Problem 56 - College Algebra with Modeling and Visualization

Question

Use translations to graph \(f .\) $$ f(x)=\sqrt[3]{x-1} $$

Step-by-Step Solution

Answer

Graph \( f(x) = \sqrt[3]{x-1} \) by shifting the cube root function \( \sqrt[3]{x} \) right by one unit.

Step 1: Identify the Parent Function

The given function is \( f(x) = \sqrt[3]{x-1} \). The parent function is \( g(x) = \sqrt[3]{x} \), which is the basic cube root function.

Step 2: Understand the Translation

Comparing \( f(x) = \sqrt[3]{x-1} \) to \( g(x) = \sqrt[3]{x} \), we observe \( x \) is replaced with \( x-1 \). This means the graph of the parent function \( g(x) \) is translated to the right by 1 unit.

Step 3: Graph the Parent Function

Begin by sketching the graph of the parent function \( g(x) = \sqrt[3]{x} \). The graph passes through the origin (0,0) and is symmetric about the origin, curving slightly upwards in the first quadrant and downwards in the third quadrant.

Step 4: Apply the Translation

Shift every point of the parent function \( g(x) \) one unit to the right. For example, the point (0,0) on \( g(x) \) moves to (1,0) on \( f(x) \), the point (1,1) moves to (2,1), and so forth.

Step 5: Sketch the Translated Function

Draw the graph of \( f(x) = \sqrt[3]{x-1} \) using the translated points. The graph will have the same shape as the parent function \( g(x) \), but it will start at (1,0) due to the rightward shift.

Key Concepts

Function Transformations

Function transformations allow us to modify the graphs of functions in specific ways, such as shifting, stretching, or reflecting them. When dealing with the cube root function, these transformations can help us understand how changes to the function's formula affect its graph.

• **Translation**: In the given problem, the transformation applied to the parent function is a translation. Specifically, we have substituted \( x \) with \( x-1 \). This represents a horizontal shift. Remember, replacing \( x \) with \( x-a \) shifts the graph to the right by \( a \) units. In contrast, \( x+a \) would shift it to the left by \( a \) units.
• **Vertical and Horizontal Stretches/Compressions**: Although not applicable to our current problem, transformations may also involve multiplying the function by a constant, which stretches or compresses its graph vertically or horizontally.
• **Reflection**: Another possible transformation is reflection, which flips the graph either over the x-axis or y-axis, but this is also not needed for our cube root focus in this exercise.

Understanding these transformations is crucial for visualizing how the graph of any function changes with its formula. For our function, the right shift gives us insight into how the \( f(x) = \sqrt[3]{x-1} \) graph is derived from the parent \( g(x) = \sqrt[3]{x} \).

Graphing Functions

Graphing functions is a fundamental skill in mathematics that helps us visualize how a function behaves. For cube root functions, noting specific characteristics of the graph can simplify understanding:

• **Symmetry**: The parent cube root function, \( g(x) = \sqrt[3]{x} \), is symmetric about the origin. This means it looks the same on both sides if you rotate it 180° around the origin.
• **Intercepts**: For \( g(x) = \sqrt[3]{x} \), it passes through the origin (0,0). After applying the transformation for \( f(x) = \sqrt[3]{x-1} \), the graph passes through (1,0).
• **Shape**: The cube root function's graph has a distinctive shape. It gradually increases in the first quadrant and decreases in the third quadrant, with each segment appearing as a mirrored curve about the origin for the parent function.

When graphing a transformed function like our \( f(x) \), starting by sketching the parent function first and then applying known transformations step-by-step is helpful. This way, you can ensure each component of the transformation is accurately depicted on your graph.

Parent Function

The parent function is a basic form of a certain family of functions. In our exercise, the parent function is \( g(x) = \sqrt[3]{x} \), the simplest form of cube root functions. Understanding the properties of the parent function is crucial before applying transformations.

• **Basic Properties**: The cube root function is defined for all real numbers. It is continuous and smooth, curving through key points like the origin.
• **Graph Characteristics**: Its graph is a gentle curve that extends indefinitely in both directions along the x-axis and y-axis. It may not pass through multiple quadrants like some other functions, but it exhibits unique behavior as part of each quadrant it occupies.
• **Template for Transformations**: By understanding the parent function well, transformations become easier to predict. All shifts, stretches, and reflections originate from the parent function's basic graph. Recognizing this template helps in systematically applying transformations to more complex functions.

With the parent function as a baseline, alterations like \( f(x) = \sqrt[3]{x-1} \) only adjust the basic graph according to the specific transformation rules applied, maintaining the signature curve of the cube root function family.