A graph that repeats after a fixed interval, which is a period, of the independent variable. Horizontal shift is obtained by determining the change being made to the x value. A phase shift represents the amount a wave has shifted horizontally from the original wave. EXAMPLE Graph one period of s(x) = -coos(xx) The “minus” sign tells me that the graph Is upside down. Since the multiplier out front is an “understood” -1, the amplitude Is unchanged. The argument (the xx Inside the cosine) Is growing three times as fast (because of the 3), so the period Is one-third as long; the period for this graph will be (2/3)TO.
Here Is the regular graph of cosine: I need to flip this upside down, so I’ll swap the +1 and -1 points on the graph: … And then I’ll fill In the rest of the graph: And now I need to change the period. Rather than trying to figure out the points for the graph on the regular axis, I’ll Instead re-number the axis, which Is a lot easier. The regular period Is from O to 211, but this graph’s period goes from O to (210/3. Then the midpoint of the period Is going to be = 11/3, and the zeroes will be midway between the peaks and troughs.
So I’ll erase the x-axis values from the regular graph, and renumber the axis: REAL LIFE SITUATION Light waves are graphs of trig functions. So are springs that are bouncing and waves In sleekly. Basically any sort of wave (Including sound waves and earthquakes) can be described by trig functions (although some are much more complicated than others). SERIES Arithmetic Sequence, Arithmetic Series Arithmetic sequence is a sequence of numbers that has a constant difference between every two consecutive terms.
If a sequence of values follows a pattern of adding a fixed amount from one term to the next, it is referred to as an arithmetic sequence. Examples Arithmetic Sequence Common Difference, d d add 3 to each term to arrive at the next term, or… The difference ah 13,16,. =3 -5, -10, d= -5 add -5 to each term to add-1/2 to each term to arrive at the next term, or…. The difference ah – al is -1/2. Arithmetic sequences and series are used every day by engineers, accountants, and builders.