Problem 69 - A Graphical Approach to Precalculus with Limits

Question

In a square viewing window, graph \(Y_{1}=\sqrt[3]{\mathrm{X}-6}\) and \(\mathrm{Y}_{2}=\mathrm{X}^{3}+6,\) an example of a pair of inverse functions. Now graph \(Y_{3}=X .\) Describe how the graph of \(Y_{2}\) can be obtained from the graph of \(Y_{1}\), using the graph \(Y_{3}=X\) as a basis for your description.

Step-by-Step Solution

Answer

\(Y_2\) is \(Y_1\) reflected over \(Y_3 = X\).

Step 1: Understand the given functions

We are asked to visualize the functions \(Y_1 = \sqrt[3]{X - 6}\) and \(Y_2 = X^3 + 6\). Recognize that \(Y_1\) and \(Y_2\) may be inverse functions, which typically reflects symmetry with respect to the line \(Y_3 = X\). The function \(Y_3 = X\) is also given to help identify this symmetry.

Step 2: Graph the first function, \(Y_1\)

Graph \(Y_1 = \sqrt[3]{X - 6}\). This function represents a horizontal shift 6 units to the right of the standard cube root function \(Y = \sqrt[3]{X}\). The graph is symmetric about the origin \((6, 0)\) and extends infinitely in both x and y directions as X values approach infinity or negative infinity.

Step 3: Graph the second function, \(Y_2\)

Graph \(Y_2 = X^3 + 6\). This function represents a vertical shift of the standard cubic function \(Y = X^3\) upwards by 6 units. The graph is also symmetric but focuses around the point \((0, 6)\).

Step 4: Graph the line \(Y_3\)

Graph the line \(Y_3 = X\). This line passes through the origin with a slope of 1. It acts as the line of symmetry between inverse functions.

Step 5: Analyze the relationship between \(Y_1\) and \(Y_2\)

Notice that the graph of \(Y_2\) can be derived by reflecting the graph of \(Y_1\) over the line \(Y_3 = X\). Since \(Y_1\) and \(Y_2\) are inverses, each point (a, b) on \(Y_1\) has a corresponding point (b, a) on \(Y_2\). This inversion demonstrates the reflected symmetry across \(Y_3 = X\).

Key Concepts

Symmetry

In the context of inverse functions, symmetry plays a vital role. Symmetry refers to the balanced proportions on opposite sides of a central line or point. When considering the functions from the exercise, symmetry is with respect to the line \(Y_3 = X\). Here, this line \(Y_3 = X\) acts as the mirror across which inverse functions reflect.

The function \(Y_1 = \sqrt[3]{X - 6}\) and its inverse \(Y_2 = X^3 + 6\) exhibit this symmetrical property. When you graph these functions, you'll notice that for any point \((a, b)\) on the graph of \(Y_1\), there is a corresponding point \((b, a)\) on the graph of \(Y_2\). Such a relationship is fundamental to understanding the behavior of inverse functions, showcasing the mirrored reflection across the line \(Y_3 = X\).

• Imagine a point on \(Y_1\),
• Flip it over the line \(Y_3 = X\).
• You find the corresponding point on \(Y_2\).

This reflection confirms the symmetrical nature and helps in visualizing the transformations necessary to transition between \(Y_1\) and \(Y_2\).

Graph Transformations

Graph transformations are changes made to the shape or position of a graph, which help to understand functions better by visual manipulation. They often involve translating, reflecting, stretching, or compressing graphs. In this exercise, transformations are key to visualizing how the two inverse functions relate.

For \(Y_1 = \sqrt[3]{X - 6}\), the graph involves a horizontal shift right, moving 6 units from the original position of the cube root function \(Y = \sqrt[3]{X}\). Simply, every point on the basic cube root curve relocates 6 places to the right on the x-axis, maintaining the same shape.

• Horizontal Shift: Moves the graph along the x-axis.
• For \(Y_2 = X^3 + 6\), a vertical shift upwards occurs, also by 6 units, adjusting from the standard cubic function \(Y = X^3\).

Each point on \(Y= X^3\) goes up by 6 units on the y-axis but otherwise retains its configuration. This showcases how simple transformations can alter graph positioning while the fundamental features remain unchanged.

Cube Root Function

The cube root function \(Y = \sqrt[3]{X}\) is a critical function to understand when learning about inverse functions and transformations. It is a unique algebraic function where every input (or x-value) is paired with a single output (or y-value), and vice versa.

This function plots a curve that passes through the origin (0, 0) and extends infinitely in both directions. Importantly, unlike square root functions, it applies to all real numbers, including negatives, due to a cube root's property of being defined for each real number, positive or negative.

• The graph curves gently across the x-axis as it approaches negative infinity and rises steeply to positive infinity.
• The cube root function inherently possesses rotational symmetry around the origin. This is evident when it graph is turned 180 degrees around the origin; the graph looks the same.

The modification seen in \(Y_1 = \sqrt[3]{X - 6}\) is a translated form, shifting this familiar curve sufficiently to the right by six units on the x-axis.

Cubic Function

The cubic function \(Y = X^3\) is a vital function to master, especially in the study of inverses and graph transformations. This polynomial function beautifully displays how outputs change rapidly as inputs vary, demonstrating a unique curve that becomes increasingly steep.

The standard cubic function has the basic form of a smooth curve passing through the origin and extending to negative and positive infinity as the x-values decrease or increase respectively. It exhibits point symmetry about the origin, meaning if you rotate the graph around the origin by 180 degrees, it remains unchanged.

• The essential property of a cubic function is that it can have up to three real zeros (or roots) where the curve intersects the x-axis.
• It approaches negative infinity swiftly on the left and positive infinity quickly on the right.

When transformed into \(Y_2 = X^3 + 6\), the entire graph shifts vertically up by 6 units. This vertical translation means while the point of intersection with the x-axis is altered, the overall curvature stays intact. Such transformations allow studying functional relationships, especially how transformations affect points of intersection or symmetry.