1. Graphing Sine and Cosine
Review
Graphing Sine and Cosine
Graph the following functions. State the amplitude, period, phase shift, vertical shift and reflection.
𝑦=2𝑐𝑜𝑠 𝑥− 3𝜋 2 −1
𝑦=−3𝑠𝑖𝑛 𝑥+ 5𝜋 6 +4

2. Finding an Equation for a Sine/Cosine Graph
Steps:
Find the midline (horizontal middle) of the graph. This is your “d” value.
Calculate the distance from the midline to the top (or bottom) of your graph. This is your “a” value.
Highlight one complete wave of your graph. Calculate the distance of the one wave. This is your period. Use the period formula, 2𝜋 𝑏 and the distance calculated to find your “b” value.
Identify the starting point of your highlight wave. This is your phase shift. Use your phase shift formula, 𝑐 𝑏 , and the “b” value found in step 3 to find your “c” value.
Place you’re a, b, c, and d values into the general form to obtain the final equation: f x =a sin (𝑏𝑥−𝑐) +𝑑
**Answer may vary

3. Review
Extension
Write the Equation that corresponds to each graph.

4. Graphing Other Trigonometric Functions
Keeper 14
Accelerated Pre-Calculus

5. Properties Tangent
Properties of the Tangent Function

6. Period of Tangent
For 𝑦=atan 𝑏𝑥+𝑐 , 𝑤ℎ𝑒𝑟𝑒 𝑏≠0, 𝑝𝑒𝑟𝑖𝑜𝑑= 𝜋 𝑏 .

7. Asymptotes of Tangent
To find the vertical asymptotes of any tangent function of the form 𝑦= atan 𝑏𝑥+𝑐 +𝑑, solve: 𝑏𝑥+𝑐=− 𝜋 2 𝑎𝑛𝑑 𝑏𝑥+𝑐= 𝜋 2

8. Examples Locate the asymptote and graph the function 𝑦=−2 tan 𝑥 2 +3

9. Graphing Secant and Cosecant
Keeper 14
Accelerated Pre-Calculus

10. Steps for Graphing Secant and Cosecant
Graph the corresponding cosine (for secant) or sine (for cosecant) wave.
Identify the midline of the graph.
Draw asymptotes through the points that cross the midline (3 asymptotes total).
Draw U’s between the asymptotes (at least 2 U’s, one up and one down).

11. Properties of the Cotangent Function
For 𝑦=a𝑐𝑜𝑡 𝑏𝑥+𝑐 , 𝑤ℎ𝑒𝑟𝑒 𝑏≠0, 𝑝𝑒𝑟𝑖𝑜𝑑= 𝜋 𝑏 .
To find the vertical asymptotes of any tangent function of the form 𝑦= a𝑐𝑜𝑡 𝑏𝑥+𝑐 +𝑑, solve:
𝑏𝑥+𝑐=−𝜋 𝑎𝑛𝑑 𝑏𝑥+𝑐=𝜋

12. Examples Locate the asymptote and graph the function 𝑦= cot 𝑥 3
𝑦=3𝑐𝑜𝑡 𝑥 2

13. Properties of Cosecant and Secant Functions

14. Examples Sketch Graphs of Cosecant and Secant Functions 𝑦=csc⁡ 𝑥+ 𝜋 2