how do you shift a sine and cosine graph

• f(x) = sin (x – 3) moves the parent graph of y = sin x to the right by 3.
• g(x) = cos (x + 2) moves the parent graph of y = cos x to the left by 2.
• k(x) = sinx + 4 moves the parent graph of y = sin x up 4.

Full Answer

How do you graph the sine function?

To easily draw a sine function, on x – axis we’ll put values from $ -2 pi$ to $ 2 pi$, and on y – axis real numbers. First, codomain of the sine is [-1, 1], that means that your graphs highest point on y – axis will be 1, and lowest -1, it’s easier to draw lines parallel to x – axis through -1 and 1 on y axis to know where is your boundary.

What exactly is sine cosine and tangent?

Sine, cosine, and tangent are three ratios of the lengths of the sides of a triangle. Since they are ratios, sine, cosine, and tangent are mathematical relationships that can be expressed as fractions. Sine, Cosine, Tangent and the Three Sides. The measurement of sine, cosine, and tangent will depend upon which angle is being measured in the ...

What is the Pythagorean property for cosine and sine?

The sum of the squares of sine and cosine functions at an angle equals to one is called the Pythagorean identity of sine and cosine functions. The sine and cosine are two functions in trigonometry. They have a direct relationship between them in square form but it represents the Pythagorean theorem.

Is sine and cosine a periodic function?

The sine and cosine functions have several distinct characteristics: They are periodic functions with a period of 2π. The domain of each function is (−∞,∞) ( − ∞, ∞) and the range is [−1,1] [ − 1, 1]. The graph of y =sinx y = sin. ⁡. x is symmetric about the origin, because it is an odd function.

How do you shift a sine function?

Phase shift is the horizontal shift left or right for periodic functions. If c = π 2 then the sine wave is shifted left by . If then the sine wave is shifted right by 3. This is the opposite direction than you might expect, but it is consistent with the rules of transformations for all functions.

How do you find the shift of a sine graph?

the horizontal shift is obtained by determining the change being made to the x-value. The horizontal shift is C. The easiest way to determine horizontal shift is to determine by how many units the "starting point" (0,0) of a standard sine curve, y = sin(x), has moved to the right or left.

How do you graph phase shifts of sine and cosine functions?

4:301:07:58Graphing Trigonometric Functions, Phase Shift, Period ... - YouTubeYouTubeStart of suggested clipEnd of suggested clipSet the X plus C equal to zero and solve for ax. So the phase shift is usually it's negative C overMoreSet the X plus C equal to zero and solve for ax. So the phase shift is usually it's negative C over B but we'll talk about that later in the video.

How do you transform sin and cos graphs?

3:1518:34Graphing Sine and Cosine Trig Functions With Transformations ...YouTubeStart of suggested clipEnd of suggested clipThe generic equation for sine or cosine wave is y is equal to a sine B. X plus C plus D. As weMoreThe generic equation for sine or cosine wave is y is equal to a sine B. X plus C plus D. As we talked about already a is the amplitude B.

How do you determine horizontal and vertical shifts?

How ToIdentify the vertical and horizontal shifts from the formula.The vertical shift results from a constant added to the output. Move the graph up for a positive constant and down for a negative constant.The horizontal shift results from a constant added to the input. ... Apply the shifts to the graph in either order.

How do you graph a sine function with a vertical shift?

5:197:06Graphing the Sine Function with a Vertical Shift - YouTubeYouTubeStart of suggested clipEnd of suggested clipPoint come back up to the x-intercept at the next critical. Point then head up to the maximum. PointMorePoint come back up to the x-intercept at the next critical. Point then head up to the maximum. Point and then go back down to the x-intercept.

What order do you do graph transformations?

Apply the transformations in this order:Start with parentheses (look for possible horizontal shift) (This could be a vertical shift if the power of x is not 1.)Deal with multiplication (stretch or compression)Deal with negation (reflection)Deal with addition/subtraction (vertical shift)

What is the formula for phase shift?

Using Phase Shift Formula, F(x) = Asin(Bx−C) + D.

How do you find the phase shift of a cosine function?

1:422:46Learn How to Find the Amplitude, Period, Phase Shift and ... - YouTubeYouTubeStart of suggested clipEnd of suggested clipX plus 3 equals 0. Minus 3 minus 3 x equals negative 3. So therefore you're shifting. Left threeMoreX plus 3 equals 0. Minus 3 minus 3 x equals negative 3. So therefore you're shifting. Left three units that's the patient.

What is the formula for phase shift?

Using Phase Shift Formula, F(x) = Asin(Bx−C) + D.

What is sine phase shift?

Any sine wave that does not pass through zero at t = 0 has a phase shift. The difference or phase shift as it is also called of a Sinusoidal Waveform is the angle Φ (Greek letter Phi), in degrees or radians that the waveform has shifted from a certain reference point along the horizontal zero axis.

How do you shift a cosine graph?

1:386:20Graphing Sine and Cosine with Phase (Horizontal) Shifts, Example 2YouTubeStart of suggested clipEnd of suggested clipSo it says takes to take the original sine graph and shift it PI units. And again even though I seeMoreSo it says takes to take the original sine graph and shift it PI units. And again even though I see a negative we go in the positive direction. So shift PI units to the right.

How do you find the phase shift of a tangent function?

4:0211:38Graphing Tangent: Period and Phase Shift - YouTubeYouTubeStart of suggested clipEnd of suggested clipSo it's different so 180 divided by 3 is 60 degree views. Okay whatever this number is you take theMoreSo it's different so 180 divided by 3 is 60 degree views. Okay whatever this number is you take the opposite. And that is the phase shift.

What is the basic equation for cosine?

Note that in the basic equation for cosine, A = 1, B = 1, C = 0, and D = 0.

What happens when a graph is negative?

If A is negative, the graph will flip over the x-axis. This is called a reflection.

How to move x value in period?

For each x-value in the period, move the x-value to the left by C/B if C/B is negative, or move each x-value to the right by C/B if C/B is positive.

Is there a scale on a graph?

Since you are only hand-drawing your graphs, there is no precise scale, but it must be accurate at certain points .

What is the horizontal shift in trigonometry?

In trigonometry, this Horizontal shift is most commonly referred to as the Phase Shift. As Khan Academy states, a phase shift is any change that occurs in the phase of one quantity. In other words, there is a change in the phase of our wave. Typically, this is the trickiest transformation for students to draw because it involves an adjustment ...

When you get to dealing with parentheses, will you notice that you are either shifting your starting point to the right?

When we get to dealing with our parentheses, we will notice that we will be either shifting our starting point to the right (positive value) or left (negative value) a specific distance , as appropriately stated by Math Bits Notebook.

How to find the x value of a graph?

The graph starts behind the maximum point. To determine this x value, subtract from the x-coordinate of the maximum:

What does a dotted line mean in a graph?

The dotted line is at , where the maximum occurs and therefore where the graph starts . This means that the graph is shifted to the right .

Is there a horizontal shift in parentheses?

Since is being added inside the parentheses, there will be a horizontal shift. The goal is to maintain zero within the parentheses so you will shift left radians.

How do you know if a graph is a sine and cosine?

You can graph sine and cosine functions by understanding their period and amplitude. Sine and cosine graphs are related to the graph of the tangent function, though the graphs look very different.

What are the four basic transformations of sine and cosine?

how to graph sine and cosine functions with the four basic transformations: amplitude, period, phase shift and vertical shift.

What is the coefficient of a graph?

The coefficients A and B in y = Asin (Bx) or y = Acos (Bx) each have a different effect on the graph. If A and B are 1, both graphs have an amplitude of 1 and a period of 2pi. For sine and cosine transformations, when A is larger than 1, the amplitude increases and is equal to the value of A; if A is negative, the graph reflects over the x-axis. When B is greater than 1, the period decreases; use the formula 2pi/B to find the period.

What happens when a graph is larger than 1?

For sine and cosine transformations, when A is larger than 1, the amplitude increases and is equal to the value of A; if A is negative, the graph reflects over the x-axis.

What is the horizontal shift in a graph?

It shifts the graph left (if h is negative) or right (if h is positive) and in the amount equal to h. The amount of horizontal shift is called the phase shift, which equals h.

How to find the equation of a trig function?

When finding the equation for a trig function, try to identify if it is a sine or cosine graph. To find the equation of sine waves given the graph, find the amplitude which is half the distance between the maximum and minimum . Next, find the period of the function which is the horizontal distance for the function to repeat. If the period is more than 2pi, B is a fraction; use the formula period=2pi/B to find the exact value. Last, find any phase shift, h.

Does the constant H change the period?

The constant h does not change the amplitude or period (the shape) of the graph. It shifts the graph left (if h is negative) or right (if h is positive) and in the amount equal to h. The amount of horizontal shift is called the phase shift, which equals h.

How to do phase shift in a sine function?

The additional term D in the function y = sin ( x + D) allows for a phase shift (moving the graph to the left or right) in the graph of the sine functions. (This also holds for the cosine function.) The phase shift is | D |. This is a positive number. It does not matter whether the shift is to the left (if D is positive) or to the right (if D is negative). The sine function is odd, and the cosine function is even. The cosine function looks exactly like the sine function, except that it is shifted π/2 units to the left (Figure 8 ). In other words,

How to find the graph of a function?

The graph of the function y = − f ( x) is found by reflecting the graph of the function y = f ( x) about the x ‐axis. Thus, Figure can also represent the graph of y = −3 sin 2 x. Specifically,

What is the period of a sine function?

The additional factor C in the function y = sin Cx allows for period variation (length of cycle) of the sine function. (This also holds for the cosine function.) The period of the function y = sin Cx is 2π/|C|. Thus, the function y = sin 5 x has a period of 2π/5. Figure 7 illustrates additional examples.

What is the additional factor B in the function y = B sin x?

The additional factor B in the function y = B sin x allows for amplitude variation of the sine function. The amplitude, | B |, is the maximum deviation from the x ‐axis—that is, one half the difference between the maximum and minimum values of the graph. This also holds for the cosine function (Figure 4 ).

What is the period of cos x?

Because cos x has a period of 2π, cos π x has a period of 2 (Figure 9 ).

Which way is the graph shifted?

Positive, the graph is shifted to the right.

What is phase shift formula?

We know that in the phase shift formulas A * sin (Bx - C) + D and A * cos (Bx - C) + D, the A determines how far the values fluctuate on either side of the centerline. The B specifies how far we extend the graph's bumps and, as a result, how fast we get to repeat the values. Also, together with C, the two describe if we've moved the function to the left or right and how much.

What is the amplitude of phase shift equations?

Yup, you guessed it: the amplitude of the phase shift equations A * sin (Bx - C) + D and A * cos (Bx - C) + D is simply equal to A.

What can affect phase shift amplitude?

Therefore, the only thing that can affect the amplitude in the phase shift formulas A * sin (Bx - C) + D and A * cos (Bx - C) + D is the non-zero A. And indeed, since sin (Bx - C) and cos (Bx - C) are all this time between -1 and 1, the multiplier A changes this range to -1 * A = -A and 1 * A = A.

What is phase shift?

The phase shift (also called the horizontal shift or horizontal translation) describes how far horizontally the graph has been moved from the regular sine or cosine. As such, the value is equal to 0 if we have the two functions unaltered.

What is the amplitude of a graph?

The amplitude is how far (either way) the values run from the graph's centerline. For a simple sine or cosine, its value is 1 since the centerline is at 0, and the function's values range from -1 to 1.

What is the period of a graph?

The period is the length on the horizontal axis, after which the function begins repeating itself. In other words, the (infinite) graph is just a bunch of period-length copies glued together at the ends. For a simple sine or cosine, the period equals 2π since sin (0) = sin (2π) = sin (4π) = ... and the parts in between are exactly the same (and similarly for the cosine).