Problem 21 - 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. $$y_{1}=\sqrt[3]{x}, \quad y_{2}=\sqrt[3]{-x}, \quad y_{3}=\sqrt[3]{-(x-1)}$$

Step-by-Step Solution

Answer

Graph \(y_{1}\) normally, \(y_{2}\) reflects \(y_{1}\) across the y-axis, and \(y_{3}\) shifts \(y_{2}\) to the right by 1 unit.

Step 1: Understand the basic graph

The graph of the function \(y_{1} = \sqrt[3]{x}\) is similar to a cubic function but flatter. It passes through the origin \((0,0)\) and is symmetric with respect to the origin. It extends infinitely in both directions and has a vertical slope as \(x\) approaches zero.

Step 2: Graph the reflection transformation

The function \(y_{2} = \sqrt[3]{-x}\) is a reflection of \(y_{1} = \sqrt[3]{x}\) across the y-axis. To graph this, reflect each point on the \(y_{1}\) graph across the y-axis. For example, the point \((1,1)\) on \(y_{1}\) becomes \((-1,1)\) on \(y_{2}\), and the point \((-2,-\sqrt[3]{2})\) becomes \((2,-\sqrt[3]{2})\).

Step 3: Apply horizontal and vertical transformation

For the function \(y_{3} = \sqrt[3]{-(x-1)}\), start with the reflection \(y_{2} = \sqrt[3]{-x}\) and apply a horizontal shift to the right by 1 unit. Each point on \(y_{2}\) moves 1 unit to the right, so the point \((0,0)\) on \(y_{2}\) becomes \((1,0)\) on \(y_{3}\), \(-1, \sqrt[3]{1}\) becomes \((0, \sqrt[3]{1})\), and similarly for other points.

Key Concepts

Cubic Root Functions

Cubic root functions describe the relationship between two variables, where one variable is the cube root of another. The general form of a cubic root function is \( y = \sqrt[3]{x} \). These functions are interesting because they smoothly pass through the origin and exhibit unlimited behavior in both the positive and negative directions along the x-axis. Cubic root functions are particularly noted for being flatter around the origin compared to quadratic functions, which makes them less steep and easier to manage near the zero point.
The basic graph of \( y = \sqrt[3]{x} \) is a curve that increases gradually from the bottom left to the top right. This curve is not symmetric in the typical sense but rather exhibits point symmetry through the origin. It appears as though the graph bends through the origin, making both sections of the curve mirror images of each other across this point.
Using cubic root functions as a base, we can explore several transformations that modify the appearance of these graphs into different forms.

Function Reflections

Reflections in graphing are transformations that create a mirror image of a function across a chosen axis. For instance, converting \(y_1 = \sqrt[3]{x} \) into \( y_2 = \sqrt[3]{-x} \) involves reflecting the graph of the function across the y-axis.
In reflections across the y-axis, every point \((x, y)\) on the original graph translates to \((-x, y)\) on the reflected graph.

• For the point \((1, 1)\) on \( y_1 \), it becomes \((-1, 1)\) on \( y_2 \).
• A point like \((-2, -\sqrt[3]{2})\) would shift to \((2, -\sqrt[3]{2})\).

These reflected points help form the inverted journey of the graph in relation to its initial form. Reflections are particularly useful in graph transformations as they offer insights into symmetry and balance within mathematical equations.

Horizontal Transformation

Horizontal transformations involve shifting the graph of a function left or right along the x-axis. For example, with the function \( y_3 = \sqrt[3]{-(x-1)} \), we apply a horizontal transformation to \( y_2 = \sqrt[3]{-x} \) by moving it one unit to the right.
This alteration affects each point on the graph. A point \( (a, b) \) gets shifted to \( (a+1, b) \) on the new graph.

• The original \( (0, 0) \) on \( y_2 \) becomes \( (1, 0) \) on \( y_3 \).
• The point \(( -1, \sqrt[3]{1})\) shifts to \((0, \sqrt[3]{1})\).

By sliding the function right, we effectively delay the input, modifying where certain features of the graph appear along the x-axis. This form of transformation is ideal when adjusting the position of the graph to meet specific requirements or conditions.

Vertical Transformation

Vertical transformations cast different adjustments, elevating or lowering the graph along the y-axis. Though this particular problem does not include explicit vertical transformations like \( y + k \), understanding that concept can aid in comprehending overall graph dynamics.
In a vertical transformation, every point \((x, y)\) translates vertically so that the new point becomes \((x, y+k)\). These transformations result in the entire graph being lifted or dropped vertically, whilst the x-coordinates remain unchanged.

• If you apply \( k = 2 \) for example, every y-coordinate in your original function shifts by adding 2.
• Thus, a point \( (3,4) \) on a graph with the transformation applied becomes \( (3,6) \).

Vertical transformations are widely used to modify the range of a graph, impacting how the graph sits relative to the x-axis. Understanding them enriches the broader mastery of graph transformations, offering versatility in manipulating and redesigning graphs to fulfill a multitude of conditions and problem-solving contexts.