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Graphing Trigonometric Functions 1. What facts do we need to know? a. KNOW YOUR FIRST QUADRANT UNIT CIRCLE VALUES FOR COSINE AND SINE!!! b. Remember that the basic period of the tangent and cotangent functions is π while the other four functions have 2π as their basic period. c. Only cosine and sine have all real numbers as their domain. The other four functions have vertical asymptotes in their graphs. d. Remember – vertical asymptotes occur when there are non-removable domain problems in a function. Whenever cosine is 0 there will be a vertical asymptote in the graph of tangent and secant. Whenever sine is 0 there will be a vertical asymptote in the graph of cotangent or cosecant. e. For functions of the form y = Asin(Bx + C) + D , the A changes amplitude and the D is a vertical shift. The B changes period and the C is a horizontal shift. Remember from our algebraic graphs that the effects of vertical transformations are exactly what they look like while the effects of horizontal transformations are the opposite of how they appear. i. For example look at the following two graphs on the same axis. The dotted graph is a plain old y = sin x while the solid graph is y = sin 3x + 2 . Notice that the +2 moves the graph up while the multiplication of 3 shortens the length of one cycle of the graph rather than increasing its length Graphing Trigonometric Functions ii. So, how would you generate this graph by hand? Recognize that 2π the basic period is and set up the following table of values: 3 x sin(3x) + 2 0 π 6 π 3 π 2 2π 3 iii. In this table, the x-values are generated by dividing the period into four equal sized pieces. Calculate the y values by yourself and see if they match the graph f. The process is the same for the other five trig functions when you are trying to graph them. Know your period, divide it into four pieces, know your unit circle values, and then plot your points for one basic cycle of the graph. g. Try this with the following examples which we will discuss in class tomorrow: ⎛1 ⎞ i. y = 2 cos ⎜ x − π ⎟ + 1 ⎝2 ⎠ π⎞ ⎛ ii. y = tan ⎜ x + ⎟ + 1 ⎝ 2⎠ iii. y = sec x − 3 iv. y = 2 cot ( 2x )