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Chapter 4: Problem 8

Identify the vertical translation for each equation. Do not sketch the graph. $$ y=-2-\sin x $$

Short Answer

Expert verified
The vertical translation is 2 units downward.

Step by step solution

01

Recognize the Parent Function

The parent function for the given equation is the sine function, which is written as \( y = \sin x \). This function oscillates between -1 and 1, with midline at 0.
02

Identify Changes from the Parent Function

The given function is \( y = -2 - \sin x \), which contains a linear term '-2' added to the whole function, compared to the parent function \( y = \sin x \).
03

Determine the Vertical Translation

The additional term '-2' indicates a vertical shift in the graph of the function. Specifically, the entire graph of \( y = \sin x \) is moved 2 units downward. This is because subtracting 2 lowers every point on the sine wave by 2 units.

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Key Concepts

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

Understanding the Sine Function
The sine function, commonly represented as \( y = \sin x \), is a fundamental trigonometric function. It is periodic, meaning it repeats its values in regular intervals or cycles. For the sine function, this cycle repeats every \( 2\pi \) radians.
  • The sine wave oscillates between -1 and 1.
  • The midline of the wave is located at 0, which means it balances equally around the horizontal axis.
  • Key points include: starting at 0 at \( x = 0 \), reaching 1 at \( x = \frac{\pi}{2} \), returning to 0 at \( x = \pi \), hitting -1 at \( x = \frac{3\pi}{2} \), and back to 0 at \( x = 2\pi \).
The understanding of this function provides the basis for interpreting modifications such as transformations or shifts.
Exploring the Parent Function
A parent function is the simplest form of a family of functions that retains the shape of that function type. For trigonometric functions like sine, the parent function is \( y = \sin x \).
  • It serves as a reference point from which transformations are applied.
  • The parent function's graph can be modified to create new functions.
  • Changes can include shifts, stretches, or compressions, affecting amplitude, period, and phase.
Recognizing the parent function before transformations allows us to easily understand how each transformation affects the overall graph.
Decoding Vertical Shift
A vertical shift in a function refers to moving the entire graph up or down along the vertical axis. In our exercise, the given function is \( y = -2 - \sin x \).
  • The term \(-2\) represents the vertical shift, moving the graph downward by 2 units.
  • This is because the graph of \( y = \sin x \) is being lowered by subtracting 2 from all its y-values.
  • Vertical shifts do not affect the shape or period of the wave, just its vertical positioning.
Identifying vertical translations quickly tells us where the wave is positioned relative to the original midline on the graph.

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Most popular questions from this chapter

Identify the amplitude for each of the following. Do not sketch the graph. $$ y=-\frac{5}{2} \sin x $$

Graph one complete cycle of each of the following. In each case, label the axes accurately and identify the period for each graph. $$ y=\sin \pi x $$

Graph one complete cycle of each of the following. In each case, label the axes accurately and identify the amplitude for each graph. $$ y=6 \sin x $$

Graph one complete cycle of each of the following. In each case, label the axes accurately and identify the period for each graph. $$ y=\sin \frac{\pi}{2} x $$

Use your graphing calculator to graph each family of functions for \(-2 \pi \leq x \leq 2 \pi\) together on a single coordinate system. (Make sure your calculator is set to radian mode.) What effect does the value of \(h\) have on the graph? $$ y=\sin (x-h) \quad \text { for } h=0, \frac{\pi}{4}, \frac{\pi}{2} $$
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