To solve this exercise, we need to match each function graph $y = f(x)$ with the correct derivative graph $y = g(x) = f'(x)$.

Solution Steps

1. Identify Key Features of $f(x)$:

   • Look for points where the slope of $f(x)$ is zero (peaks, valleys, or points where the graph changes direction). These points correspond to where $f'(x) = 0$ in the derivative graph.
   • Observe intervals where $f(x)$ is increasing or decreasing. If $f(x)$ is increasing, then $f'(x)$ should be positive; if $f(x)$ is decreasing, then $f'(x)$ should be negative.

2. Analyze Each $f(x)$ Graph:

   • For each of the graphs labeled $y = f(x)$, note where the function has a maximum or minimum (indicating a zero derivative), as well as intervals of increase or decrease.

3. Match with Corresponding $g(x) = f'(x)$:

   • After analyzing each $f(x)$ graph, find the corresponding graph in the bottom row that matches these patterns of zero crossings and signs.

Solution

Let's go through each $y = f(x)$ graph in the top row and match it with its corresponding derivative graph $y = g(x)$ in the bottom row.

1. First Graph (Top Row, Left) $y = f(x)$:

   • This function has two turning points around $x = -1$ and $x = 1.5$, where the slope is zero.
   • The derivative graph should cross the x-axis at these points.
   • Corresponding derivative graph: Middle Graph (Bottom Row).

2. Second Graph (Top Row, Center) $y = f(x)$:

   • This function has a turning point at around $x = -2$ and another at $x = 0.5$.
   • The derivative should cross the x-axis near these points.
   • Corresponding derivative graph: Right Graph (Bottom Row).

3. Third Graph (Top Row, Right) $y = f(x)$:

   • This function has a maximum point around $x = 1$.
   • The derivative graph should cross the x-axis around $x = 1$.
   • Corresponding derivative graph: Left Graph (Bottom Row).

Final Matches

The pairs for each $y = f(x)$ and its derivative $y = g(x)$ are:

• First Graph (Top Row, Left) with Middle Graph (Bottom Row).
• Second Graph (Top Row, Center) with Right Graph (Bottom Row).
• Third Graph (Top Row, Right) with Left Graph (Bottom Row).

Let me know if you would like further details on any step!

Related Questions

1. How do you determine the points where a function's derivative is zero by looking at the graph?
2. What does the shape of the derivative graph tell you about the concavity of the original function?
3. How can you tell if a function is increasing or decreasing by analyzing its derivative graph?
4. Why does a maximum or minimum in $f(x)$ translate to a zero in $f'(x)$?
5. How can inflection points in $f(x)$ be identified on the graph of $f'(x)$?

Tip

Always check where the original function $f(x)$ has turning points (maxima and minima) to find where the derivative $f'(x)$ should cross the x-axis.