You have drawn the graph. Now, by adding -k to the whole equation outside the absolut bracket, you are shifting the graph vertically. You are required to find the values of k for different number of solutions. You can do this by taking a ruler and putting it horizontally on the sketch, sliding it up and down and see where the ruler cuts how many points.

For part ii, you are required to find k value for 3 solutions. The only point the graph will have 3 solutions (cutting x axis) is below the turning point, where the solutions are the graph going up to the turning point, the graph going down from the turning point, and the right hand side of the parabola going infinity. Therefore you must shift the graph downwards by less than the x value of the turning point, and that is your k.

For 2 solutions, you have only one point where it gives 2 solutions, which is when the turning point touches x axis. This give the solutions at the turning point and the right hand side of the parabola. Therefore the value of k is the x value of the turning point.

For 1 solution, you have two ranges. The original graph already have one solution, which is the cutting point of the right hand side of the parabola. The next range will be the right side towards x=5, where it is above the turning point. Therefore the first range of k is k=0, and k more than the x value of the turning point

Hope it helps!

Thanks so much! I understand it now.
Thanks for detailed answer too!