Graphing General Sine and Cosine Functions
To graph the general function y A sin Bx C or y A cosBx C , where B 0 , follow
1. Write as y A sin B x or y A cos B x .
2. Find an interval whose length is one period 2 / B by solving the compound inequality
0 Bx C 2
3. Divide the interval into four equal parts.
4. Use the amplitude and any reflection with the basic graph to draw one period.
5. Draw the graph over additional periods, to the right or to the left, as needed.
The amplitude of the function is A . The horizontal translation (phase shift) is units to the
C C C
right if 0 , and units left if 0.
B B B
Graphing the Cosecant and Secant Functions
To graph y A cscBx C or y A secBx C , where B 0 , follow these steps.
1. Graph the corresponding reciprocal function as a guide, using a dashed curve. That is,
To Graph Use as a Guide
y A cscBx C y A sin Bx C
y A secBx C y A cosBx C
2. Sketch the vertical asymptotes. They will have equations of the form x k , where k is an
intercept of the graph of the guide function and the line y d , where d is the horizontal
translation, if any.
3. Sketch the graph of the desired function by drawing the typical U-shaped branches between
the adjacent asymptotes. The branches will be above and below the guide function and
intersect the guide function at its maximums and minimums.
Graphing the Tangent and Cotangent Functions
To graph y A tanBx C or y A cotBx C , where B 0 , follow these steps.
1. Write as y A tan B x or y A cot B x .
2. The period is . To locate two adjacent vertical asymptotes, solve the following equations
For y A tanBx C : Bx C and Bx C
For y A cotBx C : Bx C 0 and Bx C .
3. Divide the interval formed by the vertical asymptotes into four equal parts.
4. Use the any reflection with the basic graph to draw one period.
5. Draw additional periods as needed.