1. The problem asks us to sketch a graph of Karen's speed versus time with three segments: a constant slow speed, a gradual increase, and a constant peak speed. 2. We represent speed as a function of time, $y = f(t)$, where $y$ is speed and $t$ is time. 3. The first segment is a horizontal line at a low speed, say $y = s_1$, for $0 \leq t \leq t_1$. 4. The second segment is an increasing line from $s_1$ to $s_2$ over $t_1 < t \leq t_2$, which can be represented as $y = m(t - t_1) + s_1$ where $m = \frac{s_2 - s_1}{t_2 - t_1}$. 5. The third segment is a horizontal line at the peak speed $s_2$ for $t > t_2$. 6. Putting it all together, the piecewise function is: $$ f(t) = \begin{cases} s_1 & 0 \leq t \leq t_1 \\ m(t - t_1) + s_1 & t_1 < t \leq t_2 \\ s_2 & t > t_2 \end{cases} $$ 7. This graph starts flat at $s_1$, rises linearly to $s_2$, then stays flat at $s_2$. 8. This matches the description: slow constant speed, gradual increase, then constant peak speed. Final answer: The speed versus time graph is a piecewise function with two horizontal segments and one increasing linear segment as shown above.