Problem 107 - College Algebra

Question

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)=\sqrt[3]{x}+2 $$

Step-by-Step Solution

Answer

The graph of \(f(x) = \sqrt[3]{x}\) is transformed into the graph of \(g(x) = \sqrt[3]{x} + 2\) by shifting all the points on the graph of \(f(x)\) two units upwards.

Step 1: Graphing the Function \(f(x) = \sqrt[3]{x}\)

To conclusively sketch the function \(f(x) = \sqrt[3]{x}\), the cubic root function, consider points on the graph where \(x\) is a perfect cube. For instance, when \(x = -8, -1, 0, 1, 8\), the cubic root values will be \(-2, -1, 0, 1, 2\) respectively. Plot these points on the graph and then join them smoothly to create the graph of \(f(x)\).

Step 2: Understanding the transformation

The function \(g(x) = \sqrt[3]{x} + 2\) is a vertical transformation of the original function \(f(x) = \sqrt[3]{x}\). Essentially, it shifts the graph of \(f(x)\) up by two units. This is because every \(y\) value in the function \(f(x)\) is being increased by 2 in the function \(g(x)\).

Step 3: Graphing the Function \(g(x) = \sqrt[3]{x} + 2\)

Since the transformation involves shifting the graph two units upwards, the points for the function \(f(x) = \sqrt[3]{x}\) will simply move up by two units for the graph of \(g(x) = \sqrt[3]{x} + 2\). This leads to the new points: when \(x = -8, -1, 0, 1, 8\), the \(y\) values will be \(0, 1, 2, 3, 4\) respectively. Using the same approach as before, plot these points and smoothly join them to complete the graph of \(g(x)\).

Key Concepts

Graph Transformations

Graph transformations allow us to adjust and reposition graphs of functions without altering their core shape. In our exercise, we focus on transforming the graph of the cube root function. The base function is often written as \(f(x) = \sqrt[3]{x}\) and its graph sets a foundation for further transformations. Translations, reflections, stretches, and compressions are all types of graph transformations. Here, we specifically deal with a vertical translation, which is a subtype of translation transformations. By transforming the graph, we can create entirely new functions from a given basic graph by shifting, reflecting, or stretching it in different ways, making understanding graph transformations essential when analyzing functions.

Vertical Transformation

Vertical transformations involve shifting the graph of a function up or down along the y-axis. In our specific function, \(g(x) = \sqrt[3]{x} + 2\), the transformation happens vertically. This is because we are simply adding 2 to each of the \(y\) values output by the original function \(f(x) = \sqrt[3]{x}\).

As a result:

• The graph does not change shape; it only changes position along the vertical axis.
• Each point \( (x, y) \) on the original graph becomes \( (x, y+2) \) on the transformed graph.

This simple adjustment is a foundational concept in graph transformations and is powerful for manipulating the basic behavior of functions.

Graph Sketching

Graph sketching is the process of drawing a visual representation of a function's behavior on the coordinate plane. To sketch the cube root graph, start by identifying and plotting crucial points where \(x\) is a perfect cube: \(-8, -1, 0, 1, 8\). These points correspond to the cubic root values of \(-2, -1, 0, 1, 2\), respectively.

Use these points as benchmarks to sketch the curve by smoothly connecting them. When sketching the vertically transformed graph \(g(x) = \sqrt[3]{x} + 2\), simply move each point two units upward.
This sketch will give a clear visual of how the line shifts upward without altering any other aspect of its shape. Graph sketching remains a vital skill in understanding how functions behave and interact.

Cubic Root Values

Cubic root values are found by determining a number which, when raised to the power of three, equals \(x\). Mathematically speaking, this is expressed as \(f(x) = \sqrt[3]{x}\). It’s important to recognize perfect cubes such as -8, -1, 0, 1, and 8, as their roots, -2, -1, 0, 1, and 2, are easy to calculate and commonly used in graphing.

Understanding these values helps facilitate graph sketching. By focusing on basic cubic root values, one can quickly identify significant graph points and their transformations. It demystifies the cube root function by allowing pattern recognition of how input and output values change based on the cube root’s behavior.