/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 56 Find the equation for each curve... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 56

Find the equation for each curve in its final position. The graph of \(y=\sin (x)\) is shifted a distance of \(\pi / 2\) to the left, translated one unit upward, stretched by a factor of \(4,\) then reflected in the \(x\) -axis.

Short Answer

Expert verified
The final equation is \( y = -4 \times \big(\text{sin}(x + \frac{\text{蟺}}{2}) + 1\big) \).

Step by step solution

01

Horizontal Shift

The function is shifted \( \frac{\text{蟺}}{2} \) units to the left. This means we replace \( x \) with \( x + \frac{\text{蟺}}{2} \) in the function. Therefore, \( y = \text{sin}(x) \) becomes \( y = \text{sin}\big(x + \frac{\text{蟺}}{2}\big) \).
02

Vertical Translation

Next, translate the function one unit upward. This involves adding 1 to the function. Our new equation is \( y = \text{sin}\big(x + \frac{\text{蟺}}{2}\big) + 1 \).
03

Vertical Stretch

The function is then stretched by a factor of 4. To accomplish this, multiply the entire function by 4. The equation now is \( y = 4 \times \big(\text{sin}\big(x + \frac{\text{蟺}}{2}\big) + 1\big) \).
04

Reflection in the x-axis

Finally, reflect the function in the x-axis. This involves multiplying the entire function by -1, resulting in \( y = -4 \times \big(\text{sin}\big(x + \frac{\text{蟺}}{2}\big) + 1\big) \). Therefore, the final equation is \ y = -4 \times \big(\text{sin}\big(x + \frac{\text{蟺}}{2}\big) + 1 \big) \.

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Key Concepts

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

Horizontal Shift
Understanding horizontal shifts is crucial in transforming trigonometric functions. A horizontal shift means moving the graph left or right along the x-axis. To shift the function \(\text{y = sin(x)}\) by a distance of \(\frac{\pi}{2}\) units to the left, we replace \(x\) with \((x + \frac{\pi}{2})\). This results in the function \(\text{y = sin}\left(x + \frac{\pi}{2}\right)\). It's like translating all the points on the graph by \(\frac{\pi}{2}\) units to the left. Remember, a positive addition inside the function moves the graph to the left, while a negative addition moves it to the right.
Vertical Translation
Vertical translation adjusts the graph up or down along the y-axis. For the function \(\text{y = sin}\left(x + \frac{\pi}{2}\right)\), translating it one unit upward means adding 1 to the entire function. Thus, the updated function becomes \(\text{y = sin}\left(x + \frac{\pi}{2}\right) + 1\). Here, every point on the graph of the function is moved up by 1 unit. This translation does not affect the period or shape of the function, only its position vertically.
Vertical Stretch
A vertical stretch scales the graph up or down by a certain factor, changing its amplitude. For a stretch by a factor of 4, we multiply the entire function by 4. Starting with \(\text{y = sin}\left(x + \frac{\pi}{2}\right) + 1\), we now have \(\text{y = 4}\left(\text{sin}(x + \frac{\pi}{2}) + 1\right)\). This means every y-value on the graph is stretched four times their original distance from the x-axis, making the peaks and troughs more pronounced.
Reflection in the x-axis
Reflecting a function in the x-axis involves flipping it upside down. This is achieved by multiplying the entire function by -1. Our current function \(\text{y = 4}\left(\text{sin}\left(x + \frac{\pi}{2}\right) + 1\right)\) becomes \(\text{y = -4}\left(\text{sin}\left(x + \frac{\pi}{2}\right) + 1\right)\). This reflection inverts all the y-values, so positive values become negative and vice versa. This step completes the series of transformations, giving us the final graph.

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Most popular questions from this chapter

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