Vertical Translation and Phase Shift

Vertical Translation and Phase Shift Trigonometry MATH 103 S. Rook

Overview • Section 4.3 in the textbook: • Vertical translation • Phase shift • Graphing sine and cosine in general

Vertical Translation

How Vertical Translation Affects a Graph • The graphs of y = k + A sin x and y = k + A cos x are related to the graphs y = A sin x and y = A cos x: • The value of k is added to each y-coordinate of y = A sin x or y = A cos x to get the new functions y = k + A sin x or y = k + A cos x • E.g. (0, 0) on y = sin x would become (0, -2) on the graph of y = -2 + sin x • Amplitude = |A| • The maximum value is k + |A| and the minimum value is k + -|A| • The range of y = k + A sin x or y = k + A cos x is then [k + -|A|, k + |A|]

How Vertical Translation Affects a Graph (Continued) • If k > 0 y = k + A sin x or y = k + A cos x will be shifted UPk units as compared to y = A sin x or y = A cos x • If k < 0 y = k + A sin x or y = k + A cos x will be shifted DOWNk units as compared to y = A sin x or y = A cos x • The value of kaffects ONLY the y-coordinate • The value of kaffects ONLY the vertical translation

Graphing y = k + A sin x or y = k + A cos x • To graph one cycle of y = k + A sin x or y = k + A cos x: • Follow the same steps for graphing y = A sin x or y = A cos x: • Dividing intervals, creating a table of values, etc. • Must take the effect of k into account when constructing the graph: • The maximum value will be k + |A| and the minimum value will be k + -|A|

Vertical Translation (Example) Ex 1: Graph one complete cycle: y = -3 + 2sin x

Phase Shift

Phase Shift • Phase shift occurs when we move a graph horizontally • In the x-direction • We have a phase shift when we add/subtract a quantity to the variable inside of the trigonometric function • Recall that the inside of a trigonometric function is also called the argument • e.g.

How Phase Shift Affects a Graph • Consider the effects of phase shift on y = sin x and y = sin(x – h) or y = cos x and y = cos(x – h) • Graph one cycle of y = sin x and y = sin(x – π⁄2) using a table of values: • Notice that y = sin x completes one cycle in the interval 0 to 2π and y = sin(x – π⁄2) completes one cycle in the interval π⁄2 to 5π⁄2 • y = sin(x – π⁄2) has been shifted to the right by h = π⁄2

How Phase Shift Affects a Graph (Continued) • The value of the phase shift coincides with the leftmost value of a cycle • The phase shift for y = sin x is 0 in the interval 0 to 2π and the phase shift for y = sin(x – π⁄2) is π⁄2 in the interval π⁄2 to 5π⁄2 • To establish a relationship between y = sin x and y = sin(x – h) or y = cos x and y = cos(x – h): • When h = 0, the graph begins a cycle at 0 • When h = π⁄2, the graph begins a cycle at π⁄2 (Add π⁄2; )

Relationship Between h and Phase Shift • Therefore, for y = sin(x – h) or y = cos(x – h): phase shift = h • If y = sin(x – h) or y = cos(x – h) h > 0 • If y = sin(x + h) or y = cos(x + h) h < 0 • However, if the signs are confusing, we can also find the value of the phase shift using the interval method: • i.e. 0 ≤ argument ≤ 2π and then isolate the variable (e.g. x) to obtain: phase shift ≤ x ≤ end value

Graphing y = sin(x – h) or y = cos(x – h) • To graph one cycle, we repeat the same steps for graphing y = A sin x or y = A cos xEXCEPT: • Calculate the phase shift • Calculate the subinterval length (π⁄2 if the period is 2π) • Adjust the x-axis to start at the phase shift • Label the x-axis by adding increments of the subinterval until the end value of the interval is reached • Note that we have the SAME PERIOD on the phase shifted interval as we do when using • Can verify by subtracting the endpoints • This period is 2π if B = 1 (a constant of 1 multiplying the variable)

Phase Shift (Example) Ex 2: Graph one complete cycle: a) b)

Graphing Sine and Cosine in General

Graphing y = sin(Bx + C) or y = cos(Bx + C) • The process is slightly different for calculating phase shift for y = sin(Bx + C) or y = cos(Bx + C) • Use the interval method: • Recall that the phase shift coincides with the leftmost value of the interval • Phase shift is then -C⁄B and period is 2π⁄B (subtract the endpoints of the interval) • Can either derive the period and phase shift using the interval method or memorize the formulas: period = 2π⁄B and phase shift = -C⁄B

Summary of y = k + A sin(Bx + C) or y = k + A cos(Bx + C) • Given y = k + A sin(Bx + C) or y = k + A cos(Bx + C), recall that the: • Vertical translation is k • Amplitude is |A| • Period is 2π⁄B • Phase shift is -C⁄B • Note that it may be easier to use the interval method to obtain the period and phase shift • Domain is (-oo, +oo) • Range is [k + -|A|, k + |A|]

Graphing y = k + A sin(Bx + C) or y = k + A cos(Bx + C) • To graph y = k + A sin(Bx + C) or y = k + A cos(Bx + C): • Find the values for A (amplitude), period, k (vertical translation), and phase shift • “Construct the Frame” for one cycle: • Calculate the subinterval length (easiest to use period⁄4) • x-axis by the interval method: • Start the cycle at the leftmost value of the interval (phase shift) • Label the x-axis by adding increments of the subinterval until the end value of the interval is reached

Graphing y = k + A sin(Bx + C) or y = k + A cos(Bx + C) (Continued) • x-axis by the formula method: • Start the cycle at -C⁄B (phase shift) • Label the x-axis by adding increments of the subinterval until 2π⁄B – C⁄B is reached • y-axis: • Minimum value is k + -|A| • Maximum value is k + |A| • Create a table of values for the points marked on the x-axis • Connect the points by using the shape of the sine or cosine function • Extend the graph as necessary

Graphing Sine and Cosine in General (Example) Ex 3: a) identify the amplitude b) identify the vertical translation c) identify the period d) identify the phase shift e) graph one cycle

Graphing Sine and Cosine in General (Example) Ex 4: a) identify the amplitude b) identify the vertical translation c) identify the period d) identify the phase shift e) graph on the given interval

Summary • After studying these slides, you should be able to: • Identify the vertical translation, amplitude, period, and phase shift for ANY sine or cosine graph or equation • Graph an equation of the form y = k + A sin(Bx + C) or y = k + A cos(Bx + C) • Additional Practice • See the list of suggested problems for 4.3 • Next lesson • The Other Trigonometric Functions (Section 4.4)

Vertical Translation and Phase Shift