Graphing Trig Function Graphs with a phase shift
A quick Review
𝑓(𝑥) = 𝐴 ∗ 𝑡𝑟𝑖𝑔(𝐵𝑥 − 𝐶) + 𝐷
But there is a better way to write this:
𝐶
𝑓(𝑥) = 𝐴 ∗ 𝑡𝑟𝑖𝑔 𝐵(𝑥 −
+𝐷
𝐵
Trig---any of the six functions
Every function has a series of order pairs. We will use (𝜃. 𝑦) to discuss the transformations.
Amplitude (A) : the measure of the distance of peaks and troughs from the midline (center)---it
is always positive
(𝜃, 𝐴 ∗ 𝑦)
If A is negative then the function will flip across the x-axis.
(𝜃, −𝐴 ∗ 𝑦)
Period: horizontal width of a single cycle (wave)---the distance it travels before it repeats.
2π
Sine
Cosine
Cosecant
Secant
π
Tangent
Cotangent
period =
parent function period
B
The size of the period is affect by the horizontal stretch/shrink (B)
𝜃
,𝑦
𝐵
If B is negative then the function will flip across the y-axis.
−
𝜃
,𝑦
𝐵
Phase Shift
: How far the function has been shifted horizontally
𝜃+
𝐶
.𝑦
𝐵
Vertical Shift (D) is how far the function has been shifted vertically. D moves the midline.
(𝜃. 𝑦 + 𝐷)
The midline is typically y=0
Let’s look at a graph to make sense of it all
Let us look at a few without graphing them
A) 𝑦 = sin(4𝑥 − 𝜋)
Amplitude:
C) 𝑦 = 2 sin
Reflection:
Amplitude:
Period:
Reflection:
Phase Shift:
Period:
Vertical Shift:
Phase Shift:
𝑥
Vertical Shift:
B) 𝑦 = −4 sin(𝑥) + 3
Amplitude:
Reflection:
Period:
Phase Shift:
Vertical Shift:
D) 𝑦 = − sin −𝑥 +
Amplitude:
Reflection:
Period:
Phase Shift:
Vertical Shift:
+1
𝑦 = 2 sin(4𝑥) + 3
𝑦 = −3 cos 2(𝑥 − 3𝜋)
𝑦 = −3 cos(2𝑥 − 3𝜋)
𝑦 = 3 tan 𝑥 −
𝜋
2
𝑦 = −2 csc
2
𝜋
𝑥−
+1
3
2
𝑦 = 2 sec −3𝜃 −
𝜋
+2
2
𝑦 = 𝑐𝑜𝑠
𝜃 𝜋
+
−2
4 4
𝑦 = 2cot
𝜃 5𝜋
+
3
6