Problem 71 - Algebra and Trigonometry

Question

\(69-72\) . Graph the functions on the same screen using the given viewing rectangle. How is each graph related to the graph in part \((a) ?\) Viewing rectangle \([-4,6]\) by \([-4,4]\) $$ \begin{array}{ll}{\text { (a) } y=x^{6}} & {\text { (b) } y=\frac{1}{3} x^{6}} \\\ {\text { (c) } y=-\frac{1}{3} x^{6}} & {\text { (d) } y=-\frac{1}{3}(x-4)^{6}}\end{array} $$

Step-by-Step Solution

Answer

Graphs (b), (c), and (d) are vertically stretched, flipped, and translated versions of graph (a), respectively.

Step 1: Analyze the Function in Part (a)

In part (a), the function given is \( y = x^6 \). This is a symmetric even function with all its values being non-negative. Its graph is primarily shaped like a very steep "U" centered at the origin (0,0). Since the powers are high, the graph appears flat near the origin and rises quickly, forming very narrow wings as \( x \) moves away from \( 0 \).

Step 2: Analyze the Function in Part (b)

In part (b), the function is \( y = \frac{1}{3} x^6 \). This is a transformation of the function in part (a) wherein each value of \( y \) is divided by 3. Consequently, the graph will stretch vertically, making it wider than the graph of \( y = x^6 \). It remains symmetric about the y-axis and looks similar but less sharp than the graph from (a).

Step 3: Analyze the Function in Part (c)

In part (c), the function is \( y = -\frac{1}{3} x^6 \). This transformation is a vertical flip and stretch of the graph from part (b). The graph looks like the letter "W" because all positive outputs in the graph of \( y = \frac{1}{3} x^6 \) become negative. Thus, it is an upside-down version of the widened U-graph from step 2, still centered at the origin.

Step 4: Analyze the Function in Part (d)

In part (d), the function is \( y = -\frac{1}{3}(x-4)^6 \). This modifies the graph from part (c) by replacing \( x \) with \( x-4 \), which translates the entire graph 4 units to the right. The shape (inverted W) and the vertical stretch stay the same as in step 3, but the graph's center shifts from the origin to (4,0) along the x-axis.

Key Concepts

Even Functions

Even functions are an interesting class of functions in mathematics. They have a special property called symmetry. This means that if you graph an even function, you will notice it looks the same on both sides of the y-axis. This is because even functions satisfy the condition

• \( f(x) = f(-x) \) for all x in the domain.

For instance, the function \( y = x^6 \) is even. If you look at its graph, you will see it has a U-shape centered at the y-axis. Whether you look to the left or the right of this axis, the graph mirrors itself perfectly, giving it a balanced appearance.
This symmetry makes even functions useful in various mathematical analyses and applications, where predicting patterns can be crucial.

Function Transformations

Function transformations are methods to change the position or shape of a graph on a coordinate plane. Transformations can include shifts (translations), stretches, and reflections. Let's explore a few common transformations:

• **Vertical Stretch/Compression:** In the function \( y = \frac{1}{3}x^6 \), each y-value is divided by 3, making the graph stretch wider, which is less steep.
• **Vertical Reflection:** In \( y = -\frac{1}{3}x^6 \), the graph flips over the x-axis, turning the U-shape into an upside-down "W".
• **Horizontal Shift:** In the function \( y = -\frac{1}{3}(x-4)^6 \), the graph moves 4 units to the right, altering its position but maintaining its shape.

Understanding these transformations helps to predict how altering a function affects its graph. It enables easier interpretation and comparison of multiple graphs with similar origins.

Graph Symmetry

Symmetry in graphs reveals a lot about the nature of functions. It’s a property that makes analysis and computation simpler. In particular, we often deal with two types of symmetry:

• **Y-axis Symmetry:** In even functions like \( y = x^6 \), the graph has symmetry across the y-axis.
• **Origin Symmetry:** While the functions in our exercise don't exhibit this, some functions are symmetric around the origin, following \( f(x) = -f(-x) \).

Recognizing symmetry just by looking at a graph can simplify the process of finding function properties and solving related problems. This intrinsic property serves as a tool to double-check manual calculations and understand the behavior of the function at a glance.