A Vertex Form of a cubic equation is: a_o (a_i x - h)³ + k
If a ≠ 0, this equation is a cubic which has several points:
Inflection (Turning) Point
1, 2, or 3 x-intecepts
1 y-intercept
Maximum/Minimum points may occur

There is a sample charge at on the worksheet. Create a similar chart on your paper; for the sketch column, allow more room. Move the sliders a, b, and c to create a record of your work.
Examine the table to see if you can determine a pattern for the variables a, b, c, and d from the chart. I suggest testing initially with only one slider at a time. Do a second chart varying only two sliders; then vary all three.