Answer :
To describe how the graph of the function [tex]\( g(x) = -\frac{1}{5} \sqrt{x} \)[/tex] can be obtained from the basic graph, we can use the following steps:
1. Start with the graph of [tex]\( f(x) = \sqrt{x} \)[/tex]:
The function [tex]\( f(x) = \sqrt{x} \)[/tex] is a basic square root function. Its graph starts at the origin (0, 0) and increases slowly, moving to the right in the first quadrant. It passes through points like (1, 1), (4, 2), and (9, 3).
2. Vertical Compression:
Next, we modify the graph of [tex]\( \sqrt{x} \)[/tex] by applying a vertical compression. This is achieved by multiplying the function by [tex]\( \frac{1}{5} \)[/tex].
- Mathematically, this transformation can be written as [tex]\( f(x) = \frac{1}{5} \sqrt{x} \)[/tex].
- Each [tex]\( y \)[/tex]-coordinate of the points on the original square root function is multiplied by [tex]\( \frac{1}{5} \)[/tex].
For example:
- The point (1, 1) becomes (1, [tex]\( \frac{1}{5} \)[/tex])
- The point (4, 2) becomes (4, [tex]\( \frac{2}{5} \)[/tex])
- The point (9, 3) becomes (9, [tex]\( \frac{3}{5} \)[/tex])
3. Reflection Across the X-axis:
Finally, we reflect the graph across the x-axis by multiplying the function by -1. This effectively changes the sign of each [tex]\( y \)[/tex]-coordinate from positive to negative.
- Mathematically, we write this as [tex]\( g(x) = -\frac{1}{5} \sqrt{x} \)[/tex].
For example:
- The point (1, [tex]\( \frac{1}{5} \)[/tex]) becomes (1, [tex]\( -\frac{1}{5} \)[/tex])
- The point (4, [tex]\( \frac{2}{5} \)[/tex]) becomes (4, [tex]\( -\frac{2}{5} \)[/tex])
- The point (9, [tex]\( \frac{3}{5} \)[/tex]) becomes (9, [tex]\( -\frac{3}{5} \)[/tex])
So, the sequence of steps are:
1. Start with the graph of [tex]\( f(x) = \sqrt{x} \)[/tex].
2. Apply a vertical compression by multiplying each [tex]\( y \)[/tex]-coordinate by [tex]\( \frac{1}{5} \)[/tex].
3. Reflect the graph across the x-axis.
### Graphing the Function
To graph [tex]\( g(x) = -\frac{1}{5} \sqrt{x} \)[/tex]:
1. Plot key points obtained from the steps outlined above:
- (0, 0)
- (1, -[tex]\( \frac{1}{5} \)[/tex])
- (4, -[tex]\( \frac{2}{5} \)[/tex])
- (9, -[tex]\( \frac{3}{5} \)[/tex])
2. Draw a smooth curve passing through these points, reflecting the shape of the square root function but compressed vertically by a factor of 5 and flipped over the x-axis.
The resulting graph will be a smooth curve starting at the origin and moving right and downward in the fourth quadrant. Use these points to guide your drawing for an accurate representation.
Here is a rough sketch of the graph for [tex]\( g(x) = -\frac{1}{5} \sqrt{x} \)[/tex]:
```
y
↑
│
│ (1, -0.2)
│
│
│ (4, -0.4)
│
│
│ * (9, -0.6)
│ .
│ .
│ .
│-----------------------------→ x
0 1 2 3 4 ...
```
In this sketch, note that the points on the curve align with the calculated transformed points reflecting the original square root function characteristics transformed as described.
1. Start with the graph of [tex]\( f(x) = \sqrt{x} \)[/tex]:
The function [tex]\( f(x) = \sqrt{x} \)[/tex] is a basic square root function. Its graph starts at the origin (0, 0) and increases slowly, moving to the right in the first quadrant. It passes through points like (1, 1), (4, 2), and (9, 3).
2. Vertical Compression:
Next, we modify the graph of [tex]\( \sqrt{x} \)[/tex] by applying a vertical compression. This is achieved by multiplying the function by [tex]\( \frac{1}{5} \)[/tex].
- Mathematically, this transformation can be written as [tex]\( f(x) = \frac{1}{5} \sqrt{x} \)[/tex].
- Each [tex]\( y \)[/tex]-coordinate of the points on the original square root function is multiplied by [tex]\( \frac{1}{5} \)[/tex].
For example:
- The point (1, 1) becomes (1, [tex]\( \frac{1}{5} \)[/tex])
- The point (4, 2) becomes (4, [tex]\( \frac{2}{5} \)[/tex])
- The point (9, 3) becomes (9, [tex]\( \frac{3}{5} \)[/tex])
3. Reflection Across the X-axis:
Finally, we reflect the graph across the x-axis by multiplying the function by -1. This effectively changes the sign of each [tex]\( y \)[/tex]-coordinate from positive to negative.
- Mathematically, we write this as [tex]\( g(x) = -\frac{1}{5} \sqrt{x} \)[/tex].
For example:
- The point (1, [tex]\( \frac{1}{5} \)[/tex]) becomes (1, [tex]\( -\frac{1}{5} \)[/tex])
- The point (4, [tex]\( \frac{2}{5} \)[/tex]) becomes (4, [tex]\( -\frac{2}{5} \)[/tex])
- The point (9, [tex]\( \frac{3}{5} \)[/tex]) becomes (9, [tex]\( -\frac{3}{5} \)[/tex])
So, the sequence of steps are:
1. Start with the graph of [tex]\( f(x) = \sqrt{x} \)[/tex].
2. Apply a vertical compression by multiplying each [tex]\( y \)[/tex]-coordinate by [tex]\( \frac{1}{5} \)[/tex].
3. Reflect the graph across the x-axis.
### Graphing the Function
To graph [tex]\( g(x) = -\frac{1}{5} \sqrt{x} \)[/tex]:
1. Plot key points obtained from the steps outlined above:
- (0, 0)
- (1, -[tex]\( \frac{1}{5} \)[/tex])
- (4, -[tex]\( \frac{2}{5} \)[/tex])
- (9, -[tex]\( \frac{3}{5} \)[/tex])
2. Draw a smooth curve passing through these points, reflecting the shape of the square root function but compressed vertically by a factor of 5 and flipped over the x-axis.
The resulting graph will be a smooth curve starting at the origin and moving right and downward in the fourth quadrant. Use these points to guide your drawing for an accurate representation.
Here is a rough sketch of the graph for [tex]\( g(x) = -\frac{1}{5} \sqrt{x} \)[/tex]:
```
y
↑
│
│ (1, -0.2)
│
│
│ (4, -0.4)
│
│
│ * (9, -0.6)
│ .
│ .
│ .
│-----------------------------→ x
0 1 2 3 4 ...
```
In this sketch, note that the points on the curve align with the calculated transformed points reflecting the original square root function characteristics transformed as described.
