Problem 104 - Precalculus Essentials

Question

Begin by graphing the standard cubic function, \(f(x)=x^{3} .\) Then use transformations of this graph to graph the given function. $$r(x)=(x-2)^{3}+1$$

Step-by-Step Solution

Answer

To plot the function \(r(x) = (x-2)^3 + 1\) using the standard cubic function \(f(x) = x^3\), start by plotting \(f(x) = x^3\), then shift this graph 2 units to the right (because of '-2') and then 1 unit upward (because of '+1'). This results in the graph of the function \(r(x) = (x-2)^3 + 1\).

Step 1: Plot the Standard Cubic Function

Begin by plotting the standard cubic function \(f(x) = x^3\). This function simply cubes each 'x' value to give the corresponding 'y' value. The graph should show that as 'x' increases or decreases, 'y' also increases or decreases, respectively but at an accelerating rate.

Step 2: Apply the Horizontal Shift

Next step is to apply the horizontal shift on the graph. The '-2' inside the function \(r(x) = (x-2)^3 + 1\), shifts the entire graph of \(f(x) = x^3\) to the right by 2 units. Note that values inside the function will always affect the graph in opposite direction, hence '-2' shifts the graph to the right by 2 units.

Step 3: Apply the Vertical Shift

After applying the horizontal shift, apply the vertical shift to the graph. The '+1' outside of the function \(r(x) = (x-2)^3 + 1\), moves the graph of \(f(x) = x^3\) that is already shifted horizontally, now 1 unit upward.