7. Give all transformations that occur to the function y = x ^ 3 that produce the function y = 1/3 * x ^ 3 + 5 Give both the transformations and the order in which they occur.

See explanantion

Step-by-step explanation:

Solution:-

- We will start of with the given function y = x^3, and plot a small sketch.

First transformation:-

- We multiply the y = x^3, with a constant a = 1/3. Where,

                                   y1 = a*y

Where,                 a : real constant

- The cases for a:

         a > 1 > makes the function steeper

         0 < a < 1  > Flattens the graph/less steep

         a < - 1   > Reflection about x-axis and makes the function steeper.

       0 < a < -1  > Flattens the graph/less steep and reflection about x-axis

- For a = 1/3, the graph flattens or in other words with increasing value of "x" the value of "y" increases at a decreased rate.

Second Transformation:-

- We add the b = 5 to the resulting graph y = 1/3*x^3.

                                y2 = 1/3*x^3 + b

Where,                     b : Real constant

- The cases for b:

           b > 0  > Vertical upward shift of the graph by b units

           b < 0 > Vertical downward shift of the graph by b units

- For b = 5 > 0, The graph 5 units up or in other words the y-intercept shifts from y = 0 to y = 5.

- The required function y = 1/3*x^3 + 5 undergoes two transformation. a = 1/3 flatten out the starting function y = x^3 and b = 5 shifts the resulting graph vertically upwards.