Question

Make a sketch of the given pairs of functions. Be sure to draw the graphs accurately relative to each other. $$y=x^{3} \text { and } y=x^{7}$$

Short answer

Question: Sketch the graphs of the functions $$y=x^3$$ and $$y=x^7$$, and compare their behavior in different regions of the domain. Answer: The graphs of $$y=x^3$$ and $$y=x^7$$ both pass through the origin (0,0) and have opposite end behaviors due to their odd degrees. However, the graph of $$y=x^7$$ grows much faster than $$y=x^3$$ as x moves away from 0. In the range $$0<x<1$$, the function $$y=x^7$$ lies below $$y=x^3$$, and the reverse is true for the range $$-1<x<0$$.

Step-by-step solution

Identify the type of functions given

We are given two functions: $$y=x^3$$ and $$y=x^7$$. Both of these are polynomial functions, with the degree of the first function being 3 (odd) and the degree of the second function being 7 (odd).

Recall the general behavior of these types of functions

Odd-degree polynomial functions have tails that extend in opposite directions. For positive leading coefficients (as both of these functions have), the tails have the following end behaviors: the left tail goes down as x goes towards negative infinity ($$x \to -\infty$$), and the right tail goes up as x goes towards positive infinity ($$x \to \infty$$). Notice that the higher the degree of the function, the faster the function grows as x moves away from zero.

Identify the key points and characteristics of the first function

For the function $$y=x^3$$, the key points we need to consider are its x-intercept (also a y-intercept) and some additional points to assist in accurately sketching the function. The intercept occurs at point (0,0) as $$0^3 = 0$$. In addition, observe the behavior as x approaches infinity: $$\lim_{x\to\infty} x^3 = \infty$$. Similarly, as x approaches negative infinity, the function also goes towards negative infinity.

Identify the key points and characteristics of the second function

For the function $$y=x^7$$, the key points we need to consider are also its x-intercept (also a y-intercept) and some additional points to help in sketching the function. The intercept occurs at point (0,0) as $$0^7 = 0$$. As x approaches infinity, $$\lim_{x\to\infty} x^7 = \infty$$, and as x approaches negative infinity, the function also goes towards negative infinity.

Comparison of the two functions

Now, it is crucial to emphasize the relative behavior of the two polynomial functions. Since $$x^7$$ has a higher degree than $$x^3$$, the graph of $$y=x^7$$ grows much faster than $$y=x^3$$ as x moves away from 0. In the range $$0<x<1$$, the function $$y=x^7$$ lies below $$y=x^3$$, since $$x^7$$ is smaller than $$x^3$$ for this range. The reverse is true for the range $$-1<x<0$$.

Sketching the graph

Using the information gathered in the previous steps, sketch the graphs of the functions $$y = x^3$$ and $$y = x^7$$. Be sure to accurately show the relative behaviors of the two functions, their intercepts at (0,0), and the end behaviors of the functions for negative and positive infinity. The graph should represent the faster growth of $$y=x^7$$ for x>1 and x<-1, and the relationship between the two functions within the range of $$-1<x<1$$.