To tackle the questions in the image, let's break down each of the four listed features for the function $g(x) = \log_2(x + 4) - 1$:

1. Vertical Asymptote

• The vertical asymptote occurs where the argument of the logarithm is equal to zero because the logarithm function is undefined for non-positive numbers. For the function $g(x) = \log_2(x + 4) - 1$, the argument is $x + 4$.
• Setting $x + 4 = 0$, we find: $x = -4$
• Thus, the vertical asymptote is at $x = -4$.

2. X- and Y-Intercepts

• X-Intercept: Set $g(x) = 0$ and solve for $x$. $\log_2(x + 4) - 1 = 0$ $\log_2(x + 4) = 1$ $x + 4 = 2^1 = 2$ $x = -2$ The x-intercept is at $x = -2$.

• Y-Intercept: Set $x = 0$ and solve for $g(x)$. $g(0) = \log_2(0 + 4) - 1$ $g(0) = \log_2(4) - 1$ $g(0) = 2 - 1 = 1$ The y-intercept is at $y = 1$.

3. Transformations

• The function $g(x) = \log_2(x + 4) - 1$ can be considered as a transformation of the parent function $f(x) = \log_2(x)$.
• Translation Left: The $+4$ inside the logarithm translates the graph 4 units to the left.
• Vertical Shift Down: The $-1$ outside the logarithm shifts the graph down by 1 unit.

4. Domain

• The domain of the function $g(x) = \log_2(x + 4) - 1$ is determined by the argument of the logarithm, which must be positive: $x + 4 > 0$ $x > -4$
• Therefore, the domain is $x > -4$, or in interval notation, $(-4, \infty)$.

---

Summary

1. Vertical Asymptote: $x = -4$.
2. X-Intercept: $x = -2$; Y-Intercept: $y = 1$.
3. Transformations: Left by 4 units, down by 1 unit.
4. Domain: $x > -4$.

Would you like more details or have any questions?

Expanding Questions:

1. How would the vertical asymptote change if the function were $g(x) = \log_2(x - 3) - 1$?
2. What effect does changing the base of the logarithm have on the graph?
3. If the function were $g(x) = \log_2(x + 4) + 2$, how would the transformations differ?
4. How do you find the range of $g(x)$?
5. What is the relationship between the domain and the vertical asymptote?

Tip:

Always check the argument of the logarithmic function to determine where it is undefined, which will help you locate the vertical asymptote and establish the domain.