Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$f(x)=1-x^{6}$$

Step-by-Step Solution

Answer

The function \(f(x)=1-x^{6}\) is a power function with roots at \(x=-1\) and \(x=1\). It has a local minimum at the point (0,1) and both ends of the graph point downwards. The nature of the function can be verified by sketching it using a graphing utility.

Step 1: Identify the Function Type

Given the function \(f(x)=1-x^{6}\), this function is a power function with an even degree (6) and negative lead coefficient(-1). For these types of functions, the left and the right ends of the graph move in the same direction, and in this case downwards due to the negative coefficient.

Step 2: Find the Roots or Zeros of the Function

Setting \(f(x) = 0\) to find the x-intercepts (roots or zeros) of the function gives: \(0=1-x^{6}\). Solving for \(x\) involves rearranging to \(x^{6}=1\) and taking the sixth root of both sides to get \(x = \pm 1\). Thus, the function has roots at \(x = 1\) and \(x = -1\).

Step 3: Identify the Local Minima and Maxima

Given that the power is even, the function will have a maximum at both ends and a relative minimum at x=0. You can calculate the y value by substituting \(x=0\) to the function \(f(0)= 1-0^{6}= 1\). This local minimum is at the point (0,1).

Step 4: Sketch the Graph

Using the derived information, the graph can be sketched by hand. The graph should cross the x-axis at \(x = -1\) and \(x = 1\), and it should have a local minimum at (0,1). The graph should also be symmetrical about the y-axis, and the ends should point downwards.

Step 5: Confirm the Sketch with a Graphing Utility

Finally, confirm and correct the sketch by plotting the function using any graphing tool, online graph plotting software or graphic calculator.

Key Concepts

Analyzing Graphs Algebraically

Analyzing the graph of a function algebraically involves identifying its shape, behavior at extreme values, and points like the intercepts and peaks. For the power function f(x) = 1 - x^6, its even degree tells us that the function is symmetrical with respect to the y-axis and the negative leading coefficient suggests that the graph points downwards at both ends. Algebraic analysis entails evaluating function behavior as x approaches infinity or negative infinity, and identifying its intercepts by setting f(x) = 0.

The graph can be dissected into its characteristic points and intervals. For instance, the roots at x = \(\pm 1\) give us points where the graph will cross the x-axis. Symmetry about the y-axis means if we have a point on the graph at (a, b), the point (-a, b) will also belong to the graph. This algebraic groundwork provides an essential framework for sketching the graph accurately by hand.

Sketching Graphs by Hand

Once we understand the behavior of the function algebraically, we can begin sketching. Start by plotting the roots on the x-axis, at x = -1 and x = 1. Next, include the y-intercept by calculating f(0), which is the local minimum in this case. You can then sketch the smooth continuous curve of the power function, respecting the determined symmetry and asymptotic behaviors, notably that the graph falls away to negative infinity as x moves away from the origin.

Remember to pay attention to scale for a more accurate representation. For example, the fact that the function has an even power suggests that the graph is not just symmetrical, but also has a 'flatter' appearance around the minimum compared to odd-powered functions. This step is crucial for developing a strong intuition about the behavior of different types of functions.

Finding Roots or Zeros

Identifying the roots or zeros of a function is a fundamental aspect of function analysis. It gives us the x values where the function touches or crosses the x-axis. For the function f(x) = 1 - x^6, we find the roots by setting the function equal to zero and solving for x. This results in x^{6} = 1 which simplifies to x = \(\pm 1\).

Understanding how to find these zeros helps to demystify the graph's behavior and provides pivotal points around which the graph is shaped. Additionally, this information about extremities serves as a guide to the domain within which the function will assume non-negative values, which for our function is between x = -1 and x = 1.

Graphing Utility Confirmation

Once you've graphed the function by hand, using a graphing utility can help confirm the accuracy of your sketch. After inputting the function f(x) = 1 - x^6 into a graphing calculator or software, you’ll be able to compare the key features you’ve identified algebraically and sketched manually: the symmetry, roots at x = \(\pm 1\), and the local minimum at (0, 1).

The tool acts as an efficient means to check for errors and reinforce learning. With the visualization power of graphing utilities, one can immediately notice discrepancies, if any, between their manual sketch and the actual graph plot. This makes such utilities an integral part of learning and understanding the nature of algebraic functions in graphical terms.