Problem 103 - 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. $$r(x)=(x-3)^{3}+2$$

Step-by-Step Solution

Answer

The function \(r(x) = (x-3)^3 + 2\) is graphed by shifting the graph of the basic cubic function \(f(x) = x^3\) three units to the right and two units up.

Step 1: Graph the Basic Cubic Function

The first cubic function to be graphed is the basic cubic: \(f(x) = x^3\). This function represents a basic curve that passes through the origin (0,0) and extends indefinitely to both the left and right. Points of interest could include (1,1), (-1,-1), (2,8) and (-2,-8). Plot these points to form the basic cubic curve.

Step 2: Understand the Transformation

The function that needs to be graphed is \(r(x) = (x-3)^3 + 2\). This is a transformation of the basic cubic function. The '(x-3)' part denotes a horizontal shift or translation of 3 units to the right. The '+2' part implies a vertical shift or translation of 2 units upward. This means that every point on the basic cubic function f(x) is shifted 3 units right and 2 units up to obtain the corresponding point on r(x).

Step 3: Graph the Transformed Function

Now, to graph \(r(x)\), take the points from the graph of \(f(x)\), shift each one 3 units to the right and 2 units up. The origin (0,0) moves to (3,2). Other points move accordingly: (1,1) becomes (4,3), (-1,-1) becomes (2,1), (2,8) becomes (5,10) and (-2,-8) becomes (1,-6). Plot these points and draw the curve similar to the basic cubic function.