Write the explicit dependance of xi on the original

Question

Consider the transport equation u_t + 4u_x = 0. On the grid below sketch seven different characteristic curves for the equation. Let ξ be the characteristic variable. Write the explicit dependance of ξ on the original variables t,x. Set v(t, ξ) = u(t, x), and use the chain rule to compute u_x and u_y in terms of v, v_t, v_ξ, etc. Verify that the function v(t, ξ) satisfies the equation v_t = 0.