Problem 96 - A Graphical Approach to Precalculus with Limits

Question

Use transformations to explain how the graph of the given function can be obtained from the graphs of the square root function or the cube root function. $$y=\sqrt[3]{8 x-8}$$

Step-by-Step Solution

Answer

Right 1 unit, vertically stretch by 2.

Step 1: Identify the base function

The base function we begin with is the cube root function, which is the simplest form, \( y = \sqrt[3]{x} \).

Step 2: Factor the expression inside the root

Rewrite the expression inside the cube root to identify any transformations. We have the function \( y = \sqrt[3]{8(x-1)} \). This can be rewritten by factoring out 8 from \( 8x - 8 \) as: \( y = \sqrt[3]{8} \cdot \sqrt[3]{x-1} \).

Step 3: Determine the transformations

1. Horizontal translation: The inner expression \( x-1 \) represents a translation to the right by 1 unit. 2. Vertical stretch: The factor \( \sqrt[3]{8} = 2 \) indicates a vertical stretch by a factor of 2.

Step 4: Apply the transformations

Starting from the graph of \( y = \sqrt[3]{x} \): 1. Translate the graph 1 unit to the right, reflecting the transformation from \( x \) to \( x-1 \). 2. Apply a vertical stretch, multiplying all y-values by 2, due to the factor \( \sqrt[3]{8} = 2 \).

Key Concepts

Cube Root Function

The cube root function is a critical concept to grasp when studying transformations of graphs. It is represented by the mathematical expression \( y = \sqrt[3]{x} \). This function outputs the value that, when cubed, gives the input \( x \). For example, \( \sqrt[3]{8} = 2 \) because \( 2^3 = 8 \).

In terms of its graphical representation, the cube root function has a distinctive S-shaped curve that passes through the origin point \((0,0)\).

• For positive inputs, the function opens upwards to the right.
• For negative inputs, it tails downward to the left.

Unlike a square root function, which is only defined for non-negative inputs, the cube root function is defined for all numbers. This is because all real numbers have real cube roots, offering a symmetric form about the origin.

Graphing Functions

To graph the cube root function effectively, there are a few essential steps you can follow. Understanding these will help you proceed to graph transformed functions confidently.

Start with identifying a set of key points from the function's equation. For the base cube root function \( y = \sqrt[3]{x} \), some typical points include:

• \((-8, -2)\)
• \((-1, -1)\)
• \((0, 0)\)
• \((1, 1)\)
• \((8, 2)\)

Connect these points smoothly to capture the characteristic S-shape of the graph.

Plotting the transformed cube root function \( y = \sqrt[3]{8(x-1)} \), as seen in the original exercise, involves first applying the identified transformations to the base points before drawing your graph.

Following this method allows for precise and accurate graphing of not only the basic cube root function but also any transformations applied to it.

Function Transformation

Function transformation is a fascinating topic that involves changing the position and shape of a graph. Such changes are made by altering the equation of the base function. Transformations typically include translations, stretches, compressions, and reflections.

For the function \( y = \sqrt[3]{8(x-1)} \), consider the transformations:

• Horizontal Translation: The shift represented by \( x - 1 \) moves the graph 1 unit to the right. This is because the input value is effectively increased by 1, shifting every x-coordinate.


• Vertical Stretch: The factor \( \sqrt[3]{8} = 2 \) scales the graph vertically by a factor of 2. This means every y-coordinate is multiplied by 2, causing the graph to stretch upwards.

Understanding these transformations allows you to manipulate and predict how the graph will appear after changes are applied to the base function. By mastering function transformations, you can analyze and visualize complex functions more easily.