1. The problem asks to sketch a graph of a bike ride as a function of distance traveled over time. 2. To model this, we use a piecewise function $f(t)$ where $t$ is time and $f(t)$ is distance. 3. A continuous piecewise function means the end of one sub-function matches the start of the next, ensuring no jumps. 4. Example function: $$ f(t) = \begin{cases} 2t & 0 \leq t < 3 \\ 6 + (t-3) & 3 \leq t < 5 \\ 8 - 0.5(t-5) & 5 \leq t \leq 7 \end{cases} $$ 5. Explanation: - From $0$ to $3$ hours, distance increases at $2$ units/hour: $f(t) = 2t$. - From $3$ to $5$ hours, distance increases slower: $f(t) = 6 + (t-3)$. - From $5$ to $7$ hours, distance decreases as the rider returns: $f(t) = 8 - 0.5(t-5)$. 6. Check continuity at $t=3$: $$2 \times 3 = 6$$ $$6 + (3-3) = 6$$ Both equal 6, so continuous. 7. Check continuity at $t=5$: $$6 + (5-3) = 8$$ $$8 - 0.5(5-5) = 8$$ Both equal 8, so continuous. 8. This function models a bike ride with different speeds and a return segment, continuous over $[0,7]$. Final answer: The piecewise function above represents the bike ride distance over time continuously.