Problem 32 - A Graphical Approach to Precalculus with Limits

Question

The graph of \(y=0.5 \sqrt[3]{x+2}\) can be obtained from the graph of \(y=\sqrt[3]{x}\) by shifting horizontally ____________ units to the __________ and shrinking vertically by applying a factor of _________.

Step-by-Step Solution

Answer

2 units to the left and shrunk vertically by 0.5.

Step 1: Understanding the Base Function

The base function is given as \(y = \sqrt[3]{x}\). This is the graph of a cube root function, where the graph is symmetric with respect to the origin.

Step 2: Identifying the Horizontal Shift

Compare \(y = 0.5 \sqrt[3]{x+2}\) to \(y = \sqrt[3]{x}\). Note the expression \(x+2\) inside the cube root, which suggests a horizontal shift. The \(+2\) inside the cube root indicates that the graph is shifted 2 units to the left.

Step 3: Identifying the Vertical Shrink

The original function is multiplied by 0.5, meaning every y-value of \(y = \sqrt[3]{x+2}\) is half of what it would be without the multiplication. This describes a vertical shrink by a factor of 0.5.

Key Concepts

Horizontal Shift

When we talk about the horizontal shift of a function, it's all about how the graph moves left or right on the x-axis. The function given in the exercise is a modification of the cube root function, written as \(y = \sqrt[3]{x+2}\). Here, the expression inside the cube root, \(x+2\), is crucial.

• The term \(+2\) suggests that the graph is shifted horizontally.
• Specifically, because it's \(x + 2\), the graph will shift 2 units to the left.

This might seem a bit counterintuitive at first. Usually, you'd think "plus" means moving to the right. However, the standard rule is that if the term inside the function is \(x + c\), then the shift is actually \(c\) units to the left. Always remember:

• Addition inside transforms shift left.
• Subtraction inside transforms shift right.

Vertical Shrink

A vertical shrink affects how "tall" or "short" the graph appears. It involves multiplying the entire function by a constant. In our function \(y = 0.5 \sqrt[3]{x+2}\), the graph is multiplied by 0.5.

• This 0.5 tells us that the graph is vertically shrunk by a factor of 0.5.

To visualize this, imagine squeezing the graph towards the x-axis without shifting it horizontally. Each y-value in the original function \(y = \sqrt[3]{x}\) is halved in \(y = 0.5 \sqrt[3]{x+2}\). Consider this:

• If the graph reaches up to value 2 at some point, it now reaches only up to 1 after the shrink.
• The graph retains its general shape but appears squished vertically.

Cube Root Function

The cube root function, \(y = \sqrt[3]{x}\), is fundamental and distinctive. Its graph is symmetric around the origin, meaning it looks the same when you flip it along both the x-axis and y-axis. The function has a unique S-like shape.

• It's different from a square root function, which only exists for non-negative x-values.
• In contrast, the cube root function allows both positive and negative values for x, covering the whole of the x-axis.

The graph passes through the origin (0,0), indicating that the cube root of zero is also zero. There are three basic traits you often notice on such graphs:

• For positive x-values, the graph rises steadily.
• For negative x-values, the graph falls steadily.
• It always shows a smooth curvy path around the origin.

Understanding the basic cube root function helps in recognizing its transformations, such as shifts and shrinks, as seen in the modified version from the exercise. Such transformations let you manipulate the graph's position and shape effortlessly.