Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Describe the sequence of transformations from \(f(x)=\sqrt[3]{x}\) to \(y\). Then sketch the graph of \(y\) by hand. Verify with a graphing utility. \(y=\sqrt[3]{x+1}-1\)

Step-by-Step Solution

Answer

The function \(y = \sqrt[3]{x + 1} - 1\) is a transformation of the function \(f(x) = \sqrt[3]{x}\). It involves a shift of one unit to the left and one unit down. The graph of the new function can be drawn based on the graph of the original function with the identified transformations, and this can be verified using a graphing utility.

Step 1: Identify The Original Function

The original function is \(f(x) = \sqrt[3]{x}\), which is the cubic root of \(x\). It is important to remember that the graph of \(f\) increases as x increases, and it passes through the origin (0,0).

Step 2: Identify The New Function

The new function is \(y = \sqrt[3]{x + 1} - 1\). This function can be gotten from the original function through transformations.

Step 3: Transform The Original Function

The transformation can be seen by comparing the new function with the old one. The transformation consists of two parts. First, the function \(f(x)\) has been shifted one unit to the left. This is reflected in the \(x + 1\) part of the new function. Second, the function \(f(x)\) has been shifted one unit down. This is reflected in the \(- 1\) part of the new function.

Step 4: Sketch The Graphs

Start by sketching the graph of the original function \(f(x) = \sqrt[3]{x}\). This graph should pass through the origin and increase as \(x\) increases. Next, sketch the graph of the new function \(y = \sqrt[3]{x + 1} - 1\). The original graph should be shifted one unit to the left and one unit down.

Step 5: Verify With A Graphing Utility

Use a graphing tool to draw both \(f(x) = \sqrt[3]{x}\) and \(y = \sqrt[3]{x + 1} - 1\). The graph of the new function should match the hand-drawn sketch unless there is an error.

Key Concepts

Cubic Root Function

The cubic root function, represented by the equation \(f(x) = \sqrt[3]{x}\), is an interesting and unique type of function. Unlike the square root function, which only works for non-negative numbers, the cubic root can handle all real numbers.
This is because any real number, positive or negative, can be cubed, and therefore, its cubic root exists.

• The graph of the cubic root function is s-shaped, starting from the lower left quadrant, passing through the origin, and moving to the upper right quadrant.
• It is symmetric with respect to the origin, meaning if you rotate the graph 180 degrees around the origin, it remains unchanged.
• The function increases continuously as \(x\) increases, without any bounds.

Understanding the basic shape and properties of this function will help when applying transformations such as shifts in its graph.

Graphing

Graphing functions provides a visual representation of their behavior and transformations.
When you graph the cubic root function \(f(x) = \sqrt[3]{x}\), it appears smooth and passes through the origin (0,0).

• To graph this function, plot some key points like \((0,0)\), \((1,1)\), and \((-1,-1)\).
• Once these points are plotted, connect them considering the continuous curve of the cubic root function.
• Remember, it gradually rises from the left to the right, capturing the function’s continuous increase.

After understanding the graph of the original cubic root function, you can predict and verify the results of transformations like shifts, which alter the function's position on the graph without changing its shape.

Horizontal Shift

A horizontal shift in function transformations involves moving the function graph left or right on the coordinate plane.
The function \(y = \sqrt[3]{x+1}\) showcases a horizontal shift.

• In this case, the \"+1\" inside the root indicates a shift to the left by 1 unit.
• To apply this transformation, take all points on the original graph and move them 1 unit to the left.
• This doesn’t affect how the graph stretches or compresses; it merely alters its horizontal position.

Recognizing how to detect horizontal shifts from equations is crucial in sketching and graphing transformed functions accurately.

Vertical Shift

Vertical shifts adjust the position of the graph up or down along the y-axis. For the function \(y = \sqrt[3]{x+1} - 1\), there’s a clear vertical shift.

• The \"-1\" outside the root signifies a downward shift by 1 unit.
• Each point on the function graph is moved 1 unit lower than its original position.
• Like horizontal shifts, vertical shifts do not change the shape of the graph, only its position.

By combining both horizontal and vertical shifts, the graph of \(y = \sqrt[3]{x+1} - 1\) is shifted leftward and downward, altering its placement while maintaining the original function's distinctive s-shape.