To find the equation for $g(x)$ in terms of $f(x)$, we need to carefully analyze both the graphs of $y = f(x)$ and $y = g(x)$ provided in the image.

1. Graph of $f(x)$: On the left, we see the graph of $f(x)$, which involves a zigzag pattern.

   • At $x = 0$, $f(x) = 1$
   • At $x = 1$, $f(x) = 0$
   • At $x = 2$, $f(x) = -1$
   • At $x = 3$, $f(x) = 0$
   • At $x = 4$, $f(x) = 1$

2. Graph of $g(x)$: On the right side, we observe the graph of $g(x)$, which is similar in shape to $f(x)$ but transformed:

   • The shape is vertically stretched and translated, and the graph is flipped upside-down compared to $f(x)$.

Steps to Determine the Transformation:

Horizontal Reflection:

Since the graph of $g(x)$ is flipped vertically compared to $f(x)$, we know this involves a reflection in the x-axis. This suggests a transformation of the form $g(x) = -f(x)$.

Vertical Stretch:

The values of $g(x)$ appear to be a vertically stretched version of $f(x)$. Specifically, the graph of $g(x)$ extends further on the y-axis compared to $f(x)$. This suggests a scaling factor, likely of 2 or more.

Vertical Translation:

The graph of $g(x)$ is shifted downward compared to $f(x)$. This suggests a vertical translation, implying the form $g(x) = -2f(x) - 4$.

Tentative Conclusion:

The transformation applied seems to be: $g(x) = -2f(x) - 4$

Would you like further clarification on this transformation, or is there anything you would like me to explore in more detail?

Here are five related questions:

1. What are other common transformations of functions?
2. How do you reflect a function along the y-axis instead of the x-axis?
3. How can you identify a vertical stretch vs. a vertical compression?
4. What is the effect of horizontal shifts on function graphs?
5. How do transformations affect the domain and range of a function?

Tip: When graphing transformations, always track key points (like intercepts) to understand shifts and stretches more clearly.