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

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Algebra - Symmetry Example 1 Determine the symmetry of each of the following equations. (a) [Solution] (b) (c) 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! A-level Maths Trigonometry Revision - Graphs of trigonometric functions The graphs of the three major functions are very important and you need to learn the characteristics of each. The sine function This graph is continuous (there are no breaks). The range is -1 ≤ sin θ ≤ +1. Graphs of trigonometric functions The Topics | Home Zeros of a function The graph of y = sin x The period of a function The graph of y = cos x The graph of y = sin ax

The Graphs of Sine and Cosine A sine wave, or sinusoid, is the graph of the sine function in trigonometry. A sinusoid is the name given to any curve that can be written in the form (A and B are positive). Sinusoids are considered to be the general form of the sine function. Symmetry and Graphing Symmetry and Graphing (page 3 of 3) Sections: Symmetry about an axis, Symmetry about a point, Symmetry and graphing Symmetry is more of a geometrical than an algebraic concept, but the subject of symmetry does come up in a couple of algebraic contexts. 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. Graphing Trigonometric Functions Graphing Trigonometric Functions (page 1 of 3) Sections: Introduction, Examples with amplitude and vertical shift, Example with phase shift You've already learned the basic trig graphs. But just as you could make the basic quadratic x2, more complicated, such as –(x + 5)2 – 3, so also trig graphs can be made more complicated. We can transform and translate trig functions, just like you transformed and translated other functions in algebra. Let's start with the basic sine function, f(t) = sin(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.