Problem 106 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The function \(h(x)=\frac{1}{2}(x-2)^{3}-1\) is derived from the standard cubic function \(f(x)=x^3\) by shifting the graph right by 2 units, down by 1 unit and compressing it vertically by a factor of 1/2.

Step 1: Graph the standard function

Begin by graphing the standard cubic function \(f(x)=x^3\). This function represents a simple cubic curve, starting at negative infinity, crossing the origin (0,0), and extending to positive infinity.

Step 2: Observe the transformation

The function \(h(x)=\frac{1}{2}(x-2)^{3}-1\) can be thought of as the graph of \(f(x)=x^3\) that has been shifted horizontally to the right by 2 units, vertically downward by 1 unit, and vertically compressed by a factor of 1/2.

Step 3: Apply horizontal shift

The term \((x-2)\) in the function implies a horizontal shift of 2 units to the right. So, move every point on the graph of \(f(x)\) two units to the right.

Step 4: Apply vertical shift

The term -1 in the function implies a downward shift of 1 unit. So, move every point on the horizontally shifted graph down one unit.

Step 5: Apply vertical compression

The coefficient of \(\frac{1}{2}\) on the \((x-2)^3\) term means we compress the graph vertically by a factor of 1/2. Multiply each y-coordinate by 1/2 to get the final graph.

Key Concepts

Standard Cubic Function

Starting with the foundations, the standard cubic function is of the form f(x) = x^3. When plotted on a graph, it reveals a distinctive 'S'-shaped curve, known for its symmetry about the origin. The graph passes through the origin point \(0,0\) and extends infinitely in both the positive and the negative directions of the y-axis.

Its key feature is that for any positive value of \(x\), the \(y\)-value will also be positive, whereas for any negative value of \(x\), the \(y\)-value will be negative. Hence, this creates the smooth rise and fall that characterizes cubic functions. Understanding the behavior of the standard cubic function is crucial because it serves as the starting point for graphing more complex cubic equations, which are derived from it through various transformations.

Horizontal Shift

When a cubic function includes a term like \(\left(x-h\right)^3\), where \(h\) is a constant, it leads to a horizontal shift of the entire graph. In the context of \(h(x)=\frac{1}{2}(x-2)^{3}-1\), the term \(x-2\) represents a shift to the right by 2 units.

If the term had been \(x+2\), the shift would be to the left by 2 units instead. It's important to remember that the sign inside the parentheses is opposite to the direction of the shift. The horizontal shift is a sideways movement which does not alter the shape of the graph but repositions it along the x-axis. For students, a handy tip is to note that each point on the original graph is moved horizontally by the same amount, and this transformation can be visualized by imagining sliding the entire curve along the x-axis to its new location.

Vertical Shift

A vertical shift, as the name implies, moves the graph up or down along the y-axis. This transformation is denoted by a constant added or subtracted from the entire function, such as the '-1' in the function \(h(x)=\frac{1}{2}(x-2)^{3}-1\). Here, after applying the horizontal shift, a vertical shift of one unit downward is made. This doesn't influence the graph's symmetry or stretch, only its position relative to the y-axis.

It is the simpler kind of transformation, where if the number is positive, the shift is upwards, and if negative, like in our case, the shift is downwards. It's like picking up the graph and moving it straight up or down without changing its orientation or size.

Vertical Compression

Lastly, the concept of vertical compression deals with how 'tall' or 'short' the graph appears. A cubic function is vertically compressed when it is multiplied by a factor between 0 and 1, as seen in \(\frac{1}{2}(x-2)^{3}\).

Vertical compression squashes the graph towards the x-axis, making it less steep. In contrast, a factor greater than 1 would result in a vertical stretch, making the graph steeper. For the given function \(h(x)\), every y-coordinate of the transformed graph \(\left(x-2\right)^3\) is halved, reducing the height of each point and giving the impression that the graph has been pressed down from above. Students should imagine this as if the y-coordinates are weighted and the invisible 'weights' are pulling the graph closer to the x-axis.