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## Algebra 2 (Eureka Math/EngageNY)

### Course: Algebra 2 (Eureka Math/EngageNY)>Unit 2

Lesson 3: Topic B: Lesson 11: Graphing sinusoidal functions

# Transforming sinusoidal graphs: vertical stretch & horizontal reflection

Sal graphs y=2*sin(-x) by considering it as a vertical stretch and a horizontal reflection of y=sin(x). Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• Are there videos on translation of sine and cosine functions? (vertical shift and phase shift)
• no,there aren't but i can tell you what vertical and phase shift are

HORIZONTAL SHIFT
To shift a graph horizontally, a constant must be added to the function within parentheses--that is, the constant must be added to the angle, not the whole function

VERTICAL SHIFT
To shift such a graph vertically, one needs only to change the function to f (x) = sin(x) + c , where c is some constant. Thus the y-coordinate of the graph, which was previously sin(x) , is now sin(x) + 2 . All values of y shift by two.
PHASE SHIFT
Phase shift is any change that occurs in the phase of one quantity, or in the phase difference between two or more quantities.
hope it does not confuses you (also you can think diagramaticaly the graphs of sine and cosine graphs,it would hep a bit)
• What is the difference between -2sin(x) and 2sin(-x)?
• Now sinx is an odd function. This means that sin(-x) = -sin(x). Thus
-2sin(x) - 2sin(-x) = -2sinx - (-2sinx) = -2sinx+2sinx = 0.
• Sal says near the end of the video that the negative flips it over the y-axis, is it also correct to say that it is reflected over the x-axis.
• yes, because the -x is the same as -1*x, which effects the amplitude(how far the graph line varies from 0) if that is change by the amplitude, it makes the circle wider, and also taller, so that it works for both sine and cosine. the center of the circle is 0 so if the y is longer, the x has to match it. if the y is effected by the -1, so is the x
• I feel all of this is confuse because, out of nowhere, with no explanation whatsoever, we were asked to deal with the equation in the form of y=acos(bx+c)+d. Everything was going smoothly, and then came the "Finding amplitude and midline of sinusoidal functions from their formulas" playlist, where it is assumed that we already know that type of equation. Now we have to join bits and pieces of information from the hints of different exercises and next videos, but it feels all too muddy. Please, post a video explaining the y=acos(bx+c)+d equation, like you guys did with the slope-intercept equation.
• Yeah I agree... very badly executed on Khan Academy's part. I have no problem understanding it since I'm already familiar with the subject, but I can feel your frustration.
• i should know this, but is f(-x) same as -f(x) ?
• Not necessarily. It is true for some functions, such as sin x, but not generally.

(Functions for which f(-x) = -f(x) is true are called odd functions. If f(-x) = f(x), then the function is said to be even.)
• Will I ever use this is a real life scenario? Is this truly necessary to be taught to everyone in school? Can't I trade in a specific class like this for something that would help me to be a useful contribution to society?
• Knowing what causes transformations in sinusoidal graphs is extremely useful in real life as many signals travel in sinusoidal patterns. Radio signals, microwaves, even light itself can be modeled by sinusoidal functions. Knowing the period (horizontal stretch) can help you find the frequency of the function which can help you analyze radio signals, what color of light you're looking at etc.. Vertical stretch gives you amplitude and (though I don't know exactly how it works) the AM radio stations use Amplitude Modulation to transmit their signals. If we didn't know how to analyze these types of functions, the world as we know it wouldn't exist...
Hope this helps! :)
• Hi, I was just wondering if you are given what would generally appear to be a vertical shift but it is in parentheses what do you do? For example: y = -π/2(cos(√3x+2)-.5)
What does the -.5 mean in this situation?
• If the -.5 is in the parentheses, this means it is affecting the output of the function, therefore resulting in a horizontal shift. In this specific case, the graph would shift to the right -.5 units.
• Could anyone please point me to a lesson which explains how to calculate the phase shift.
• I think he's covered this, but I haven't located it.

In "asin(bx+c)+d", c is the phase shift.

In sin(x+c), when you are at x radians on a graph, your y is sin(x+c). So if c is positive, you look ahead c radians to find y. Therefore your graph is shifted to the left by c. And if c is negative you use y for an angle c radians less than where you are (x), so the graph is shifted to the right by c.