Answer :
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
```
