Question

Numerical and Graphical Analysis \(\quad\) In Exercises 79 and \(80,\) (a) use a graphing utility to complete the table, (b) plot the points in the table and graph the function by hand, (c) use a graphing utility to graph the function and compare the result with your hand-drawn graph in part (b), and (d) determine any intercepts and symmetry of the graph. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & -1 & -0.8 & -0.6 & -0.4 & -0.2 & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1 \\ \hline \boldsymbol{y} & & & & & & & & & & & \\ \hline \end{array} $$ $$ y=\arccos x $$

Short answer

There is only one intercept for the given function, at \(x = 0\), and the graph has no symmetry.

Step-by-step solution

Filling the Table

This Step requires a graphing utility or calculator that can calculate the arccosine of given values of \(x\). Fill each blank under the \(y\) row with the arccosine of the corresponding \(x\) value. For example, for \(x = -1\), \( \arccos(-1) = \pi \), so fill the first empty cell in the \(y\) row with \(\pi\). Complete the rest of the table similarly.

Plotting the Points and Hand Graphing

On a graph paper, plot the points corresponding to \( (x, y) \) values from the table. Then, draw a smooth curve that goes through all the points to represent the function \(y = \arccos x\).

Graphing with a Graphing Utility & Comparison

Next, use a graphing utility to graph the function \(y = \arccos x\). Compare this graph to the hand-drawn graph for similarity. They should be the same.

Determining the Intercepts and Symmetry

The x-intercept is the x-value when \(y = 0\). In this case, \( \arccos 0 = \pi/2 \), so \(x = 0\) is the x-intercept. The y-intercept is the \(y\) value when \(x = 0\), which is also \( \pi/2 \). This means the graph doesn't intersect the x-axis, but only intersects the y-axis at \( \pi/2 \). As for symmetry, since the graph of \(y = \arccos x\) is neither even nor odd, then it has no symmetry with respect to the y-axis or the origin.

Key concepts

Arccosine Function

The arccosine function, denoted as \( y = \arccos x \), is the inverse of the cosine function. It takes an input, usually the cosine of an angle, and outputs the angle in radians, where the range is from \( 0 \) to \( \pi \). This function is key when dealing with trigonometric equations as it helps find the original angle when the cosine value is known.

When graphing the arccosine function, one should be aware that it is defined only for \( x \) values between \( -1 \) and \( 1 \) inclusive. The output \( y \) is restricted to the range \( [0, \pi] \). Understanding the domain and range is essential for correctly plotting the arccosine function on a graph.

Graphical Analysis

The process of graphing functions by hand involves several steps that build upon each other. After completing a numerical table, each pair of \(x\) and \(y\) coordinates should be plotted on a graph paper, ensuring that the scale fits the range of the function values. For the arccosine function, this means using a scale that can accommodate angles from \(0\) to \(\pi\) radians.

In graphical analysis, it's important to observe the shape and behavior of the function curve. The arccosine graph has a unique 'hill' shape, decreasing from \(y=\pi\) when \(x=-1\) to \(y=0\) when \(x=1\). Insights gleaned from graphical analysis are not only related to the visual look of the function but also to its mathematical properties, such as asymptotes, intercepts, and intervals of increase or decrease.

Numerical Table Completion

A numerical table serves as a support tool for graphing and understanding functions. Initially, the table comes with a set of \(x\) values and blank spaces for the corresponding \(y\) values. Completion of this table requires calculating the \(y\) value for every \(x\) using the function's formula. For the arccosine function, this means using a calculator or a graphing utility to find \( \arccos x \) for each value.

Completing a numerical table accurately is critical, as any mistakes will be reflected in the graph. Precision in this step leads to a better understanding of the function's nature and provides a more reliable foundation for graphical analysis.

Function Symmetry

Symmetry in mathematical functions refers to their geometric relation about a certain axis or point. Even functions are symmetric about the y-axis, while odd functions are symmetric about the origin. The arccosine function, however, does not exhibit symmetry of these kinds. It is not an even function since \( \arccos x eq \arccos (-x) \), nor is it an odd function as it does not satisfy \( -\arccos x eq \arccos (-x) \).

Understanding the symmetry, or the lack thereof, in trigonometric functions like the arccosine is useful for analyzing the function's behavior across its domain. Recognizing a function has no symmetry helps in predicting the function's output values more precisely, especially when dealing with negative inputs.