Problem 16 - College Algebra and Calculus: An Applied Approach

Question

Use the graph of \(y=x^{4}\) to sketch the graph of the function. $$f(x)=\frac{1}{2}(x-1)^{4}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\frac{1}{2}(x-1)^{4}\) is obtained from the graph of \(y=x^{4}\) by shifting it 1 unit to the right along the x-axis and scaling it vertically downwards by a factor of 1/2. This makes the graph appear slightly to the right and flatter compared to the original graph.

Step 1: Sketch the Original Function

Begin by sketching the graph of the original function \(y=x^{4}\). It's a simple curve that starts from the origin (0,0), rises to the right, and rises to the left. Due to the even power, it is symmetric with respect to the y-axis.

Step 2: Apply the Horizontal Shift

Next, apply the horizontal shift to the graph. The modification \(x-1\) in the function indicates a shift 1 unit to the right. Hence, every point on the original graph \(y=x^{4}\) is moved 1 unit to the right.

Step 3: Apply the Vertical Scaling

Finally, apply the vertical scaling to the graph. The coefficient \(\frac{1}{2}\) in front of the function scales the vertical values by a half. As a result, points on the graph will be half as tall as they were on the original graph. This will make the graph appear flatter.

Step 4: Combine the Transformations

By combining the horizontal shift and vertical scaling, you obtain the graph of the given function. The graph will appear slightly to the right and flatter compared to the original graph of \(y=x^{4}\).

Key Concepts

Horizontal Shift

When transforming graphs, a horizontal shift is a crucial concept to understand. It involves moving the entire graph left or right along the x-axis without changing its shape. This is especially useful when dealing with functions like polynomial functions, where shifts can help in repositioning the graph to better fit a particular set of data or context.
A horizontal shift is achieved by altering the input of the function. In mathematical terms, if you have a function \(y = f(x)\) and want to shift it, you modify it to \(y = f(x - c)\). Here, \(c\) represents how many units you would like to move the graph.

• If \(c\) is positive, the graph shifts to the right.
• If \(c\) is negative, the graph shifts to the left.

In our exercise, the function \(f(x) = (x-1)^4\) is derived from \(y = x^4\) and includes a shift of 1 unit to the right due to the \(x-1\) subtraction. This shift maintains the integrity of the function's shape while simply adjusting its position on the graph. Keeping track of these shifts becomes second nature with practice, ensuring that you can easily reposition graphs.

Vertical Scaling

Vertical scaling fundamentally alters the appearance of a graph by stretching or compressing it along the y-axis. This transformation involves multiplying the y-values of a function by a coefficient, which can amplify or diminish the graph's heights and dips.
The coefficient \(a\) in front of a function \(f(x)\), resulting in \(y = a f(x)\), determines the vertical scaling. Here's how it works:

• If \(a > 1\), the graph stretches, making it taller.
• If \(0 < a < 1\), the graph compresses, becoming shorter, or flatter.
• If \(a < 0\), besides scaling, the graph is also flipped over the x-axis.

In our given exercise, the function \(f(x) = \frac{1}{2}(x-1)^4\) involves multiplying the original function by \(\frac{1}{2}\). This flattens the graph to half its original height, while retaining the overall shape and orientation. Imagine gently pressing down on a curve to reduce its height—this captures the essence of vertical scaling. Such transformations allow you to adjust the intensity of a function's peaks and troughs to fit different requirements.

Polynomial Functions

Polynomial functions form a foundational aspect of algebra and involve expressions with variables raised to whole number powers. They are expressed in the form \(y = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\). Each term is defined by a coefficient \(a_i\) and a power \(n\), where \(n\) is a non-negative integer.
These functions can create various graph shapes depending on their degree, which is the highest power of \(x\). For instance:

• Linear functions (degree 1) result in straight lines.
• Quadratic functions (degree 2) depict parabolas, which are U-shaped curves.
• Cubic functions (degree 3) can produce S-shaped curves.
• Quartic functions (degree 4), like in our exercise, create W-shaped curves.

The quartic function \(y = x^4\), used in the exercise, exhibits symmetry around the y-axis and showcases a more flattened central peak compared to a quadratic.
These polynomial forms are particularly useful because they can approximate many forms of data, model natural phenomena, and are easily transformed. When graph transformations like scaling or shifting are applied, as seen in the given exercise, they allow these polynomial functions to model even more varied behaviors, adapting more closely to real-world scenarios.