The problem involves graphing the logarithmic function:
$g(x) = 2 \log_3(x + 4) - 1$

Let me break this down step by step so you can graph it:

1. Identify key components of the function:

• Parent Function: The parent function is $\log_3(x)$.
  This function has a vertical asymptote at $x = 0$ and passes through $(1, 0)$ since $\log_3(1) = 0$.

• Transformations:

  • $(x + 4)$: This shifts the graph 4 units to the left. So, the vertical asymptote moves to $x = -4$.
  • $2$: This stretches the graph vertically by a factor of 2.
  • $-1$: This shifts the graph 1 unit downward.

2. Plotting points:

To graph, we need at least two points.

• First, rewrite the function for calculations:
  $g(x) = 2 \log_3(x + 4) - 1$

Step 1: Choose $x$-values and compute $g(x)$.

Since $\log_3(y)$ is harder to compute, use simple values of $x + 4$ that make the logarithm easy. For example:

1. $x + 4 = 3$ (since $\log_3(3) = 1$):
   $x = -1 \quad \text{and} \quad g(x) = 2 \cdot 1 - 1 = 1$
   So, one point is $(-1, 1)$.

2. $x + 4 = 9$ (since $\log_3(9) = 2$):
   $x = 5 \quad \text{and} \quad g(x) = 2 \cdot 2 - 1 = 3$
   So, another point is $(5, 3)$.

3. Vertical asymptote:

The vertical asymptote is at $x = -4$.

4. Sketch the graph:

1. Plot the two points $(-1, 1)$ and $(5, 3)$.
2. Draw the vertical asymptote at $x = -4$.
3. Connect the points smoothly, ensuring the curve approaches the vertical asymptote as $x$ approaches $-4$ from the right.

Would you like help with more details or examples? Let me know! Here are some related questions to explore:

Follow-up Questions:

1. How do you compute logarithms with bases other than 10 or $e$ on a calculator?
2. What are the steps to identify transformations of logarithmic functions?
3. What happens if the coefficient of the logarithm (like the 2 in this example) changes to a fraction?
4. How can we determine the domain of this function?
5. How does changing the base of the logarithm ($\log_3$ to $\log_5$) affect the graph?

Tip:

Always start by identifying the vertical asymptote when graphing logarithmic functions—it helps anchor the transformations.