Problem 36 - A Graphical Approach to Precalculus with Limits

Question

The graph of \(y=\sqrt[3]{x}\) is shifted 2 units to the left. This graph is then vertically stretched by applying a factor of 1.5. Finally, the graph is shifted 8 units upward.

Step-by-Step Solution

Answer

The final equation is \( y = 1.5 \cdot \sqrt[3]{x + 2} + 8 \).

Step 1: Identify the base function

The base function given is \( y = \sqrt[3]{x} \). This is the cube root function, which is the template we will modify according to the transformations described.

Step 2: Apply horizontal shift

To shift the graph of a function 2 units to the left, you replace \( x \) with \( x + 2 \) in the function. Thus, the new function becomes \( y = \sqrt[3]{x + 2} \).

Step 3: Apply vertical stretch

A vertical stretch involves multiplying the entire function by a factor. Here, the stretching factor is 1.5. Therefore, the function becomes \( y = 1.5 \cdot \sqrt[3]{x + 2} \).

Step 4: Apply vertical shift

To shift the function 8 units upwards, simply add 8 to the entire expression. The resulting function is \( y = 1.5 \cdot \sqrt[3]{x + 2} + 8 \).

Key Concepts

Horizontal Shift

When we talk about a *horizontal shift* in the context of function transformations, we're essentially moving the graph left or right. This transformation affects the *x*-value of each point on the graph. A positive shift means the graph moves right, while a negative shift implies a leftward shift.

In the equation, replace each instance of the variable *x* with \( x + a \). The sign of \( a \) will determine the direction of the shift:

• For a shift to the left, add to \( x \), which effectively means \( a < 0 \).
• For a shift to the right, subtract from \( x \), indicating \( a > 0 \).

In our example, shifting the cube root function \( y = \sqrt[3]{x} \) 2 units left involves replacing \( x \) with \( x + 2 \). This gives the new equation \( y = \sqrt[3]{x + 2} \).A key reason for using *horizontal shifts* is to reposition functions without altering their basic shape. It's important to realize that these shifts are counterintuitive due to their direction and mathematical sign.

Vertical Stretch

A *vertical stretch* transformation alters the *y*-values of a function, effectively modifying how "tall" or "compressed" the graph appears. This transformation's impact is huge, as it scales the graph vertically either outward or inward, depending on the factor used.

For vertical stretching, every output is multiplied by a constant factor. If you have a factor \( k \):

• If \( k > 1 \), the graph stretches away from the *x*-axis, becoming taller.
• If \( 0 < k < 1 \), the graph compresses towards the *x*-axis, appearing shorter.

In the given problem, a vertical stretch factor of 1.5 is applied to the previously shifted function. Consequently, the equation becomes \( y = 1.5 \cdot \sqrt[3]{x + 2} \). This stretch causes each *y*-value of the function to become 1.5 times its original amount.
Vertical stretches are crucial for adjusting how functions fit or overlap in graphs, without modifying their horizontal position or the core shape of the function.

Vertical Shift

A *vertical shift* raises or lowers a graph by a constant value, impacting the *y*-coordinate of every point. While some transformations involve variable adjustment, vertical shifts are straightforward, adding or subtracting a constant from the entire function.

How it functions:

• Adding a constant moves the graph up.
• Subtracting a constant moves it down.

Vertical shifts preserve the shape and orientation of the graph but can help in aligning it to particular points on the graphing plane.
From our problem, the function \( y = 1.5 \cdot \sqrt[3]{x + 2} \) is vertically shifted 8 units up. This transformation updates the equation to \( y = 1.5 \cdot \sqrt[3]{x + 2} + 8 \), effectively aligning the graph upwards by 8 units.
Understanding vertical shifts simplifies reading and transforming graphs, enabling better visual alignment with various axes or other functions.