Problem 39 - Algebra for College Students

Question

Graph \(f(x)=\sqrt[3]{x}\). Now predict the graph for each of the following, and check each prediction with your graphing calculator. (a) \(f(x)=5+\sqrt[3]{x}\) (b) \(f(x)=\sqrt[3]{x+4}\) (c) \(f(x)=-\sqrt[3]{x}\) (d) \(f(x)=\sqrt[3]{x-3}-5\) (e) \(f(x)=\sqrt[3]{-x}\)

Step-by-Step Solution

Answer

Each graph transformation of \( f(x) = \sqrt[3]{x} \) involves shifting or reflecting the base graph according to the specified function. Confirm predictions with a graphing calculator.

Step 1: Understand the Base Function

The given function is \( f(x) = \sqrt[3]{x} \). This is a cube root function. The graph of \( \sqrt[3]{x} \) has a shape similar to an S that is symmetric around the origin, crossing the origin at \((0,0)\). It is increasing and passes through all quadrants.

Step 2: Analyze Transformation (a)

The function \( f(x) = 5 + \sqrt[3]{x} \) translates the base function vertically upwards by 5 units. Every point on \( \sqrt[3]{x} \) moves up by 5 units.

Step 3: Analyze Transformation (b)

The function \( f(x) = \sqrt[3]{x+4} \) translates the base function horizontally to the left by 4 units. Every point on \( \sqrt[3]{x} \) moves 4 units left.

Step 4: Analyze Transformation (c)

The function \( f(x) = -\sqrt[3]{x} \) reflects the base function across the x-axis. This flips the graph upside down.

Step 5: Analyze Transformation (d)

The function \( f(x) = \sqrt[3]{x-3} - 5 \) translates the base function horizontally to the right by 3 units and vertically downwards by 5 units.

Step 6: Analyze Transformation (e)

The function \( f(x) = \sqrt[3]{-x} \) reflects the base function across the y-axis. This mirrors the graph horizontally.

Step 7: Verify Using Graphing Calculator

Check each transformed graph using a graphing calculator to confirm that the predicted transformations are correct.

Key Concepts

Graphing Functions

When graphing functions, we are essentially plotting points on a coordinate plane that represents the function's behavior.
The goal is to visually understand how a function behaves as its input values change. The graph of the cube root function, represented by \( f(x) = \sqrt[3]{x} \), is particularly interesting.
Its characteristic "S"-shape is symmetric around the origin, indicating its overall symmetry and how it crosses both axes at the point \((0,0)\).
This symmetry tells us that for any \(x\), the function returns values on both sides of the origin.

• Graphs help in predicting the behavior of functions.
• They allow us to see the increases and decreases as \(x\) changes.
• Visual aid can confirm calculations and simplify complex concepts.

Understanding graphing is crucial to visualizing transformations and predictions in functions.

Function Transformations

Function transformations involve modifying a function's graph in various ways, such as translating, reflecting, stretching, or compressing.
For the function \( f(x) = \sqrt[3]{x} \), transformations can be applied to understand how the graph shifts and changes its shape.
For instance, any function \( g(x) = f(x) + c \) results in translating the function vertically by \(c\) units. Similarly, \( g(x) = f(x+c) \) shifts it horizontally.

• Vertical shifts: Changes the function up or down.
• Horizontal shifts: Moves the graph left or right.
• Reflections: Flips the graph across an axis.
• Stretch/Compression: Alters the graph's width.

Identifying these transformations helps build an intuitive understanding of how complex functions work.

Cube Root Function

The cube root function \( f(x) = \sqrt[3]{x} \) is defined for all real numbers, unlike the square root function which is only defined for non-negative numbers.
The cube root of \(x\) is the number that, when raised to the third power, gives \(x\). The graph of \( f(x) = \sqrt[3]{x} \) is central to understanding transformations because of its inherent properties.

• It passes through the origin \((0,0)\).
• It has points such as \((-1,-1)\) and \((1,1)\), showing how it crosses quadrants.
• The function is continuous and defined for both negative and positive values of \(x\).

The cube root function is a straightforward yet powerful example when learning about graph transformations and reflections.

Reflection and Translation in Graphs

Reflections and translations are two critical types of transformations in graphs.
Reflections involve flipping a graph over an axis, which can completely change its orientation.
For \( f(x) = \sqrt[3]{x} \), reflecting over the x-axis simply multiplies every y-value by -1, resulting in an upside-down graph.

• Reflection over x-axis: Changes the sign of all outputs \(f(x)\).
• Reflection over y-axis: Affects the inputs, as seen in \(f(x) = \sqrt[3]{-x}\).

Translation shifts the graph without altering its shape. For example, translating \( \sqrt[3]{x} \) by a vector \((h, k)\) involves shifting left or right by \(h\) units, and up or down by \(k\) units.

• Examples of Translation: Vertically by "5" in \(f(x) = 5 + \sqrt[3]{x}\), or horizontally by "-3" in \(f(x) = \sqrt[3]{x-3}\).

Understanding these effects helps predict how complex functions will behave based on transformations and shifts.