Problem 28 - Thomas Calculus

Question

Exercises \(25-34\) tell how many units and in what directions the graphs of the given equations are shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation. $$ y=x^{2 / 3} \quad \text { Right } 1, \text { down } 1 $$

Step-by-Step Solution

Answer

The graph shifts by 1 unit right and 1 unit down. New equation: \( y = (x-1)^{2/3} - 1 \).

Step 1: Identify the Original Function

The original function given is \( y = x^{2/3} \).

Step 2: Determine the Shifts

The problem states that the graph should be shifted right by 1 unit and down by 1 unit.

Step 3: Apply the Horizontal Shift

To shift the graph of \( y = f(x) \) to the right by 1 unit, we replace \( x \) with \( x - 1 \). Thus, the equation becomes \( y = (x-1)^{2/3} \).

Step 4: Apply the Vertical Shift

To shift the graph of \( y = f(x) \) down by 1 unit, we subtract 1 from the function. Thus, the equation modifies to \( y = (x-1)^{2/3} - 1 \).

Step 5: Write the Equation for the Shifted Graph

The equation for the shifted graph is \( y = (x-1)^{2/3} - 1 \).

Step 6: Sketch the Graphs

Draw two graphs on the same set of axes: one for the original equation \( y = x^{2/3} \) and the other for the shifted equation \( y = (x-1)^{2/3} - 1 \). Label them accordingly to show their equations.

Key Concepts

Understanding Horizontal Shift

A horizontal shift moves a function graph left or right along the x-axis. It involves changing the input variable, usually denoted by an alteration to the function's equation. Specifically, to shift a graph:

• Right by a certain number of units, subtract the number from the input variable. For example, replace \(x\) with \(x - c\) to shift right \(c\) units.
• Left by a certain number of units, add the number to the input variable. For instance, replace \(x\) with \(x + c\) to shift left \(c\) units.

In the problem you were given, the original function \(y = x^{2/3}\) was shifted to the right by 1 unit. This adjustment involved replacing \(x\) with \(x-1\), resulting in the new equation \(y = (x-1)^{2/3}\). The visual effect of this shift is that every point on the graph moves 1 unit to the right. Whenever shifting graphs horizontally, it’s essential to focus on the function’s input, ensuring that the change reflects the desired movement on the x-axis.

Interpreting Vertical Shift

A vertical shift moves a function graph up or down along the y-axis. This transformation changes the function’s output, meaning the entire equation is adjusted upward or downward:

• To move the graph up, add the number to the function. For example, go from \(y = f(x)\) to \(y = f(x) + c\) to go up \(c\) units.
• To move the graph down, subtract the number from the function. Change \(y = f(x)\) to \(y = f(x) - c\) to go down \(c\) units.

In this exercise, the function was to be shifted down by 1 unit. This was achieved by subtracting 1 from the function: \(y = (x-1)^{2/3} - 1\). This shift impacts every point vertically on the graph, translating each point downs along the y-axis by one unit. Understanding vertical shifts helps in precisely modifying graph placement without affecting the basic shape.

Basic Principles of Function Graphing

Function graphing involves plotting points that satisfy a function on the coordinate plane. It’s foundational for visualizing how functions behave and how graph transformations like shifts and stretches affect them. When graphing any function:

• Start by identifying key points like intercepts and critical values that outline the function's behavior.
• Use symmetries of the function to reduce the amount of computation needed to plot points on both sides of the y-axis.
• Understand the baseline transformations: horizontal shifts, vertical shifts, compressions, stretches, and reflections to predict how a graph will look post-transformation.

For this specific example, two graphs are plotted: the original \(y = x^{2/3}\) and the transformed graph \(y = (x-1)^{2/3} - 1\). By labeling these graphs clearly, one can compare how the shifts modified the position of the graph while retaining its original shape. Correct graphing aids in a deeper comprehension of the behavior of functions and the impact of transformations.