Problem 128 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. The graph of \(y=4-8 \sin ^{2} x\) has a maximum at \((\pi, 4)\).

Step-by-Step Solution

Answer

The statement is true. The function \(y=4-8 \sin ^{2} x\) indeed has a maximum at \((\pi, 4)\).

Step 1: Substitute the minimum value of \(sin^2 x\) into the function

The minimum of \(sin^2 x\) occurs at 0. So we substitute \(0\) into \(y=4-8 sin ^{2} x\) which gives us \(y = 4 - 8*0 = 4\). So, the maximum of \(y\) is 4, which occurs when \(sin^2 x = 0\).

Step 2: Find the x-value at which this maximum y-value occurs

The maximum of \(y\) occurs at the minimum of \(sin^2 x\) i.e., when \(sin^2 x = 0\), which happens if \(x = n\pi \) where \(n\) is any integer.

Step 3: Compare the obtained maximum value coordinate with the given coordinate

Our obtained maximum is at (n\(\pi\),4) for any integer \(n\). The given coordinate is (\(\pi\),4) which fits into the solution set as n=1. So the answer is True

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that establish relationships between trigonometric functions and are true for every value of the occurring variables where both sides of the equation are defined. These identities are crucial in simplifying expressions and solving trigonometric equations. Some of the most well-known identities include the Pythagorean identity \( \sin^{2}x + \cos^{2}x = 1 \) and angle sum identities like \( \sin(x + y) = \sin x \cos y + \cos x \sin y \). In the context of our exercise, recognizing the identity \( \sin^{2}x = \frac{1 - \cos 2x}{2} \) could transform the given function into a more recognizable form, aiding in the analysis of its properties, such as identifying maxima and minima.

Maxima and Minima in Trigonometry

Understanding maxima and minima is fundamental in the study of trigonometric functions. For the sine function, the maximum value is 1, and the minimum value is -1. The square of the sine function, as seen in \( \sin^{2}x \), ranges between 0 and 1, inclusive. These values are critical when trying to find the highest or lowest points on a graph of a transformed trigonometric function. In our exercise, by substituting the minimum value of \( \sin^{2}x \) into \( y=4-8\sin^{2}x \), we find the maximum value of \( y \). This approach leverages the concept of maxima and minima in trigonometry to answer questions about the graph's peak points.

Sine Function Properties

The sine function, \( \sin x \), has several important properties that make it unique. It's a periodic function with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) radians. Its range is between -1 and 1, and it's symmetric about the origin, which reflects its odd function nature. For \( \sin^{2}x \), although the range is still between 0 and 1, the function is now even, as \( \sin^{2}(-x) = (\sin(-x))^{2} = (\sin x)^{2} = \sin^{2}x \). These properties tell us that for every \( \pi \), \( \sin^{2}x \) resets to 0 - crucial for solving our textbook exercise and locating the maxima of our function.

Transformations of Trigonometric Graphs

Transformations alter the appearance of the basic trigonometric graphs without changing their fundamental properties. Common transformations include vertical shifts, horizontal shifts, reflections, stretches, and compressions. For the exercise in question, the graph of \( y = 4 - 8\sin^{2}x \) is a vertical shift and reflection of the graph of \( \sin^{2}x \), coupled with a vertical stretch by a factor of 8, and then shifted up by 4 units. As a result, the graph still has the periodic properties of the sine square function but has been adjusted according to the transformations applied. Understanding how these transformations work allows us to predict the behavior of the graph and correctly identify characteristics such as the location of its maxima and minima.