Sketch the graph of each function. $$ y=\frac{1}{4} \cdot\left(\frac{1}{3}\right)^{x-1}-1 $$

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To sketch the graph of the function $$ y=\frac{1}{4} \cdot\left(\frac{1}{3}\right)^{x-1}-1 $$, we need to consider several key aspects:

1. **Horizontal and Vertical Asymptotes:**
   - The horizontal asymptote is found by considering the behavior of the function as $$ x $$ approaches infinity or negative infinity. Since the base of the exponent is between 0 and 1, the function will approach 0 as $$ x $$ approaches infinity or negative infinity. Therefore, the horizontal asymptote is $$ y = -1 $$.
   - There is no vertical asymptote because the function is defined for all real numbers.

2. **Intercepts:**
   - To find the y-intercept, set $$ x = 0 $$:
     $$ y = \frac{1}{4} \cdot\left(\frac{1}{3}\right)^{0-1}-1 = \frac{1}{4} \cdot\left(\frac{1}{3}\right)^{-1}-1 = \frac{1}{4} \cdot 3 - 1 = \frac{3}{4} - 1 = -\frac{1}{4} $$
     So, the y-intercept is $$ (0, -\frac{1}{4}) $$.
   - To find the x-intercept, set $$ y = 0 $$:
     $$ 0 = \frac{1}{4} \cdot\left(\frac{1}{3}\right)^{x-1}-1 $$
     $$ \frac{1}{4} \cdot\left(\frac{1}{3}\right)^{x-1} = 1 $$
     $$ \left(\frac{1}{3}\right)^{x-1} = 4 $$
     $$ x-1 = \log_{\frac{1}{3}}(4) $$
     $$ x = \log_{\frac{1}{3}}(4) + 1 $$
     Since $$ \log_{\frac{1}{3}}(4) $$ is a negative number, the x-intercept will be a negative number. However, it is not an integer, so there is no exact x-intercept.

3. **Behavior of the Function:**
   - The function is decreasing because the base of the exponent is between 0 and 1.
   - The function is concave down because the second derivative is negative.

4. **Sketching the Graph:**
   - Start by plotting the y-intercept $$ (0, -\frac{1}{4}) $$.
   - Draw a smooth curve that approaches the horizontal asymptote $$ y = -1 $$ as $$ x $$ approaches infinity or negative infinity.
   - The curve should be decreasing and concave down.

Here is a rough sketch of the graph:

```
y
|
|       *
|      /
|     /
|    /
|   /
|  /
| /
|---------------------
x
```

Note that this is a rough sketch, and the actual graph may look slightly different.
The graph has a horizontal asymptote at $$ y = -1 $$, a y-intercept at $$ (0, -\frac{1}{4}) $$, and is decreasing and concave down.

Sketch the graph of each function.

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