91Ó°ÊÓ

To find the position function, integrate the velocity function with respect to time: $$ s(t) = \int v(t) dt = \int (2t + 4) dt $$

Integrating the velocity function and adding a constant of integration \(C\): $$ s(t) = t^2 + 4t + C $$

Using the initial condition \(s(0) = 0\), we can determine the value of \(C\): $$ 0 = 0^2 + 4(0) + C $$ From this, we can see that \(C = 0\), and our position function becomes: $$ s(t) = t^2 + 4t $$

Now that we have both the velocity function, \(v(t) = 2t + 4\), and the position function, \(s(t) = t^2 + 4t\), we can graph them. For this step, use graphing tools or software to plot the graphs for both functions. The horizontal axis represents time (t) and the vertical axis represents the function values. The velocity function, \(v(t)\), is a linear function with a slope of 2 and intersects the vertical axis at 4. The position function, \(s(t)\), is a quadratic function with a vertex at the origin and is symmetric about the line \(t=2\). By analyzing and comparing these two graphs, we can visualize how the velocity of the object changes with respect to time and how this affects the object's position along a line.