Problem 47 - A Graphical Approach to Precalculus with Limits

Question

Use translations of one of the basic functions \(y=x^{2}, y=x^{3}\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=(x-1)^{3}$$

Step-by-Step Solution

Answer

Translate the graph of \(y=x^3\) one unit right.

Step 1: Identify the Basic Function

The given function is \(y = (x-1)^3\). The basic function is \(y = x^3\), which is a cubic function known for its S-shaped curve that passes through the origin (0,0).

Step 2: Determine the Translation

Compared to the basic function \(y = x^3\), the function \(y = (x-1)^3\) involves a horizontal translation. Specifically, it translates the graph of \(y = x^3\) one unit to the right. This is because the term \((x-1)\) indicates a shift to the positive x-direction.

Step 3: Sketch the Translated Graph

Start by sketching the basic graph of \(y = x^3\). This graph is symmetric about the origin with points (0,0), (1,1), and (-1,-1). To sketch \(y = (x-1)^3\), shift each of these points one unit to the right, resulting in the new points (1,0), (2,1), and (0,-1). Draw the S-shaped curve through these points to complete the graph.

Key Concepts

Translations in Graphs

Understanding translations in graphs is key to graph sketching. Translations help us shift the entire graph of a function in the Cartesian plane without changing its shape. There are two types of translations: horizontal and vertical.

• A horizontal translation moves the graph left or right. This happens when a constant is added or subtracted from the variable inside the function. For example, if you have the function \(y = (x-1)^3\), the "-1" inside the parentheses means the graph moves one unit to the right.
• Vertical translation, on the other hand, moves the graph up or down. It occurs when a constant is added or subtracted to the function as a whole. For instance, in \(y = x^3 + 2\), "+2" moves the graph two units up.

The important thing about translations is that they maintain the geometry of the graph. This means that the shape and orientation remain the same, only the position changes. So, when you see a function that's modified like \(y = (x-1)^3\), think about it as taking the basic graph \(y = x^3\) and sliding it to the right by one unit.

Cubic Functions

Cubic functions are polynomial functions with the highest degree of three, expressed as \(y = x^3\). These functions are known for their distinctive S-shaped curve when graphed. The key features of cubic functions include:

• Symmetry: A basic cubic function, \(y = x^3\), is symmetric about the origin, meaning if you rotate the graph 180 degrees around the origin, it will look the same.
• Intercepts: The graph passes through the origin at the point (0,0), making it an important reference point.
• End Behavior: As \(x\) approaches infinity, \(y\) also approaches infinity, and as \(x\) approaches negative infinity, \(y\) approaches negative infinity. This behavior creates the S-curve shape.

When sketching cubic functions like \(y = (x-1)^3\), remember it's just a horizontal translation of \(y = x^3\). The curve retains its symmetry and S-shape, but the entire graph shifts to the right. Understanding these basic characteristics helps in correctly sketching these functions by hand.

Graphing by Hand

Graphing by hand is a valuable skill that helps you understand the function's behavior more deeply. When graphing a function like \(y = (x-1)^3\) without using technology, follow these steps:

• Identify the Basic Graph: Begin by recognizing the basic form of the function, in this case, \(y = x^3\). This gives you a starting point to sketch the graph.
• Determine the Translation: Note any translations from the basic function. For \(y = (x-1)^3\), there's a horizontal shift one unit to the right.
• Sketch Key Points: On graph paper or a coordinate plane, first plot the key points of the basic graph \(((0,0)\), \((1,1)\), \((-1,-1)\))\ and adjust them based on your translation \(((1,0)\), \((2,1)\), \((0,-1)\))\.
• Draw the Curve: Finally, draw the S-shaped curve, ensuring continuity through your plotted points. Check that the symmetry and general shape of the cubic curve are maintained.

By manually drawing the graph, you practice observing how changes in the equation translate into changes on the graph. This skill is essential for effectively analyzing functions and interpreting translations.