Pure: Trigonometric Graphs

Method

1. Learn the key facts of each parent graph: y = sin(x) and y = cos(x) both have amplitude 1, period 360 degrees, and range -1 ≤ y ≤ 1; y = cos(x) is the graph of y = sin(x) translated 90 degrees to the left.Learn the key facts of each parent graph: y = sin(x) and y = cos(x) both have amplitude 1, period 360 degrees, and range -1 <= y <= 1; y = cos(x) is the graph of y = sin(x) translated 90 degrees to the left.
2. y = tan(x) has period 180 degrees, an unbounded range (all real numbers), and a vertical asymptote wherever cos(x) = 0, i.e. at x = 90 + 180n degrees for integer n.
3. For y = a sin(bx) (or the cos or tan equivalent), the amplitude becomes |a| and the period becomes 360/b degrees (or 180/b degrees for tan), since a larger b squeezes the same number of cycles into a smaller x-range.
4. For y = sin(x - c) + d (or the cos or tan equivalent), the graph is translated c to the right (or |c| to the left, if c is negative) and d upward; the vertical translation shifts the whole wave, including its maximum and minimum values, by d, but never changes the amplitude or the period.
5. To combine these into y = a sin(b(x-c)) + d, find the amplitude and period first from a and b, then shift the resulting wave c to the right and d up; locate a maximum, a minimum or an intercept by starting from the equivalent point on the standard sine wave and applying the same working.
6. The reciprocal trig graphs are built from sin, cos and tan: cosec(x) = 1/sin(x), sec(x) = 1/cos(x), and cot(x) = 1/tan(x); each has a vertical asymptote wherever its parent function is zero, and cosec(x) and sec(x) both have range |y| ≥ 1, since one divided by a number no bigger than 1 in size is always at least 1 in size.The reciprocal trig graphs are built from sin, cos and tan: cosec(x) = 1/sin(x), sec(x) = 1/cos(x), and cot(x) = 1/tan(x); each has a vertical asymptote wherever its parent function is zero, and cosec(x) and sec(x) both have range |y| >= 1, since one divided by a number no bigger than 1 in size is always at least 1 in size.
7. A tan-based graph's asymptotes transform like any other vertical feature: for y = tan(b(x-c)), the asymptotes of y = tan(x) at x = 90 + 180n degrees map to x = c + (90+180n)/b degrees; a constant added outside the tan function does not move them, since an asymptote is about the INPUT being undefined, not the output's value.
8. To count how many solutions an equation such as a sin(bx) + d = k has over a given range, compare k with the wave's maximum (d + |a|) and minimum (d - |a|) values first: no solution exists at all if k lies outside this interval, whatever the range of x, and otherwise the number of solutions can be found by picturing the horizontal line y = k crossing the wave.
9. Check whether the parent function is odd (sin, tan, cosec, cot: f(-x) = -f(x)) or even (cos, sec: f(-x) = f(x)) before assuming a reflection in the y-axis changes anything; reflecting an even function in the y-axis leaves its graph completely unchanged.

Practice questions

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