Problem 31 - A Graphical Approach to Precalculus with Limits

Question

The graph of \(y=6 \sqrt[3]{x-3}\) can be obtained from the graph of \(y=\sqrt[3]{x}\) by shifting horizontally __________ units to the _________ and stretching vertically byand stretching vertically by __________ applying a factor of _______.

Step-by-Step Solution

Answer

Shift 3 units right; stretch vertically by a factor of 6.

Step 1: Understand the Transformation Types

The graph of a function can undergo various transformations such as horizontal shifts, vertical shifts, reflections, and stretches or compressions. In this exercise, we are focusing on a horizontal shift and a vertical stretch.

Step 2: Identify the Horizontal Shift

The given function is \( y = 6 \sqrt[3]{x-3} \). The transformation from \( y = \sqrt[3]{x} \) to \( y = \sqrt[3]{x-3} \) involves a horizontal shift. The expression \( x-3 \) indicates that the graph of the original function is shifted 3 units to the right. This is because replacing \( x \) with \( x-3 \) shifts the graph to the right by 3 units.

Step 3: Recognize the Vertical Stretch

The equation \( y = 6 \sqrt[3]{x-3} \) includes a coefficient of 6 in front of the cubed root function. This indicates a vertical stretch. The original graph \( y = \sqrt[3]{x} \) is stretched vertically by a factor of 6. This means every point on the graph is pulled away from the x-axis by a factor of 6.

Step 4: Combine the Transformations

Combine the knowledge from steps 2 and 3: The graph of \( y = \sqrt[3]{x} \) is shifted horizontally 3 units to the right and then stretched vertically by a factor of 6.

Key Concepts

Horizontal Shift

A horizontal shift involves moving a graph left or right on the Cartesian plane. For the function transformation given in the exercise, we're examining the shift from \( y = \sqrt[3]{x} \) to \( y = \sqrt[3]{x-3} \). This expression reflects a horizontal movement of the graph. When you see \( x - 3 \), it indicates that each x-coordinate of the original function \( y = \sqrt[3]{x} \) is increased by 3 units. This means the entire graph moves 3 units to the right.

• To shift right: Substitute \( x \) with \( x - c \).
• To shift left: Substitute \( x \) with \( x + c \).

So, in this case, since the expression is \( x-3 \), the graph shifts 3 units to the right. Remember, subtracting a number from \( x \) moves the graph in the positive direction horizontally.

Vertical Stretch

A vertical stretch involves elongating the graph away from the x-axis. Looking at the transformation \( y = \sqrt[3]{x} \) to \( y = 6\sqrt[3]{x-3} \), there's a coefficient of 6 in front of the cube root function. This coefficient is key for indicating a stretch.

• A vertical stretch occurs when the absolute value of the coefficient is greater than 1.
• A vertical compression happens when this coefficient is between 0 and 1.

With the coefficient in this example being 6, each point on the graph is pulled six times further from the x-axis than in the original graph. In simple terms, if a point \((x, y)\) on the original graph had a y-coordinate of 1, it now has a y-coordinate of 6. The graph gets stretched vertically because every y-value is multiplied by 6, making it taller and steeper without altering the x-coordinates.

Cube Root Function

The cube root function, represented as \( y = \sqrt[3]{x} \), forms the basis of our transformations. This fundamental function has a unique shape that is symmetric around the origin, and it passes through the point (0,0). The general shape is like an elongated "S" curve, stretching infinitely along both the positive and negative x and y axes. Some notable characteristics of the cube root function include:

• It can accept both positive and negative numbers because every real number has a cube root.
• It is an odd function, thus symmetric about the origin, meaning \( f(-x) = -f(x) \).

When transformations like shifting and stretching are applied, this unique shape stretches and shifts along the axes. In this example, after applying a horizontal shift and a vertical stretch, the function \( y = 6\sqrt[3]{x-3} \) still maintains the cube root function's core characteristics but appears differently due to its modified position and scale on the graph.