PreCalc 2 · Trigonometric Functions
Secant has a vertex on the y-axis. Cosecant has an asymptote on the y-axis. Then read stretch, period, and shifts.
Which graph is it?
Vertex on the y-axis at (0, ±1) or (±A) → secant (from cosine / clouds).
Asymptote on the y-axis → cosecant (from the sine snake).
Period and B
Distance from one pair of matching U-shapes to the next is the period. B = 2πperiod.
Stretch A
|A| is the distance from the midline to a vertex (the bottom of an upper U or the top of a lower U).
Shifts
D moves the whole picture up or down. C slides it left or right, just like sine and cosine.
This is a parent graph. Is it secant or cosecant? Write the equation.
Vertex at (0, 1), asymptotes at π2.
That is parent secant.
y = sec x
Answer: y = sec x
This is a parent graph. Is it secant or cosecant? Write the equation.
Asymptote on the y-axis.
That is parent cosecant.
y = csc x
Answer: y = csc x
The U-shape on the y-axis opens down from (0, −1). Write the equation.
Flipped secant.
y = −sec x
Answer: y = −sec x
Parent period, vertex at (0, 2). Write a secant equation.
A = 2.
y = 2 sec x
Answer: y = 2 sec x
Parent csc, flipped. Write the equation.
y = −csc x
Answer: y = −csc x
Secant, parent period, vertex at (0, 3). Write the equation.
y = 3 sec x
Answer: y = 3 sec x
Secant, period π, vertex at (0, 1). Write the equation.
Period π, so B = 2.
y = sec(2x)
Answer: y = sec(2x)
Cosecant, period π, parent orientation. Write the equation.
y = csc(2x)
Answer: y = csc(2x)
Secant, parent period, shifted up 2. Write the equation.
y = sec x + 2
Answer: y = sec x + 2
Cosecant, stretch 2, parent period. Write the equation.
y = 2 csc x
Answer: y = 2 csc x
Secant, period 4π, vertex at (0, 1). Write the equation.
B = 2π4π = 12.
y = sec(12x)
Answer: y = sec(12x)
Flipped secant, period 2π, vertex at (0, −2). Write the equation.
y = −2 sec x
Answer: y = −2 sec x
Secant, period π, up 1, vertex on the y-axis at (0, 3). Write the equation.
D = 1.
Vertex 3 is 2 above the midline, so |A| = 2.
B = 2.
Starts like +sec (vertex above the midline on the y-axis).
y = 2 sec(2x) + 1
Answer: y = 2 sec(2x) + 1
Cosecant, period 2π, shifted right π2. Write the equation.
y = csc(x − π2)
Answer: y = csc(x − π2)
Secant, period π, flipped, no vertical shift. Write the equation.
y = −sec(2x)
Answer: y = −sec(2x)
Cosecant, stretch 3, down 1, parent period. Write the equation.
y = 3 csc x − 1
Answer: y = 3 csc x − 1
Secant, period 2π, left π4. Write the equation.
y = sec(x + π4)
Answer: y = sec(x + π4)
Cosecant, period π, stretch 2, up 1. Write the equation.
y = 2 csc(2x) + 1
Answer: y = 2 csc(2x) + 1
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