How to Change the Amplitude, Period, and Position of a Tangent or Cotangent Graph You can transform the graph for tangent and cotangent vertically, change the period, shift the graph horizontally, or shift it vertically. However, you should take each transformation one step at a time. For example, to graph follow these steps: Sketch the parent graph for tangent. This graph doesn't shift horizontally, because no constant is added inside the grouping symbols (parentheses) of the function. Now that you've graphed the basics, you can graph a function that has a period change, as in the function You see a lot of pi in that one. Sketch the parent graph for cotangent.Shrink or stretch the parent graph.No constant is multiplying the outside of the function; therefore, you can apply no shrink or stretch.Find the period change.You factor out thewhich affects the period. The transformed graph of y(x) = cot 2pi(x + 1/4).
1. Graphs of y = a sin x and y = a cos x by M. Bourne The Sine Curve y = a sin t The sine curve occurs naturally when we are examining waves. When waves have more energy, they go up and down more vigorously. Let's investigate the shape of the curve y = a sin t and see what the concept of "amplitude" means. Have a play with the following interactive. Sine curve Interactive Run the animation first (click "Start"). You can also drag the dot around the circle to create a sine curve (do it slowly). The scale for this is radians. t = θ = 0 y = 100 sin(0) = 0 Amplitude = radius = 100.0 2π π 23π 2π t fps = 0 Copyright © www.intmath.com Did you notice? The shape of the sine curve forms a regular pattern (the curve repeats after the wheel has gone around once). [Credits: The above animation is loosely based on this graph by HubleSoftware.] Amplitude The "a" in the expression y = a sin x represents the amplitude of the graph. Amplitude is always a positive quantity. amplitude =∣a∣ Graph of Sine x - with varying amplitudes We start with y = sin x.
Amplitude, Period, Phase Shift and Frequency Some functions (like Sine and Cosine) repeat forever and are called Periodic Functions. The Period is the length from one peak to the next (or from any point to the next matching point): The Amplitude is the height from the center line to the peak (or to the trough). Or we can measure the height from highest to lowest points and divide that by 2. The Phase Shift is how far the function is horizontally to the right of the usual position. The Vertical Shift is how far the function is vertically from the usual position. All Together Now! We can have all of them in one equation: y = A sin(Bx + C) + D amplitude is A period is 2π/B phase shift is −C/B vertical shift is D Example: sin(x) This is the basic unchanged sine formula. So amplitude is 1, period is 2π, there is no phase shift or vertical shift: Example: 2 sin(4x − 2) + 3 amplitude A = 2 period 2π/B = 2π/4 = π/2 phase shift −C/B = −(−2)/4 = 1/2 vertical shift D = 3 In words: Note the Phase Shift formula −C/B has a minus sign: And we get: Frequency
Trigonometry: Graphs: Horizontal and Vertical Shifts Among the variations on the graphs of the trigonometric functions are shifts--both horizontal and vertical. Such shifts are easily accounted for in the formula of a given function. Take function f , where f (x) = sin(x) . The graph of y = sin(x) is seen below. Figure %: The Graph of sine(x) Vertical Shifts To shift such a graph vertically, one needs only to change the function to f (x) = sin(x) + c , where c is some constant. Figure %: By adding a constant to a function, like sine, the graph is shifted vertically Horizontal Shifts 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. Figure %: Horizontal shift The graph of sine is shifted to the left by units. .
Trigonometric Equations Remember to first solve for the trig function and then solve for the angle value. Solution: If there is more than one trig function in the equation, identities are needed to reduce the equation to a single function for solving. Example: Solution: There are trig equations, just like there are normal equations, where factoring does not work!! Example: Solution: Since there are two trig functions in this problem, we need to use an identity to eliminate one of them. Using the quadratic formula, we get: Standard Grade Bitesize Maths II - Graphs : Revision 3. Graphs of tan by M. Bourne The graphs of tanx, cotx, secx and cscx are not as common as the sine and cosine curves that we met earlier in this chapter. However, they do occur in engineering and science problems. They are interesting curves because they have discontinuities. For certain values of x, the tangent, cotangent, secant and cosecant curves are not defined, and so there is a gap in the curve. [For more on this topic, go to Continuous and Discontinuous Functions in an earlier chapter.] Recall from Trigonometric Functions, that tanx is defined as: tanx=cos xsin x For some values of x, the function cosx has value 0. When this happens, we have 0 in the denominator of the fraction and this means the fraction is undefined. The same thing happens with cotx, secx and cscx for different values of x. The Graph of y = tan x Sketch y = tan x. Solution As we saw above, tanx=cosxsinx This means the function will have a discontinuity where cos x = 0. Recall that −23π=−4.7124 and −2π=1.5708. π 2π t
Amplitude and Period Amplitude and Period Learning Objective(s) · Understand amplitude and period. · Graph the sine function with changes in amplitude and period. · Graph the cosine function with changes in amplitude and period. · Match a sine or cosine function to its graph and vice versa. You know how to graph the functions and . or , where a and b are constants. We used the variable previously to show an angle in standard position, and we also referred to the sine and cosine functions as . for the input (as well as to label the horizontal axis). . You know that the graphs of the sine and cosine functions have a pattern of hills and valleys that repeat. . (or ) on the interval looks like the graph on the interval . The graph below shows four repetitions of a pattern of length . is on the interval is one cycle. You know from graphing quadratic functions of the form that as you changed the value of a you changed the “width” of the graph. and see how changes to b will affect the graph. periodic, and if so, what is the period?
Solving Trigonometric Equations Solving Trigonometric Equations (page 1 of 2) Solving trig equations use both the reference angles you've memorized and a lot of the algebra you've learned. Be prepared to need to think! Solve sin(x) + 2 = 3 for 0° < x < 360° Just as with linear equations, I'll first isolate the variable-containing term: sin(x) + 2 = 3 sin(x) = 1 Now I'll use the reference angles I've memorized: x = 90° Solve tan2(x) + 3 = 0 for 0° < x < 360° There's the temptation to quickly recall that the tangent of 60° involves the square root of 3 and slap down an answer, but this equation doesn't actually have a solution: tan2(x) = –3 How can the square of a trig function evaluate to a negative number? no solution Solve on 0° < x < 360° To solve this, I need to do some simple factoring: Now that I've done the algebra, I can do the trig. x = 30°, 90°, 270°, 330° Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved Solve sin2(x) – sin(x) = 2 on 0° < x < 360° Only one of the factor solutions is sensible. x = 270° Hmm...