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. For instance, when you're graphing quadratics, you may be asked for the parabola's axis of symmetry. This is usually just the vertical line x = h, where "h" is the x-coordinate of the vertex, (h, k). That is, a parabola's axis of symmetry is usually just the vertical line through its vertex. The other customary context for symmetry is judging from a graph whether a function is even or odd.
Warning: By definition, no function can be symmetric about the x-axis (or any other horizontal line), since anything that is mirrored around an horizontal line will violate the Vertical Line Test. For each of the following graphs, list any symmetries, and state whether the graph shows a function. Graph G: This parabola is lying on its side. Algebra - Symmetry. Example 1 Determine the symmetry of each of the following equations. (a) [Solution] (b) (c) (d) (e) Solution (a) We’ll first check for symmetry about the x-axis. Now, this is not an equivalent equation since the terms on the right are identical to the original equation and the term on the left is the opposite sign.
Next, let’s check symmetry about the y-axis. After simplifying we got exactly the same equation back out which means that the two are equivalent. Finally, we need to check for symmetry about the origin. So, as with the first test, the left side is different from the original equation and the right side is identical to the original equation. [Return to Problems] (b) We’ll not put in quite as much detail here. We don’t have symmetry here since the one side is identical to the original equation and the other isn’t. Next, check for symmetry about the y-axis. Remember that if we take a negative to an odd power the minus sign can come out in front.
(c) Now, check for symmetry about the y-axis. 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. Graphing Tangent Function. 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 Simple Trigonometric Equations. Solving General Trigonometric Equations Objective: Given a trigonometric equation of one of the following forms: A * sin (ax+b) = k A * cos (ax+b) = k A * tan (ax+b) = k A * cot (ax+b) = k A * sec (ax+b) = k A * csc (ax+b) = k where k is some constant, A, a and b are real numbers, the learner will: state the number of solutions, and determine those values of x that satisfy the equation, over all the real numbers. Practice Discussion After mastering the solution to the simpler trigonometric equation [ex: sin(x) = k] over the interval [0, 2π) the next step is to learn how to solve more complicated expressions. The easiest way to solve these problems is as follows: (the example function we shall employ is 4 sin (2x - 3) = 3) Write down the expression: 4 sin (2x - 3) = 3. Using trigonometric inverses: Set u = 2x - 3.
Examples In the problems below, and in the program, it is assumed that A has been eliminated by division. Helps to Solving Without The Calculator Instructions Using The Program Top. Lesson HOW TO - Solve Trigonometric equations. Introduction The solution of trigonometric equations is one topic that students have particular problems with. There are a few reasons for this: 1. there is usually a simplify part first that requires use of some TRIG identities. 2. there is the use of RADIANS rather than degrees, for which some students are not at ease with. 3. there is the repetitive aspect of TRIG functions that students find bewildering.
All in all, a potentially daunting topic. To be honest though, there is nothing that you need fear here, so long as you take it methodically and slowly. Also, you have to understand what it is you are doing. Solving the TRIG Equation Of the 3 topics listed above, I am concentrating on part 3, here in this Lesson. First thing, when solving a TRIG equation, is to understand or accept that each of SINE, COSINE and TANGENT have 2 angles that will satisfy the given equation within any 360 degree range. Basics This is difficult to explain, with no real picture to use, so bear with me. Done! SOLVING TRIGONOMETRIC EQUATIONS. Note: If you would like a review of trigonometry, click on trigonometry.
Example 1: Solve for x in the following equation. There are an infinite number of solutions to this problem. First isolate the cosine term. To solve for x, we have to isolate x. How do we isolate the x? We could take the inverse (arccosine) of both sides. However, inverse functions can only be applied to one-to-one functions and the cosine function is not one-to-one. Let's restrict the domain so the function is one-to-one on the restricted domain while preserving the original range. If we restrict the domain of the cosine function to that interval , we can take the arccosine of both sides of each equation. The angle x is the reference angle. Therefore, if , then The period of equals and the period of , this means other solutions exists every units.
Where n is an integer. The approximate values of these solutions are You can check each solution algebraically by substituting each solution in the original equation. Left Side: Math.