This image shows graphs of several functions labeled $y = f(x)$ and $y = g(x)$. The task is to match each function $y = f(x)$ with its corresponding derivative $y = g(x)$ graph.

To match these correctly, let's analyze the features of each graph, specifically focusing on:

1. The slope behavior of $f(x)$: The derivative $g(x)$ will have zeroes where $f(x)$ has local maxima, minima, or points of inflection, as these represent points where the slope of $f(x)$ changes.
2. Positive and negative slopes: In regions where $f(x)$ is increasing, $g(x)$ will be positive, and where $f(x)$ is decreasing, $g(x)$ will be negative.

Let's proceed by matching each $y = f(x)$ graph (top row) to its corresponding $y = g(x)$ graph (bottom row). Here is the step-by-step approach:

1. First $f(x)$ graph (far left):

   • It has a minimum around $x = -2$ and a maximum near $x = 2$.
   • The derivative $g(x)$ should pass through zero around $x = -2$ and $x = 2$.
   • The best match for this graph appears to be the second $g(x)$ graph in the bottom row.

2. Second $f(x)$ graph (second from left):

   • It has a minimum near $x = -3$ and a maximum near $x = 1$.
   • The derivative $g(x)$ should have zeroes at these points.
   • The best match seems to be the fourth $g(x)$ graph in the bottom row.

3. Third $f(x)$ graph (third from left):

   • Shows a parabola opening downwards with a maximum point around $x = 0$.
   • The derivative $g(x)$ should be a linear graph crossing the x-axis at $x = 0$.
   • The best match is the first $g(x)$ graph in the bottom row.

4. Fourth $f(x)$ graph (far right):

   • This function has a maximum near $x = -1$ and a minimum near $x = 3$.
   • The derivative $g(x)$ should cross the x-axis at these points.
   • The best match is the third $g(x)$ graph in the bottom row.

By matching based on these observations, here is the solution:

• First $f(x)$ goes with Second $g(x)$
• Second $f(x)$ goes with Fourth $g(x)$
• Third $f(x)$ goes with First $g(x)$
• Fourth $f(x)$ goes with Third $g(x)$

Would you like further clarification on any of these steps, or an explanation of the derivative properties?

Related Questions

1. How can we identify points of inflection in a function's graph?
2. What does the derivative graph tell us about the original function's concavity?
3. How do the critical points of $f(x)$ correspond to the zeroes of $g(x)$?
4. How can we tell if a function is concave up or concave down based on its derivative?
5. What role do asymptotes play in matching functions with their derivatives?

Tip

Always check where the slope of $f(x)$ is zero to find potential matches for $g(x)$, as these are key indicators of derivative behavior.