, -1)\)[/tex]
- Additional Points: [tex]\((3, 0)\)[/tex], [tex]\((5, 0)\)[/tex], [tex]\((2, 3)\)[/tex], and [tex]\((6, 3)\)[/tex]
- The equation of the parabola is [tex]\( y = (x-4)^2 - 1 \)[/tex].
This should guide you in graphing the parabola with the specified points.
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Thank you for visiting IDNLearn.com. We’re here to provide accurate and reliable answers, so visit us again soon.
