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Chapter 5: Problem 112

In Exercises \(111-114\), determine whether each statement is true or false. ( \(A\) and \(B\) are positive real numbers.) The graph of \(y=A \sin (-B x)\) is the graph of \(y=A \sin B x\) reflected about the \(x\) -axis.

Short Answer

Expert verified
True, because the graph is visually similar to a reflection about the x-axis.

Step by step solution

01

Understanding the Reflection

Reflecting a graph about the x-axis changes the sign of the output values (the y-coordinates). The equation for reflecting a function about the x-axis is done by negating the function: if you start with a function, say, \(y=f(x)\), reflecting it about the x-axis gives you \(y=-f(x)\). This means the sine function would become \(y=-A \sin(Bx)\) if it were reflected about the x-axis.
02

Compare with Given Equation

The problem states the function \(y=A \sin(-Bx)\). This is actually a transformation that involves the negation of the angle within the sine function. The transformation does not affect the y-values sign as an x-axis reflection would.
03

Understanding the Effect of Negative Inside Sine Function

For the function \(y = \sin(-Bx)\), using the identity \(\sin(-\theta) = -\sin(\theta)\), it can be rewritten as \(y = -A \sin(Bx)\). This equation resembles an x-axis reflection since it negates the sine function, but that transformation directly results from the angle negation, not a vertical reflection process.
04

Check if the Statements Match

The transformed equation \(y = -A \sin(Bx)\) arises directly from the angle negation identity, not an intentional vertical reflection across the x-axis of the function \(y=A\sin(Bx)\). However, its graphical output visually mirrors what reflecting about the x-axis would produce.
05

Determine Truthfulness

Since the transformation of \(y = A \sin(-Bx)\) directly results in what would visually match an x-axis reflection, even though that's not the change described, mathematically, visually it fits the description of an x-axis reflection.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reflection of Graphs
Reflecting a graph is akin to flipping it over a specific axis, either the x-axis or the y-axis. When we reflect a graph about the x-axis, each point on the graph has its y-value sign shifted. Basically, if you have a point \(x, y\), it becomes \(x, -y\). This change effectively turns the graph upside down along the x-axis.
  • For a function \(y = f(x)\), the reflection about the x-axis gives us \(y = -f(x)\).
This principle is crucial when interpreting and transforming graphs, as it visually alters how the entire function appears on the coordinate plane. When applied to trigonometric functions like sine and cosine, it affects their characteristic wave patterns, essentially flipping peaks and troughs relative to the axis used for reflection.
Graph Transformations
Understanding graph transformations requires two key aspects: transformations within the function's argument and outside it. When modifying the input variable (x) inside the function, it often translates into horizontal changes (shifting left/right, for instance) or stretches/compressions along the x-axis.
  • Transformations outside the function generally impact the y-values, causing vertical shifts or reflections.
When negative signs are introduced, either within the input variable or with the function itself, the graph undergoes specific transformations:
  • A negative sign within the angle (e.g., \(y = (x)\) to \(y = f(-x)\)) typically reflects the graph across the y-axis, impacting horizontal positions.
  • A negative sign placed directly with the function (e.g., \(y = -f(x)\)) changes how high or low a function's output appears, flipping it vertically.
Within trigonometric functions, these transformations sometimes harness trigonometric identities, such as \(\sin(-胃) = -\sin(胃\), to effectively understand and predict these visual outcomes.
Sine Function
The sine function is foundational in trigonometry and has unique properties, primarily noted for its smooth, periodic wave-like behavior. Its basic form is \(y = \sin(x)\), where the function oscillates between -1 and 1. Each complete cycle of this function spans from 0 to \(2\pi\).
  • Graphs of sine are often altered by changing the amplitude, period, or by reflecting it.
  • Amplitude changes adjust the height of the wave, observed as the coefficient \(A\) in \(y = A\sin(x)\).
  • The period is altered by modifying the input variable as observed in \(B\) in \(y = A \sin(Bx)\), where the new period becomes \((2\pi/B)\).
When the argument of the sine function is negated, as in \(y = \sin(-Bx)\), the behavior of the function initially suggests a reflection. Since \(\sin(-x) = -\sin(x\), this seemingly alters its peaks and valleys similarly to an x-axis reflection, though fundamentally it's due to an angle transformation within the sine identity itself.

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Most popular questions from this chapter

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=-\sin \left(\frac{\pi}{4} x\right)-3 \sin \left(\frac{5 \pi}{4} x\right), 0 \leq x \leq 4$$

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=-\frac{1}{4} \cos \left(\frac{\pi}{6} x\right)-\frac{1}{2} \cos \left(\frac{\pi}{3} x\right), 0 \leq x \leq 12$$

For Exercises \(87-90\), refer to the following: Set the calculator in parametric and radian modes and let $$ \begin{array}{l} X_{1}=\cos T \\ Y_{1}=\sin T \end{array} $$ (TABLE CANNOT COPY) Set the window so that \(0 \leq \mathrm{T} \leq 2 \pi,\) step \(=\frac{\pi}{15},-2 \leq \mathrm{X} \leq 2\) and \(-2 \leq Y \leq 2 .\) To approximate the sine or cosine of a T value, use the \([\text { TRACE }]\) key, type in the T value, and read the corresponding coordinates from the screen. Approximate \(\cos \left(\frac{5 \pi}{4}\right)\) to four decimal places.

In Exercises \(115-118, A\) and \(B\) are positive real numbers. Find the \(y\) -intercept of the function \(y=A \sin B x\)

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=8 \cos x-6 \cos \left(\frac{1}{2} x\right),-2 \pi \leq x \leq 2 \pi$$
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