Answer: To determine the transformation needed to change the green graph to the purple graph, we need to compare the two graphs and identify the differences. The green graph has a positive slope and a negative y-intercept, while the purple graph has a positive slope and a positive y-intercept. Therefore, we need to perform a vertical translation of the green graph by adding a positive constant to all of its y-coordinates.
In order to write this transformation in the form f(x) = a(bx + c) + d, we can use the following steps:
Since we are performing a vertical translation, we know that b = 1 and c = 0.
Since we are adding a positive constant to all of the y-coordinates, we know that a > 0 and d > 0.
We can determine the value of d by looking at the difference between the y-intercepts of the two graphs. In this case, the difference is 4 units, so d = 4.
We can determine the value of a by looking at any point on both graphs and finding the ratio of the vertical distance between them to the corresponding horizontal distance. For example, if we choose the point (2, 2) on both graphs, we can see that the vertical distance between them is 6 units and the horizontal distance is 2 units. Therefore, a = 3.
Putting it all together, we can write the transformation as f(x) = 3x + 4.
I hope this helps!