Problem 42 - Thomas Calculus

Question

Graph the functions in Exercises \(35-54\) $$ y=(x-8)^{2 / 3} $$

Step-by-Step Solution

Answer

The graph of \( y = (x-8)^{2/3} \) is the graph of \( y = x^{2/3} \), shifted 8 units to the right.

Step 1: Recognize the Function Type

The function given is \( y = (x-8)^{2/3} \). This is a transformed version of the function \( y = x^{2/3} \), which is a power function.

Step 2: Identify the Transformation

The transformation involves replacing \( x \) with \( x-8 \). This indicates a horizontal shift of the graph of \( y = x^{2/3} \) to the right by 8 units.

Step 3: Graph the Parent Function

First, graph the parent function \( y = x^{2/3} \). This function has domain \( x \geq 0 \) and possesses a point at the origin (0,0). As \( x \) increases, \( y \) increases but at a decreasing rate. The graph is symmetrical to the x-axis in the first quadrant due to the even power in the numerator.

Step 4: Plot Key Points for the Transformed Graph

To graph \( y = (x-8)^{2/3} \), we can determine key points by shifting points from the graph of \( y = x^{2/3} \). For example, the point (0,0) on the parent function becomes (8,0) on the transformed function.

Step 5: Draw the Transformed Graph

Using the key points identified, plot the transformed graph with its focal point now at (8,0) instead of (0,0). Since this is a horizontal shift, the shape remains the same—flattened near the x-axis and broader as x moves to the right.

Key Concepts

Power Functions

Power functions are characterized by expressions in the form of \( y = x^n \), where \( n \) is a real number. In our exercise, we have the power function \( y = x^{2/3} \). Here, \( n \) is a fraction, which means the graph will have a distinct shape. For fractional powers, the graph starts at the origin and begins to rise, hugging closely to the x-axis, and eventually expanding outward as x increases. These functions often have properties like symmetry and a particular curvature.
The power \( 2/3 \) tells us that while the graph rises as x increases, the rate at which it rises decreases.
That is why, even if there's no limit to which y can grow, the growth slows down as x becomes larger. This kind of function is often referred to as a root function because it involves extracting roots of x.

Horizontal Transformations

Horizontal transformations involve shifting a graph left or right on the coordinate plane.
This is done by adjusting the variable inside the function's parentheses. For example, if we replace \( x \) with \( x-8 \), as seen in \( y = (x-8)^{2/3} \), we shift the graph to the right by 8 units. This transformation doesn't affect the shape of the graph, only its position.
Imagine you have the graph of the function \( y = x^{2/3} \), which naturally passes through the origin (0,0). When we perform a horizontal shift to the right by 8 units, each point on the graph moves 8 units to the right, including the point (0,0) moving to (8,0). Keep in mind that only the x-values change, while the y-values remain untouched.

Plotting Key Points

Plotting key points involves identifying certain critical points on a graph that define its shape and position.
These points help in sketching the transformed graph quickly and accurately. To plot key points for a transformed function like \( y = (x-8)^{2/3} \), start by finding key points from the parent function \( y = x^{2/3} \).
For instance, one simple point is (0,0) on the parent graph. After the transformation of shifting horizontally to the right by 8, this point becomes (8,0) on the new graph. Identifying such points ensures that the graphed function is accurate.
To confirm the graph's shape, you can plot additional points by choosing other values of x, substituting them into the transformed function, and plotting the corresponding y-values.