Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
To plot the axis of symmetry and the vertex for the function [tex]\( h(x) = (x-5)^2 - 7 \)[/tex], let's follow these steps:
### Step 1: Identifying the Vertex
The given function is in the standard form of a parabola [tex]\( h(x) = (x - h)^2 + k \)[/tex], where [tex]\( h \)[/tex] and [tex]\( k \)[/tex] represent the coordinates of the vertex.
Here, [tex]\( h(x) = (x-5)^2 - 7 \)[/tex].
- The term [tex]\((x - 5)\)[/tex] indicates that [tex]\( h = 5 \)[/tex].
- The constant term [tex]\(-7\)[/tex] indicates that [tex]\( k = -7 \)[/tex].
Thus, the vertex of the function is at [tex]\( (5, -7) \)[/tex].
### Step 2: Identifying the Axis of Symmetry
The axis of symmetry of a parabola given by [tex]\( (x - h)^2 + k \)[/tex] is the vertical line that passes through the vertex.
Since the vertex is at [tex]\( (5, -7) \)[/tex], the axis of symmetry is the vertical line [tex]\( x = 5 \)[/tex].
### Step 3: Plotting the Function, Vertex, and Axis of Symmetry
To visualize this, consider the following steps for manual plotting:
1. Plot the Vertex:
- The vertex is at the point [tex]\( (5, -7) \)[/tex]. Mark this point on the graph.
2. Plot the Axis of Symmetry:
- The axis of symmetry is the vertical line [tex]\( x = 5 \)[/tex]. Draw a vertical dashed line through [tex]\( x = 5 \)[/tex].
3. Sketching the Parabola:
- The function [tex]\( h(x) = (x - 5)^2 - 7 \)[/tex] opens upwards because the coefficient of the [tex]\( (x - 5)^2 \)[/tex] term is positive.
- Plot points on either side of the vertex to get the shape of the parabola. Calculate a few [tex]\( y \)[/tex]-values for [tex]\( x \)[/tex]-values around the vertex.
For example:
- For [tex]\( x = 4 \)[/tex]: [tex]\( h(4) = (4 - 5)^2 - 7 = 1 - 7 = -6 \)[/tex].
- For [tex]\( x = 6 \)[/tex]: [tex]\( h(6) = (6 - 5)^2 - 7 = 1 - 7 = -6 \)[/tex].
- For [tex]\( x = 3 \)[/tex]: [tex]\( h(3) = (3 - 5)^2 - 7 = 4 - 7 = -3 \)[/tex].
- For [tex]\( x = 7 \)[/tex]: [tex]\( h(7) = (7 - 5)^2 - 7 = 4 - 7 = -3 \)[/tex].
Using these points, draw the curve of the parabola.
### Summary
- Vertex: [tex]\( (5, -7) \)[/tex].
- Axis of Symmetry: [tex]\( x = 5 \)[/tex].
By marking the vertex and drawing the parabola and vertical axis of symmetry, you will have a clear visual representation of the function [tex]\( h(x) = (x-5)^2 - 7 \)[/tex].
### Graph Illustration:
```
^
|
| (vertex)
|
|
| ______________Axis of Symmetry_______________
| | | |
| | | |
|_________________________|__________________*__________> x
3 5 7
```
### Step 1: Identifying the Vertex
The given function is in the standard form of a parabola [tex]\( h(x) = (x - h)^2 + k \)[/tex], where [tex]\( h \)[/tex] and [tex]\( k \)[/tex] represent the coordinates of the vertex.
Here, [tex]\( h(x) = (x-5)^2 - 7 \)[/tex].
- The term [tex]\((x - 5)\)[/tex] indicates that [tex]\( h = 5 \)[/tex].
- The constant term [tex]\(-7\)[/tex] indicates that [tex]\( k = -7 \)[/tex].
Thus, the vertex of the function is at [tex]\( (5, -7) \)[/tex].
### Step 2: Identifying the Axis of Symmetry
The axis of symmetry of a parabola given by [tex]\( (x - h)^2 + k \)[/tex] is the vertical line that passes through the vertex.
Since the vertex is at [tex]\( (5, -7) \)[/tex], the axis of symmetry is the vertical line [tex]\( x = 5 \)[/tex].
### Step 3: Plotting the Function, Vertex, and Axis of Symmetry
To visualize this, consider the following steps for manual plotting:
1. Plot the Vertex:
- The vertex is at the point [tex]\( (5, -7) \)[/tex]. Mark this point on the graph.
2. Plot the Axis of Symmetry:
- The axis of symmetry is the vertical line [tex]\( x = 5 \)[/tex]. Draw a vertical dashed line through [tex]\( x = 5 \)[/tex].
3. Sketching the Parabola:
- The function [tex]\( h(x) = (x - 5)^2 - 7 \)[/tex] opens upwards because the coefficient of the [tex]\( (x - 5)^2 \)[/tex] term is positive.
- Plot points on either side of the vertex to get the shape of the parabola. Calculate a few [tex]\( y \)[/tex]-values for [tex]\( x \)[/tex]-values around the vertex.
For example:
- For [tex]\( x = 4 \)[/tex]: [tex]\( h(4) = (4 - 5)^2 - 7 = 1 - 7 = -6 \)[/tex].
- For [tex]\( x = 6 \)[/tex]: [tex]\( h(6) = (6 - 5)^2 - 7 = 1 - 7 = -6 \)[/tex].
- For [tex]\( x = 3 \)[/tex]: [tex]\( h(3) = (3 - 5)^2 - 7 = 4 - 7 = -3 \)[/tex].
- For [tex]\( x = 7 \)[/tex]: [tex]\( h(7) = (7 - 5)^2 - 7 = 4 - 7 = -3 \)[/tex].
Using these points, draw the curve of the parabola.
### Summary
- Vertex: [tex]\( (5, -7) \)[/tex].
- Axis of Symmetry: [tex]\( x = 5 \)[/tex].
By marking the vertex and drawing the parabola and vertical axis of symmetry, you will have a clear visual representation of the function [tex]\( h(x) = (x-5)^2 - 7 \)[/tex].
### Graph Illustration:
```
^
|
| (vertex)
|
|
| ______________Axis of Symmetry_______________
| | | |
| | | |
|_________________________|__________________*__________> x
3 5 7
```
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
