Problem 48 - Algebra and Trigonometry

Question

\(45-54=\) A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=\sqrt[3]{x},\) shift 1 unit to the right

Step-by-Step Solution

Answer

The final equation is \( f(x) = \sqrt[3]{x-1} \).

Step 1: Recognize the Basic Transformation

In this problem, we have been given a cubic root function, which is a known base function. The simplest form of a cubic root function is given by \( f(x) = \sqrt[3]{x} \). We are asked to perform a transformation on this function.

Step 2: Identify the Transformation Needed

The transformation requires the graph of the function \( f(x) = \sqrt[3]{x} \) to be shifted 1 unit to the right. Shifting a graph horizontally to the right by \( c \) units is expressed by replacing \( x \) with \( x-c \) in the function.

Step 3: Apply the Transformation

Since we need to shift the function 1 unit to the right, we replace \( x \) with \( x-1 \). Therefore, the transformed function is \( f(x) = \sqrt[3]{x-1} \).

Key Concepts

Cubic Root Function

The cubic root function is a fundamental function in mathematics characterized by the expression \( f(x) = \sqrt[3]{x} \). It is the inverse of the cubic function, where a number \( y \) is the cube of a number \( x \), that is, \( y = x^3 \). In this reverse relationship, the cubic root seeks the number which, when cubed, gives \( x \).

• The graph of the cubic root function is a curve that smoothly extends from the bottom left to the top right quadrant, unlike quadratic functions that create parabolas or lines that maintain linearity.
• Because it involves cubes, which can result in negatives or positives, the cubic root function covers all real numbers as its domain and range.

Exploring how this function interacts with transformations such as shifts is essential to understanding its flexibility and behavior.

Horizontal Shift

A horizontal shift in the graph of a function alters its position along the x-axis. It's a critical transformation used to tune the graph's location without modifying its shape.Horizontal shifts are performed by changing the input variable of the function:

• To shift the graph to the right by \( c \) units, replace each instance of \( x \) with \( x-c \).
• Conversely, a shift to the left by \( c \) units involves switching \( x \) for \( x+c \).

In the cubic root function context, shifting 1 unit to the right is achieved by substituting \( x \) with \( x-1 \), which seamlessly transitions the entire graph rightward along the x-axis without distorting its cubic root properties.

Graph Transformation

Graph transformation includes a variety of operations that can modify the position, size, or orientation of a graph. They allow functions to be easily adjusted to fit particular conditions or data sets, making them versatile tools in both theoretical and applied mathematics. Key transformations include:

• Translations: This involves shifting the graph up, down, left or right, without affecting its shape.
• Reflections: A graph might be flipped over a specific axis, altering its vertical or horizontal orientation.
• Scaling: In which the graph stretches or compresses, either vertically or horizontally.

In the problem at hand, a simple horizontal translation was used to adjust the cubic root function, demonstrating how even basic transformations can significantly alter the graph's position and impact its analytical representation.