Problem 15 - A Graphical Approach to College Algebra

Question

Use transformations of graphs to sketch the graphs of \(y_{1}, y_{2},\) and \(y_{3}\) by hand. Check by graphing in an appropriate viewing window of your calculator. $$y_{1}=\sqrt[3]{x}, \quad y_{2}=-\sqrt[3]{x}, \quad y_{3}=-2 \sqrt[3]{x}$$

Step-by-Step Solution

Answer

Reflect \(y_1\) for \(y_2\); reflect and stretch by 2 for \(y_3\).

Step 1: Understand the basic graph

The base graph is \(y = \sqrt[3]{x}\). This is the cubic root function, which has a characteristic shape passing through the origin (0,0) with points (1,1), (8,2), (-1,-1), and (-8,-2), approaching a straight line diagonally on both sides of the y-axis, primarily in the first and third quadrants.

Step 2: Transformation for \(y_2 = -\sqrt[3]{x}\)

The graph of \(y_2 = -\sqrt[3]{x}\) is a reflection of \(y = \sqrt[3]{x}\) across the x-axis. This negation causes all y-values to become their opposites: points such as (1,1) become (1,-1) and (-1,-1) become (-1,1). The new curve will appear flipped vertically compared to the base graph.

Step 3: Transformation for \(y_3 = -2\sqrt[3]{x}\)

The graph of \(y_3 = -2\sqrt[3]{x}\) involves two transformations. First, reflect \(\sqrt[3]{x}\) across the x-axis, as in \(y_2\). Second, scale the graph vertically by a factor of 2, meaning each y-value from the base graph is multiplied by 2. The result is a graph that stretches away from the x-axis, remaining in the second and fourth quadrants similar to \(y_2\), but steeper.

Key Concepts

Cubic Root Function

The cubic root function, represented by the equation \(y = \sqrt[3]{x}\), is a fundamental mathematical function with unique properties. It is different from the square root function in that it allows all real numbers, both positive and negative, as inputs. This property makes its graph cross both axes, providing an S-shaped curve. The curve is symmetric around the origin (0,0), covering the first and third quadrants in a Cartesian plane.

The cubic root function is characterized by its gentle slope and slow increase in magnitude. To understand the graph instinctively, let's look at some key points:

• All curves of this function pass through the origin where \(x = 0\).
• Points like \((1,1)\) and \((-1,-1)\) fall on the curve, illustrating its direct relationship.
• The positive values of \(x\) correspond to positive \(y\) values, and vice versa.
• As \(x\) increases, \(y\) increases slowly, indicating the curve's distinctive shape.

Understanding these characteristics helps in visualizing how the function looks and behaves on a graph.

Reflection Across Axis

To manipulate the graph of a function like the cubic root, one often uses transformations such as reflection. Reflection across the x-axis is a direct way to transform these curves into new functions.

For instance, the transformation of \(y = \sqrt[3]{x}\) into \(y_2 = -\sqrt[3]{x}\) is a classic example of reflection across the x-axis. Here's how it works:

• Each \(y\)-value of the function \(y = \sqrt[3]{x}\) is negated.
• This causes every point above the x-axis to move below it and vice versa, flipping the graph vertically.
• Points such as \((1,1)\) transform to \((1,-1)\) and \((-1,-1)\) transform to \((-1,1)\).

In essence, a reflection across the x-axis inverts the graph's orientation, creating a mirror image centered on the x-axis itself. Understanding this concept is key to sketching transformed graphs accurately.

Vertical Stretching

Vertical stretching is another important graph transformation concept used to reshape functions. In the context of the cubic root function, vertical stretching amplifies the graph's height without altering its horizontal span or direction.

Consider the transformation from \(y_2 = -\sqrt[3]{x}\) to \(y_3 = -2\sqrt[3]{x}\), which involves a vertical stretch by a factor of 2:

• This transformation multiplies each function's \(y\)-value by the stretching factor, in this case, 2.
• Such scaling makes the curve appear steeper, pulling away from the x-axis.
• For example, a point \((1,-1)\) on \(y_2\) becomes \((1,-2)\) on \(y_3\), doubling in distance from the x-axis.

Vertical stretching accentuates the graph's vertical characteristics without affecting its symmetry or horizontal layout. This is a crucial technique for achieving desired graph shapes and dynamics in complex mathematical modeling.