Problem 22 - Thomas Calculus

Question

Exercises \(19-28\) tell how many units and in what directions the graphs of the given equations are to be 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

Shift the graph: \( y = (x - 1)^{2/3} - 1 \).

Step 1: Identify the Transformation

The original graph is described by the equation \( y = x^{2/3} \). We need to shift it 1 unit to the right and 1 unit down.

Step 2: Horizontal Shift

To shift the graph to the right by 1 unit, replace \( x \) with \( x - 1 \) in the original equation. This changes the equation to \( y = (x - 1)^{2/3} \).

Step 3: Vertical Shift

To shift the graph down by 1 unit, subtract 1 from the entire equation. Therefore, the equation becomes \( y = (x - 1)^{2/3} - 1 \). This represents the new shifted graph.

Step 4: Sketch the Graphs

Start by sketching the original graph of \( y = x^{2/3} \). This graph passes through the origin, rising both to the right and left symmetrically as a kind of smooth 'v' shape. Then, draw the shifted graph \( y = (x - 1)^{2/3} - 1 \) by moving the original graph 1 unit to the right and 1 unit downward. Label each graph with its respective equation.

Key Concepts

Understanding Horizontal Shifts

A horizontal shift in graph transformations involves moving the entire graph of a function left or right on the coordinate plane. This is a critical concept in understanding how graphs behave when they are altered by certain transformations. When looking at horizontal shifts for functions like cubic root functions, these shifts can easily be transitioned through the x-variable.

To apply a horizontal shift, for example shifting right by 1 unit, you would replace every instance of \(x\) in the function with \(x - 1\). This is because intuitively, shifting right involves subtracting a number from \(x\), causing the graph to move right.

Horizontal shifts are essential for modifying the position of the graph while retaining its overall shape. Here's how you can think about it for your learning:

• Recognize that shifting right means subtraction inside the function, like \( (x-1)\).
• Understand that the shape and orientation of the graph remain unchanged, only its position is adjusted.
• If shifting left was needed, replace \(x\) with \(x + a\) where \(a\) is the number of units to shift.

Exploring Vertical Shifts

Vertical shifts in transformations occur when a graph is moved up or down along the y-axis. Unlike horizontal shifts, vertical shifts do not affect the x-values directly, but adjust the entire graph's y-values. This means every point on the graph is moved a certain number of units up or down, parallel to the y-axis.

In this scenario, if you're instructed to shift the graph down by one unit, you simply subtract 1 from the entire function's output. For a function like \(y = (x-1)^{2/3}\), you complete the vertical shift by altering the equation to \(y = (x-1)^{2/3} - 1\). Here are a few pointers to help with vertical shifts:

• Shifting down is achieved by subtracting from the function, while shifting up involves addition.
• Vertical shifts change the function's output values but not its domain or shape.
• It's crucial to perform any vertical shifts after completing any horizontal shifts to maintain clarity.

Cubic Root Functions and Their Graphs

Cubic root functions, like \(y = x^{2/3}\), have distinctive characteristics that make their graphing interesting yet manageable. These functions often create graphs that remind one of square root functions but involve the cube root instead, giving them a unique curvature.

The graph of a basic cubic root function \(y = x^{2/3}\) typically forms a gentle 'v' shape and passes through the origin. When transformed via horizontal shifts and vertical shifts, these graphs maintain their general shape but are repositioned within the coordinate plane.

Understanding cubic root functions is made easier by:

• Focusing on how the \(2/3\) exponent causes the symmetry of the curve.
• Notice that despite transformations, the cubic root graph retains the 'v' shape, signifying its constant increase or decrease.
• When transformed, the simplicity in the graphing of its new position helps in visualizing and comprehending modifications.

By mastering these transformations, you'll gain greater insight into how intricate functions like the cubic root transform under various conditions.