Question

Make a sketch of the given pairs of functions. Be sure to draw the graphs accurately relative to each other. $$y=x^{1 / 3} \text { and } y=x^{1 / 5}$$.

Short answer

Question: Compare the properties of the graphs of the functions \(y=x^{1/3}\) and \(y=x^{1/5}\). Answer: Both graphs pass through the origin, are symmetric with respect to the origin, and have a horizontal asymptote on the negative x-axis. However, the graph of \(y=x^{1/3}\) rises more steeply and moves away from the x-axis more quickly than the graph of \(y=x^{1/5}\).

Step-by-step solution

Create table of values for both functions

We will create a table of values for the functions \(y=x^{1/3}\) and \(y=x^{1/5}\) to guide us in sketching the graphs. We will analyze the values for x from -8 to 8.

Plot the points

Using the table of values, we will now plot the points of both functions on the same graph. Create a coordinate system with x-axis from -8 to 8 and y-axis from -2 to 2. Plot each point and label each function on the graph.

Connect the points and analyze the graphs

Now that we have plotted the points, we will connect them to form the graph of each function. Notice the following properties of each function's graph: For \(y=x^{1/3}\): 1. The graph passes through the origin (0,0). 2. The graph is strictly increasing and has a horizontal asymptote on the negative x-axis. 3. The graph is symmetric with respect to the origin (due to the odd power). For \(y=x^{1/5}\): 1. The graph also passes through the origin (0,0). 2. The graph is strictly increasing and has a horizontal asymptote on the negative x-axis. 3. This graph is symmetric with respect to the origin as well (also due to the odd power).

Comparing the graphs

Now that we have sketched the graphs accurately relative to each other, we can compare their properties: 1. Both graphs pass through the origin and are symmetric with respect to the origin (these are key properties of a function with an odd power exponent). 2. Both graphs have a horizontal asymptote on the negative x-axis. 3. The graph of \(y=x^{1/3}\) rises more steeply and appears to move away from the x-axis more quickly than the graph of \(y=x^{1/5}\). In conclusion, we have successfully made a sketch of the given pairs of functions and analyzed their properties in relation to each other.

Key concepts

Plotting Points

Understanding the mechanics of plotting points is a foundational skill in graphing functions. To plot points for a given function, such as y=x^{1/3} or y=x^{1/5}, we begin by choosing a range of x-values and calculating the corresponding y-values. Usually, we start with easy values, including zero and some positive and negative numbers, to get a sense of the function's behavior.

For example, with y=x^{1/3}, if we substitute x=8, we find y=2 because 2^3=8. By plotting several points, each represented as an ordered pair (x, y), and carefully placing them on a graph with a coordinate system, we can start to see the shape and direction of the function's graph.

It's crucial to be accurate with our points, as they serve as the backbone of our graph. Once these points are plotted, we can connect them to reveal the curve they form, showing the function's behavior over a continuous range of values. This step-by-step approach helps create a visual representation that makes complex functions more understandable.

Function Symmetry

Function symmetry is a visually appealing and mathematically important property that can simplify graphing and understanding functions. A function is symmetric if you can reflect, rotate, or translate the graph and end up with the same graph. The functions in our example, y=x^{1/3} and y=x^{1/5}, both show symmetry with respect to the origin, which is known as rotational symmetry.

An easy way to check for this kind of symmetry is to test if replacing x with -x yields the original function negated or vice versa. For instance, for y=x^{1/3}, if we substitute x with -x, we get y=(-x)^{1/3}, which simplifies to y=-(x^{1/3}). This demonstrates that the function is odd and has rotational symmetry about the origin. Therefore, if we know one side of the graph, we can easily determine the other by reflecting it across the origin. This property is visually identified in a graph and provides an important clue to the function’s behavior.

Horizontal Asymptote

Understanding horizontal asymptotes is crucial for sketching graphs, as they describe the behavior of a function as it extends towards infinity. A horizontal asymptote is a horizontal line that the graph of a function approaches as x becomes large in either the positive or negative direction, but does not necessarily ever reach.

For the given functions, y=x^{1/3} and y=x^{1/5}, the textbook solution indicates horizontal asymptotes along the negative x-axis. However, this might be a misinterpretation, as these cubed and quintic root functions do not possess horizontal asymptotes. Instead, they continue to increase (albeit slowly) as x moves towards negative infinity.

Recognizing whether a function has a horizontal asymptote and determining its equation is a valuable skill. Typically, horizontal asymptotes are found in rational functions where the degree of the denominator exceeds the degree of the numerator, leading to the value of y getting closer and closer to zero but never reaching the asymptote as x approaches infinity.