From the image, you are asked to determine the equation of the function $f(x)$ in terms of $g(x)$, based on the given graph and the provided transformations (stretches, reflections, and translations).

Key Observations:

• $g(x)$ is a piecewise linear function that appears to be composed of two distinct line segments.
• $f(x)$ is also a linear function but seems to be transformed in relation to $g(x)$.

Steps to Determine $f(x)$ in Terms of $g(x)$:

1. Identify the equations of the given functions:

   • The graph shows two lines, $y = f(x)$ (red) and $y = g(x)$ (blue).
   • You should first write the explicit equation for both lines.
   • Based on the graph, it looks like $g(x)$ has a slope of 1 and passes through $(0, 2)$, so we hypothesize: $g(x) = x + 2$
   • The function $f(x)$ appears to have a negative slope and crosses the x-axis at $x = -4$ and y-axis at $y = 2$, so: $f(x) = -\frac{1}{2}x + 2$

2. Look at potential transformations:

   • Stretch: $f(x)$ seems like it might involve a vertical stretch, as its slope differs from $g(x)$.
   • Reflection: $f(x)$ has a negative slope, which could indicate a reflection over the x-axis relative to $g(x)$.
   • Translation: Both functions appear to have the same y-intercept, so no vertical translation is present, but there might be a horizontal shift.

3. Transformations applied to $g(x)$ to get $f(x)$:

   • First, apply a reflection over the x-axis, which would change the slope sign: $g(x)$ becomes $-g(x)$, so: $f(x) = -g(x)$
   • Next, apply a stretch by a factor of $\frac{1}{2}$, affecting the slope, so: $f(x) = -\frac{1}{2}g(x)$

Conclusion:

Thus, the equation of $f(x)$ in terms of $g(x)$ is: $f(x) = -\frac{1}{2}g(x)$

Would you like more details, or do you have any questions?

Expanding Questions:

1. Can you verify the equation of $g(x)$ from the graph?
2. How would you calculate the slope of $f(x)$ from the graph?
3. What are the effects of reflecting a function over the x-axis?
4. How do vertical stretches affect the slope of a function?
5. Can you identify horizontal translations between the two functions?

Tip:

When performing transformations, always apply reflections and stretches first before considering translations.