The given function is still $y = 5^x - 6$. Let's graph this function and confirm the key features step by step.

Steps to Graph:

1. Start with the base exponential function $y = 5^x$, which has a horizontal asymptote at $y = 0$ and passes through the point $(0, 1)$.
2. Shift the graph down by 6 units because of the subtraction. This changes the horizontal asymptote to $y = -6$, and the graph now passes through $(0, -5)$.
3. Plot a few more points for accuracy:
   • When $x = 1$, $y = 5^1 - 6 = -1$ (point: $(1, -1)$).
   • When $x = -1$, $y = 5^{-1} - 6 = \frac{1}{5} - 6 = -5.8$ (point: $(-1, -5.8)$).
   • As $x \to \infty$, $y \to \infty - 6 = \infty$.
   • As $x \to -\infty$, $y \to 0 - 6 = -6$ (approaching the asymptote).

Key Features:

1. Domain: $(-\infty, \infty)$, since exponential functions are defined for all real values of $x$.
2. Range: $(-6, \infty)$, as the graph starts at $y > -6$ and grows infinitely upwards.
3. Asymptote: $y = -6$, because the graph approaches this value as $x \to -\infty$.
4. Y-Intercept: $(0, -5)$, calculated when $x = 0$.

Would you like me to generate a visual graph for you to confirm this analysis?

Expanding Questions:

1. How can the base of an exponential function (e.g., 5 in $5^x$) affect the graph's steepness?
2. What changes would occur to the graph if $-6$ was replaced with another constant, like $+3$?
3. What is the general method to identify the asymptote of an exponential function after shifts?
4. How can we find an exact point where $y = 10$ in the given function $y = 5^x - 6$?
5. Can we rewrite the equation in logarithmic form to solve for $x$?

Tip:

Always look for transformations (shifts, stretches, or compressions) when graphing exponential functions. This makes it easier to sketch and analyze their features.