The graph of f(x) = (x - 1)^3 - 1 is a cubic function shifted to the right by 1 unit and down by 1 unit from the parent function y = x^3. Start by plotting the inflection point at (1, -1) and additional points by substituting x-values into the function, then draw the smooth cubic curve.
To sketch the graph of f(x) = (x - 1)^3 - 1, you'll follow a series of steps. Firstly, recognize that this function is a transformation of the parent function y = x^3. The graph of f(x) will be shifted to the right by 1 unit and down by 1 unit from the original cubic function.
Begin by plotting the point of inflection at (1, -1), which is the transformed origin of the parent function. Next, as with all cubic functions, as x increases or decreases away from this point, the cubic curve gets steeper. To better illustrate this, plot additional points by choosing x-values and calculating the corresponding y-values using the function. For example:
When x = 0, f(x) = (-1)^3 - 1 = -2, so plot the point (0, -2).
When x = 2, f(x) = (1)^3 - 1 = 0, so plot the point (2, 0).
Continue plotting points for a range of x-values to clearly define the curve.
After plotting the points, draw the smooth curve that passes through them, reflecting the nature of a cubic function, which goes to negative infinity as x goes to negative infinity and positive infinity as x goes to positive infinity.