Problem 60 - Beginning and Intermediate Algebra

Question

Graph \(f(x)=\sqrt[3]{x}\) by plotting points. (Hint: Make a table of values and choose \(0,\) positive, and negative numbers for \(x\) ) Then, use the transformation techniques discussed in this section to graph each of the following functions. $$\text{a)}\quad g(x)=\sqrt[3]{x}+4\quad\text{b)}\quad h(x)=-\sqrt[3]{x}\quad\text{c)} \quad k(x)=\sqrt[3]{x-2}\quad\text{d)}\quad $r(x)=-\sqrt[3]{x}-3$$

Step-by-Step Solution

Answer

Graph \(f(x)=\sqrt[3]{x}\) by plotting points in a table, then use transformation techniques to graph: a) \(g(x)=\sqrt[3]{x}+4\) - vertical shift of 4 units upward b) \(h(x)=-\sqrt[3]{x}\) - reflection across the x-axis c) \(k(x)=\sqrt[3]{x-2}\) - horizontal shift of 2 units to the right d) \(r(x)=-\sqrt[3]{x}-3\) - reflection across the x-axis and a vertical shift of 3 units downward

Step 1: Choose values for \(x\) and create a table.

Select different values for \(x\), such as \(x=-3, -2, -1, 0, 1, 2\) and \(3\). Calculate the cube root of each value and create a table.

Step 2: Calculate \(f(x)\)

Using the values chosen for \(x\), calculate \(f(x) = \sqrt[3]{x}\) for each value and fill the table. \[ \begin{array}{|c|c|c|c|c|c|c|} \hline x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline f(x) & \sqrt[3]{-3} & \sqrt[3]{-2} & \sqrt[3]{-1} & \sqrt[3]{0} & \sqrt[3]{1} & \sqrt[3]{2} & \sqrt[3]{3} \\ \hline \end{array} \]

Step 3: Graph f(x)

Plot the points from the table, and then draw the graph of the function \(f(x)=\sqrt[3]{x}\). 2. Then, use transformation techniques to graph the functions

Step 4: Plot g(x)

Since \(g(x) = \sqrt[3]{x} + 4\), a vertical shift of 4 units upward is applied to the graph of f(x). Obtain the points by adding 4 to the y-values of the previous table, plot them, and draw the graph of \(g(x)=\sqrt[3]{x} + 4\).

Step 5: Plot h(x)

Since \(h(x) = -\sqrt[3]{x}\), a reflection across the x-axis is applied to the graph of f(x). Obtain the points by multiplying the y-values of the table by -1, plot them, and draw the graph of \(h(x)=-\sqrt[3]{x}\).

Step 6: Plot k(x)

Because \(k(x) = \sqrt[3]{x-2}\), a horizontal shift of 2 units to the right is applied to the graph of f(x). Obtain the points by adding 2 to the x-values of the table, plot them, and then draw the graph of \(k(x)=\sqrt[3]{x-2}\).

Step 7: Plot r(x)

Similarly, with \(r(x) = -\sqrt[3]{x} - 3\), a reflection across the x-axis combined with a vertical shift of 3 units downward is applied to the graph of f(x). Obtain the points by multiplying the y-values of the table by -1 and then subtracting 3, plot them, and then draw the graph of \(r(x)=-\sqrt[3]{x}-3\).

Key Concepts

Cube Root Function

The cube root function, denoted as \( f(x) = \sqrt[3]{x} \), is a fascinating one to explore. Unlike the square root function, it can take both positive and negative values of \( x \). This is because the cube of a negative number is still negative, making the cube root a real number.
To graph this function, a simple approach is to create a table of values. You can choose a range of \( x \) values, including negative numbers, zero, and positive numbers to get a sense of the curve. For example:

• \( x = -8, \sqrt[3]{x} = -2 \)
• \( x = -1, \sqrt[3]{x} = -1 \)
• \( x = 0, \sqrt[3]{x} = 0 \)
• \( x = 1, \sqrt[3]{x} = 1 \)
• \( x = 8, \sqrt[3]{x} = 2 \)

Plotting these points will reveal a smooth curve passing through the origin, illustrating how the function behaves across its domain.

Function Transformations

Function transformations allow us to manipulate the appearance of graphs without altering the fundamental nature of the function. These transformations include:

• Translations (shifts up, down, left, right)
• Reflections (flipping across the axes)
• Stretches & compressions (changing the steepness and widening/narrowing effect)

Applying these to our cube root function can create new graphs with familiar shapes but different positions or orientations. This is helpful to understand how each transformation affects the graph visually and numerically. Let's delve into specific types of transformations: vertical and horizontal shifts.

Vertical Shifts

Vertical shifts are straightforward transformations where each point of a function is moved up or down the same number of units. Typically, an equation like \( g(x) = \sqrt[3]{x} + c \) indicates a shift.

• For \( g(x) = \sqrt[3]{x} + 4 \), the function moves 4 units up. Each y-coordinate increases by 4, affecting the vertical position.
• In contrast, \( r(x) = -\sqrt[3]{x} - 3 \) shifts 3 units downward. Here, the vertical shift combines with a reflection across the x-axis, which will invert the graph vertically before shifting it down.

Vertical shifts are simple yet have a profound impact, allowing functions to be relocated up and down the plane while retaining their characteristic shape.

Horizontal Shifts

Horizontal shifts involve moving a graph to the left or right along the x-axis. The general form for horizontal shifts involves terms like \( k(x) = \sqrt[3]{x - b} \).

• When the function is \( k(x) = \sqrt[3]{x - 2} \), the entire graph moves 2 units to the right. Each x-coordinate increases by 2 but keeps the y-coordinate unchanged.
• If the term was \( x + b \) instead of \( x - b \), the graph would shift \( b \) units to the left.

Horizontal shifts, while slightly more complex in notation, are just as simple in execution as vertical shifts. They allow us to tailor the position of the function horizontally on the graph.