We are tasked with graphing the transformation of the function $f(x) = 3^x$ into $g(x) = -f(-x) + 8$, which translates to:

$g(x) = -3^{-x} + 8$

This involves several transformations. Let’s break it down step by step:

1. Original Function: $f(x) = 3^x$

• This is a standard exponential function that grows rapidly as $x$ increases.
• The horizontal asymptote is at $y = 0$.

2. Transformation Analysis:

• Reflection over the $y$-axis ($-x$):
  Changing $x$ to $-x$ reflects the graph horizontally, giving $3^{-x}$.

• Reflection over the $x$-axis ($-f$):
  Adding a negative sign before the function reflects the graph vertically. This produces $-3^{-x}$.

• Vertical Translation Upward ($+8$):
  Adding $+8$ shifts the entire graph 8 units upward. This changes the horizontal asymptote to $y = 8$.

3. Key Features of $g(x) = -3^{-x} + 8$:

• Horizontal Asymptote: $y = 8$ (the graph will approach but never touch this line).
• Behavior:
  • As $x \to \infty$, $3^{-x} \to 0$ so $g(x) \to 8$.
  • As $x \to -\infty$, $3^{-x} \to \infty$, so $g(x) \to -\infty$.

4. Points to Plot:

To confirm the graph, calculate two key points:

1. At $x = 0$: $g(0) = -3^{-0} + 8 = -1 + 8 = 7$ So, $(0, 7)$.

2. At $x = -1$: $g(-1) = -3^{-(-1)} + 8 = -3^1 + 8 = -3 + 8 = 5$ So, $(-1, 5)$.

Final Steps:

• Plot the points: $(0, 7)$ and $(-1, 5)$.
• Draw the horizontal asymptote: $y = 8$.
• Sketch the reflected, translated curve: It will slope downward as $x \to -\infty$ and approach $y = 8$ as $x \to \infty$.

Let me know if you want further clarification or need me to illustrate more steps.

Related Questions:

1. What is the horizontal asymptote of an exponential function after transformations?
2. How do reflections affect exponential functions like $3^x$?
3. How do you determine key points on a transformed exponential graph?
4. What role does vertical translation play in graphing $g(x)$?
5. How do transformations interact (order of operations) for exponential graphs?

Tip: Always apply reflections and translations step by step to avoid errors in graphing!