Problem 21 - A Graphical Approach to Precalculus with Limits

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

Graphs of \(y_1\), \(y_2\), and \(y_3\) are transformations of \(y_1 = \sqrt[3]{x}\); \(y_2\) is a reflection over the y-axis, and \(y_3\) is a shifted version of \(y_2\).

Step 1: Understand the Base Graph

The base graph we will use is that of the function \(y_1 = \sqrt[3]{x}\). This is the cube root function, which has an S-shape and passes through the origin (0,0). It is symmetrical about the origin because it is an odd function.

Step 2: Graphing \(y_1 = \sqrt[3]{x}\)

Plot the points for \(y_1 = \sqrt[3]{x}\). Some key points to plot are: \((-8, -2), (-1, -1), (0, 0), (1, 1), (8, 2)\). Connect these points smoothly to form the cube root S-shaped curve. This graph passes through the origin and is symmetrical around it.

Step 3: Applying Transformation to Graph \(y_2 = \sqrt[3]{-x}\)

The function \(y_2 = \sqrt[3]{-x}\) is the reflection of the base function \(y_1\) over the y-axis. This is because every x value is replaced by its negative. Reflect all the points of the \(y_1\) graph over the y-axis. For instance, the point \((1, 1)\) reflects to \((-1, 1)\) and \((-1, -1)\) reflects to \((1, -1)\). Re-sketch the S-shaped curve with this reflection.

Step 4: Applying Transformation to Graph \(y_3 = \sqrt[3]{-(x-1)}\)

The graph of \(y_3 = \sqrt[3]{-(x-1)}\) is obtained by first reflecting the function over the y-axis (as with \(y_2\)), then shifting the function 1 unit to the right. Take the entire \(y_2\) reflection and move each point 1 unit to the right, transforming points such as from \((-1, 1)\) to \((0, 1)\) and from \((1, -1)\) to \((2, -1)\). Sketch this shifted curve, maintaining the S-shape.

Step 5: Check Using a Graphing Calculator

Confirm the transformed graphs using a graphing calculator. Enter each function: \(y_1 = \sqrt[3]{x}\), \(y_2 = \sqrt[3]{-x}\), and \(y_3 = \sqrt[3]{-(x-1)}\), and inspect the curves in an appropriate window to ensure they match the sketches.

Key Concepts

Cube Root Function

The cube root function is a unique mathematical function represented by \( y = \sqrt[3]{x} \). It features a distinctive S-shaped curve that spans both quadrants I and III on the Cartesian plane. At its core, the cube root function is defined for all real numbers because you can take the cube root of any real number, positive or negative.

This function passes through the origin, meaning it will intersect the origin at point \((0,0)\). Additionally, it is an odd function, which tells us it shows symmetry about the origin. To be detailed:

• For positive values of \(x\), \(y = \sqrt[3]{x}\) rises smoothly through points like \((1,1)\) and \((8,2)\).
• For negative values of \(x\), \(y = \sqrt[3]{x}\) falls symmetrically through points like \((-1,-1)\) and \((-8,-2)\).

The cube root function plays an important role in transformations, allowing us to explore reflections and shifts of this base function.

Reflections

Reflections of a graph are transformations that flip the graph over a specified axis. In our exercise, we explore the reflection of the cube root function over the y-axis.

This transformation is given by the function \(y = \sqrt[3]{-x}\). Here, replacing \(x\) with \(-x\) means every x-value of the original function is negated. Thus, all points shift horizontally:

• The point \((1,1)\) on the original graph becomes \((-1,1)\).
• Similarly, \((-1,-1)\) becomes \((1,-1)\).

By reflecting across the y-axis, the entire cube root curve flips direction, while maintaining its intrinsic S-shape. Reflecting functions is a vital tool to manipulate and explore transformations in graphs effectively.

Graph Shifting

Graph shifting involves moving the entire graph along the coordinate plane without altering its shape or orientation. In transforming the cube root function to \(y = \sqrt[3]{-(x-1)}\), we incorporate both reflection and shift.

This transformation starts by using \(y = \sqrt[3]{-x}\), reflecting the base function over the y-axis. Then, by modifying to \(y = \sqrt[3]{-(x-1)}\), we're shifting this reflection horizontally:

• Every point from \(y = \sqrt[3]{-x}\) moves 1 unit to the right.
• Example transformations include moving \((-1, 1)\) to \((0, 1)\) and \((1, -1)\) to \((2, -1)\).

Horizontal shifts adjust the x-coordinates of all points on the graph by adding or subtracting a constant. By shifting the graph, we can position it anywhere on the plane, allowing for comprehensive analysis of a function's behavior.