Let's work through the problem step by step, starting with the exponential function $g(x) = 2^x - 3$.

1. Graphing the Exponential Function:

The function $g(x) = 2^x - 3$ represents an exponential function with a vertical shift. Here’s what we need to do:

• Parent Function: The parent function is $f(x) = 2^x$. This is a standard exponential curve.
• Shift: The term $-3$ means the graph is shifted down by 3 units. So, instead of approaching $y = 0$ as $x \to -\infty$, it approaches $y = -3$.

Key steps:

1. Asymptote: The horizontal asymptote is at $y = -3$.

2. Two Points: Let's find two points by choosing convenient $x$-values:

   • For $x = 0$: $g(0) = 2^0 - 3 = 1 - 3 = -2$ So, one point is $(0, -2)$.

   • For $x = 1$: $g(1) = 2^1 - 3 = 2 - 3 = -1$ So, another point is $(1, -1)$.

2. Asymptote:

The asymptote is a horizontal line that the graph approaches but never touches. For this function, it is at $y = -3$.

3. Domain and Range:

• Domain: The domain of any exponential function is all real numbers, so $(-\infty, \infty)$.
• Range: Since the function has been shifted down by 3 units and never reaches or goes below the asymptote, the range is $(-3, \infty)$.

Summary:

• Two points on the graph: $(0, -2)$ and $(1, -1)$
• Asymptote: $y = -3$
• Domain: $(-\infty, \infty)$
• Range: $(-3, \infty)$

Would you like more details on any part of this, or a graph to visualize it?

Here are five questions to expand your understanding:

1. What happens to the graph if we change the base of the exponential function?
2. How does adding or subtracting a constant outside the exponential function affect its graph?
3. What is the effect of multiplying the function by a constant, such as $g(x) = 3 \cdot 2^x - 3$?
4. Can you describe the behavior of the function as $x \to -\infty$ and $x \to \infty$?
5. How would the graph change if the shift was upwards, say $g(x) = 2^x + 2$?

Tip:

When graphing exponential functions, always start by identifying the asymptote and the general shape of the parent function. Then apply any shifts or stretches accordingly.