To solve the problem of finding $g(x)$ in terms of $f(x)$ based on the given graph, let's analyze the situation.

Observations from the graph:

• The graph of $f(x)$: The solid line represents the function $f(x)$, while the dashed line represents $g(x)$.
• Key points:
  • Point $A(1, 6)$ is on $f(x)$, meaning $f(1) = 6$.
  • Point $A'(7, 2.5)$ is on $g(x)$, so $g(7) = 2.5$.
  • Point $B(-2, -30)$ is on $g(x)$, so $g(-2) = -30$.
  • Point $B'(-8, -6.5)$ is on $f(x)$, so $f(-8) = -6.5$.

Step-by-Step Reasoning:

1. Vertical shifts: $g(x)$ appears to be a vertically shifted version of $f(x)$, as both graphs exhibit similar shapes but are shifted along the vertical axis. To determine the exact shift, let's check the relationship between the corresponding points on the two graphs.

2. Horizontal shifts: There may also be a horizontal translation, as the points along the x-axis do not perfectly align between $f(x)$ and $g(x)$.

3. Finding the transformation: Let’s check the x- and y-values at key points to identify the transformation. For example:

   • $f(1) = 6$ and $g(7) = 2.5$. This suggests a horizontal shift from $x = 1$ to $x = 7$, so $g(x)$ might be shifted by $+6$ units horizontally.
   • The vertical change between the two points is $6 - 2.5 = 3.5$, suggesting a vertical shift of $-3.5$ units.

Hypothesis:

It seems like $g(x)$ is a horizontally shifted and vertically shifted version of $f(x)$. Based on the points, the transformation rule for $g(x)$ in terms of $f(x)$ could be:

$g(x) = f(x - 6) - 3.5$

This implies that $g(x)$ is obtained by shifting $f(x)$ 6 units to the right and 3.5 units downward.

Explanation:

• Horizontal shift: The $x - 6$ represents a rightward shift by 6 units.
• Vertical shift: The $-3.5$ accounts for the downward shift.

Would you like more details or have any specific questions about this solution?

Here are 5 questions related to this concept:

1. How does a horizontal shift affect the equation of a function?
2. What is the effect of a vertical shift on a function’s graph?
3. Can you identify other transformations in functions based on graph observations?
4. What is the general form for expressing a combination of horizontal and vertical shifts in a function?
5. How would the equation change if there were a reflection involved?

Tip: Always check key points on graphs to deduce transformations like shifts, stretches, or reflections.