1. The problem involves extending the slope-field tick mark from $x=0$ to $x=0.25$ and identifying the $y$-coordinate of the newest dot at $x=0.25$. 2. A slope field represents the slopes of solutions to a differential equation at various points $(x,y)$. 3. To find the $y$-coordinate at $x=0.25$, we typically use the differential equation governing the slope field and apply an initial condition or use numerical methods like Euler's method. 4. Since the exact differential equation and initial conditions are not provided, we cannot compute the exact $y$-coordinate here. 5. However, the process involves: - Using the slope at the previous point (e.g., at $x=0$) to estimate the change in $y$ over the interval $0$ to $0.25$. - Applying the formula $y_{new} = y_{old} + m \Delta x$, where $m$ is the slope at the previous point. 6. Without the slope or initial $y$ value, the $y$-coordinate at $x=0.25$ remains unknown. Therefore, the problem as stated cannot be solved without additional information about the differential equation or initial values.