PreCalc 2 · Trigonometric Functions
Sketch sine or cosine first. Dashed lines go where that graph hits the ground. Then draw U-shapes outside the wave.
Jump to section:
y = sec x
Secant x is cosine’s friend.
1. Graph y = cos x.
2. Draw vertical asymptotes where the graph hits the midline.
3. The “U” of the secant graph touches the tops and bottoms of the cosine graph.
y = csc x
Cosecant x is sine’s friend.
1. Graph y = sin x.
2. Draw vertical asymptotes where the graph hits the midline.
3. The “U” of the cosecant graph touches the tops and bottoms of the sine graph.
Four key parts
Period of y = sec x and y = csc x
Period = 2π
There are four parts of a cycle, so we divide 2π4.
Each part is π2 long.
The four letters
y = A sec(B(x − C)) + D or y = A csc(B(x − C)) + D.
|A| = amplitude (distance from the midline to a vertex of a U-shape).
Period = 2π|B| (how long one cycle is).
C = phase shift (slide left or right). D = vertical shift (move the midline).
Example:
y = 2 sec 4(x − 1) + 7
A amplitude = 2
B = 4, so period =
Period = 2πB
Period = 2π4
Period = π2
(One cycle is π2 long.)
C phase shift = +1
The phase shift is the opposite sign of inside the parenthesis.
D midline = 7
Sketch one cycle of y = sec x. List the amplitude, midline, period, phase shift, and asymptotes.
Secant follows cosine. Sketch cosine first, then U-shapes.
y = A sec(B(x − C)) + D
A − amplitude = 1
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = π2 + k·π
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 0, period 2π, phase shift none, asymptotes x = π2 + k·π
Dotted cosine: sketch this first, then the U-shapes.
Sketch one cycle of y = csc x. List the amplitude, midline, period, phase shift, and asymptotes.
Cosecant follows sine. Sketch the sine snake first, then U-shapes.
y = A csc(B(x − C)) + D
A − amplitude = 1
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = k·π
Sketch the sine snake first. Dashed lines where sine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 0, period 2π, phase shift none, asymptotes x = k·π
Dotted sine: sketch this first, then the U-shapes.
Sketch one cycle of y = sec x. List the amplitude, midline, period, phase shift, and asymptotes.
Secant follows cosine. Sketch cosine first, then U-shapes.
y = A sec(B(x − C)) + D
A − amplitude = 1
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = π2 + k·π
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 0, period 2π, phase shift none, asymptotes x = π2 + k·π
Dotted cosine: sketch this first, then the U-shapes.
Sketch one cycle of y = 2 sec x. List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 2
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = π2 + k·π
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 2, midline y = 0, period 2π, phase shift none, asymptotes x = π2 + k·π
Dotted cosine: sketch this first, then the U-shapes.
Sketch one cycle of y = 3 csc x. List the amplitude, midline, period, phase shift, and asymptotes.
y = A csc(B(x − C)) + D
A − amplitude = 3
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = k·π
Sketch the sine snake first. Dashed lines where sine hits the ground. Then U-shapes.
Answer: amplitude 3, midline y = 0, period 2π, phase shift none, asymptotes x = k·π
Dotted sine: sketch this first, then the U-shapes.
Sketch one cycle of y = −sec x. List the amplitude, midline, period, phase shift, and asymptotes.
The negative flips the U-shapes over the midline.
y = A sec(B(x − C)) + D
A − amplitude = 1
The negative flips the graph.
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = π2 + k·π
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 0, period 2π, phase shift none, asymptotes x = π2 + k·π
Dotted cosine: sketch this first, then the U-shapes.
Sketch one cycle of y = 12 sec x. List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 12
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = π2 + k·π
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 12, midline y = 0, period 2π, phase shift none, asymptotes x = π2 + k·π
Dotted cosine: sketch this first, then the U-shapes.
Sketch one cycle of y = −4 csc x. List the amplitude, midline, period, phase shift, and asymptotes.
The negative flips the U-shapes.
y = A csc(B(x − C)) + D
A − amplitude = 4
The negative flips the graph.
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = k·π
Sketch the sine snake first. Dashed lines where sine hits the ground. Then U-shapes.
Answer: amplitude 4, midline y = 0, period 2π, phase shift none, asymptotes x = k·π
Dotted sine: sketch this first, then the U-shapes.
Sketch one cycle of y = 12 csc x. List the amplitude, midline, period, phase shift, and asymptotes.
y = A csc(B(x − C)) + D
A − amplitude = 12
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = k·π
Sketch the sine snake first. Dashed lines where sine hits the ground. Then U-shapes.
Answer: amplitude 12, midline y = 0, period 2π, phase shift none, asymptotes x = k·π
Dotted sine: sketch this first, then the U-shapes.
Sketch one cycle of y = sec x + 2. List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 1
D − midline = 2
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = π2 + k·π
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 2, period 2π, phase shift none, asymptotes x = π2 + k·π
Dotted cosine: sketch this first, then the U-shapes.
Sketch one cycle of y = csc x + 4. List the amplitude, midline, period, phase shift, and asymptotes.
y = A csc(B(x − C)) + D
A − amplitude = 1
D − midline = 4
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = k·π
Sketch the sine snake first. Dashed lines where sine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 4, period 2π, phase shift none, asymptotes x = k·π
Dotted sine: sketch this first, then the U-shapes.
Sketch one cycle of y = sec x − 3. List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 1
D − midline = −3
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = π2 + k·π
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = −3, period 2π, phase shift none, asymptotes x = π2 + k·π
Dotted cosine: sketch this first, then the U-shapes.
Sketch one cycle of y = csc x − 1. List the amplitude, midline, period, phase shift, and asymptotes.
y = A csc(B(x − C)) + D
A − amplitude = 1
D − midline = −1
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = k·π
Sketch the sine snake first. Dashed lines where sine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = −1, period 2π, phase shift none, asymptotes x = k·π
Dotted sine: sketch this first, then the U-shapes.
Sketch one cycle of y = sec(2x). List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 1
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = π4 + k·π2
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 0, period π, phase shift none, asymptotes x = π4 + k·π2
Dotted cosine: sketch this first, then the U-shapes.
Sketch one cycle of y = csc(2x). List the amplitude, midline, period, phase shift, and asymptotes.
y = A csc(B(x − C)) + D
A − amplitude = 1
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = k·π2
Sketch the sine snake first. Dashed lines where sine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 0, period π, phase shift none, asymptotes x = k·π2
Dotted sine: sketch this first, then the U-shapes.
Sketch one cycle of y = sec(12x). List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 1
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 12
Period = 2π12
Period = 4π
To graph, divide the period by 4
4π4 = π
The graph changes every π.
Asymptotes: x = π + k·2π
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 0, period 4π, phase shift none, asymptotes x = π + k·2π
Dotted cosine: sketch this first, then the U-shapes.
Sketch one cycle of y = csc(4x). List the amplitude, midline, period, phase shift, and asymptotes.
y = A csc(B(x − C)) + D
A − amplitude = 1
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 4
Period = 2π4
Period = π2
To graph, divide the period by 4
π24 = π8
The graph changes every π8.
Asymptotes: x = k·π4
Sketch the sine snake first. Dashed lines where sine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 0, period π2, phase shift none, asymptotes x = k·π4
Dotted sine: sketch this first, then the U-shapes.
The phase shift is the opposite sign of inside the parenthesis. A shift left 3 is −3. When B is not 1, factor B out before you read C.
Sketch one cycle of y = sec(x − π2). List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π2
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = π + k·π
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 0, period 2π, phase shift π2, asymptotes x = π + k·π
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = sec(x + 3). List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −3
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = −1.4292 + k·π
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 0, period 2π, phase shift −3, asymptotes x = −1.4292 + k·π
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = csc(x − π). List the amplitude, midline, period, phase shift, and asymptotes.
y = A csc(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = π + k·π
Sketch the sine snake first. Dashed lines where sine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 0, period 2π, phase shift π, asymptotes x = π + k·π
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = sec(x + π2). List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π2
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = 0 + k·π
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 0, period 2π, phase shift −π2, asymptotes x = 0 + k·π
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = sec(2x − π). List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Inside: 2x − π
Factor out B = 2.
2x − π = 2(x − π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π2
Period = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = 3π4 + k·π2
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 0, period π, phase shift π2, asymptotes x = 3π4 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = csc(2x + π). List the amplitude, midline, period, phase shift, and asymptotes.
y = A csc(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Inside: 2x + π
Factor out B = 2.
2x + π = 2(x + π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π2
Period = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = −π2 + k·π2
Sketch the sine snake first. Dashed lines where sine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 0, period π, phase shift −π2, asymptotes x = −π2 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = sec(2x − π2). List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Inside: 2x − π2
Factor out B = 2.
2x − π2 = 2(x − π4)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π4
Period = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = π2 + k·π2
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 0, period π, phase shift π4, asymptotes x = π2 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = csc(2x + π3). List the amplitude, midline, period, phase shift, and asymptotes.
y = A csc(B(x − C)) + D
A − amplitude = 1
D − midline = 0
Inside: 2x + π3
Factor out B = 2.
2x + π3 = 2(x + π6)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π6
Period = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = −π6 + k·π2
Sketch the sine snake first. Dashed lines where sine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 0, period π, phase shift −π6, asymptotes x = −π6 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Two of the letters have changed.
Sketch one cycle of y = 2 sec x + 1. List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 2
D − midline = 1
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = π2 + k·π
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 2, midline y = 1, period 2π, phase shift none, asymptotes x = π2 + k·π
Dotted cosine: sketch this first, then the U-shapes.
Sketch one cycle of y = 2 csc(2x). List the amplitude, midline, period, phase shift, and asymptotes.
y = A csc(B(x − C)) + D
A − amplitude = 2
D − midline = 0
C − phase shift = none
Period = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = k·π2
Sketch the sine snake first. Dashed lines where sine hits the ground. Then U-shapes.
Answer: amplitude 2, midline y = 0, period π, phase shift none, asymptotes x = k·π2
Dotted sine: sketch this first, then the U-shapes.
Sketch one cycle of y = sec(2x) + 3. List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 1
D − midline = 3
C − phase shift = none
Period = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = π4 + k·π2
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 3, period π, phase shift none, asymptotes x = π4 + k·π2
Dotted cosine: sketch this first, then the U-shapes.
Sketch one cycle of y = 3 sec(x − π2). List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 3
D − midline = 0
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π2
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = π + k·π
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 3, midline y = 0, period 2π, phase shift π2, asymptotes x = π + k·π
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = csc(x + π) − 1. List the amplitude, midline, period, phase shift, and asymptotes.
y = A csc(B(x − C)) + D
A − amplitude = 1
D − midline = −1
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = −π + k·π
Sketch the sine snake first. Dashed lines where sine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = −1, period 2π, phase shift −π, asymptotes x = −π + k·π
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = −sec x + 4. List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 1
The negative flips the graph.
D − midline = 4
C − phase shift = none
Period = 2π|B|
B = 1
Period = 2π1
Period = 2π
To graph, divide the period by 4
2π4 = π2
The graph changes every π2.
Asymptotes: x = π2 + k·π
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 1, midline y = 4, period 2π, phase shift none, asymptotes x = π2 + k·π
Dotted cosine: sketch this first, then the U-shapes.
All four letters have changed. Factor out B before you read the phase shift.
Sketch one cycle of y = 2 sec(2x − π) + 1. List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 2
D − midline = 1
Inside: 2x − π
Factor out B = 2.
2x − π = 2(x − π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π2
Period = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = 3π4 + k·π2
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 2, midline y = 1, period π, phase shift π2, asymptotes x = 3π4 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = 3 csc(2x + π) − 2. List the amplitude, midline, period, phase shift, and asymptotes.
y = A csc(B(x − C)) + D
A − amplitude = 3
D − midline = −2
Inside: 2x + π
Factor out B = 2.
2x + π = 2(x + π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π2
Period = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = −π2 + k·π2
Sketch the sine snake first. Dashed lines where sine hits the ground. Then U-shapes.
Answer: amplitude 3, midline y = −2, period π, phase shift −π2, asymptotes x = −π2 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = −2 sec(2x − π2) + 3. List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 2
The negative flips the graph.
D − midline = 3
Inside: 2x − π2
Factor out B = 2.
2x − π2 = 2(x − π4)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π4
Period = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = π2 + k·π2
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 2, midline y = 3, period π, phase shift π4, asymptotes x = π2 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = 4 csc(2x − π) + 1. List the amplitude, midline, period, phase shift, and asymptotes.
y = A csc(B(x − C)) + D
A − amplitude = 4
D − midline = 1
Inside: 2x − π
Factor out B = 2.
2x − π = 2(x − π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = π2
Period = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = π2 + k·π2
Sketch the sine snake first. Dashed lines where sine hits the ground. Then U-shapes.
Answer: amplitude 4, midline y = 1, period π, phase shift π2, asymptotes x = π2 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = −3 sec(2x + π) − 1. List the amplitude, midline, period, phase shift, and asymptotes.
y = A sec(B(x − C)) + D
A − amplitude = 3
The negative flips the graph.
D − midline = −1
Inside: 2x + π
Factor out B = 2.
2x + π = 2(x + π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π2
Period = 2π|B|
B = 2
Period = 2π2
Period = π
To graph, divide the period by 4
π4 = π4
The graph changes every π4.
Asymptotes: x = −π4 + k·π2
Sketch cosine first. Dashed lines where cosine hits the ground. Then U-shapes.
Answer: amplitude 3, midline y = −1, period π, phase shift −π2, asymptotes x = −π4 + k·π2
Dotted graph: before the phase shift. Solid graph: after the phase shift.
Sketch one cycle of y = 2 csc(4x + 2π) + 5. List the amplitude, midline, period, phase shift, and asymptotes.
y = A csc(B(x − C)) + D
A − amplitude = 2
D − midline = 5
Inside: 4x + 2π
Factor out B = 4.
4x + 2π = 4(x + π2)
Phase shift is the opposite sign of inside the parenthesis.
C − phase shift = −π2
Period = 2π|B|
B = 4
Period = 2π4
Period = π2
To graph, divide the period by 4
π24 = π8
The graph changes every π8.
Asymptotes: x = −π2 + k·π4
Sketch the sine snake first. Dashed lines where sine hits the ground. Then U-shapes.
Answer: amplitude 2, midline y = 5, period π2, phase shift −π2, asymptotes x = −π2 + k·π4
Dotted graph: before the phase shift. Solid graph: after the phase shift.
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