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Algebra Examples
y=sec(x)y=sec(x)
Step 1
Step 1.1
For any y=sec(x)y=sec(x), vertical asymptotes occur at x=π2+nπx=π2+nπ, where nn is an integer. Use the basic period for y=sec(x)y=sec(x), (-π2,3π2)(−π2,3π2), to find the vertical asymptotes for y=sec(x)y=sec(x). Set the inside of the secant function, bx+cbx+c, for y=asec(bx+c)+dy=asec(bx+c)+d equal to -π2−π2 to find where the vertical asymptote occurs for y=sec(x)y=sec(x).
x=-π2x=−π2
Step 1.2
Set the inside of the secant function xx equal to 3π23π2.
x=3π2x=3π2
Step 1.3
The basic period for y=sec(x)y=sec(x) will occur at (-π2,3π2)(−π2,3π2), where -π2−π2 and 3π23π2 are vertical asymptotes.
(-π2,3π2)(−π2,3π2)
Step 1.4
Find the period 2π|b|2π|b| to find where the vertical asymptotes exist. Vertical asymptotes occur every half period.
Step 1.4.1
The absolute value is the distance between a number and zero. The distance between 00 and 11 is 11.
2π12π1
Step 1.4.2
Divide 2π2π by 11.
2π2π
2π2π
Step 1.5
The vertical asymptotes for y=sec(x)y=sec(x) occur at -π2−π2, 3π23π2, and every πnπn, where nn is an integer. This is half of the period.
πnπn
Step 1.6
There are only vertical asymptotes for secant and cosecant functions.
Vertical Asymptotes: x=3π2+πnx=3π2+πn for any integer nn
No Horizontal Asymptotes
No Oblique Asymptotes
Vertical Asymptotes: x=3π2+πnx=3π2+πn for any integer nn
No Horizontal Asymptotes
No Oblique Asymptotes
Step 2
Use the form asec(bx-c)+dasec(bx−c)+d to find the variables used to find the amplitude, period, phase shift, and vertical shift.
a=1a=1
b=1b=1
c=0c=0
d=0d=0
Step 3
Since the graph of the function secsec does not have a maximum or minimum value, there can be no value for the amplitude.
Amplitude: None
Step 4
Step 4.1
The period of the function can be calculated using 2π|b|2π|b|.
2π|b|2π|b|
Step 4.2
Replace bb with 11 in the formula for period.
2π|1|2π|1|
Step 4.3
The absolute value is the distance between a number and zero. The distance between 00 and 11 is 11.
2π12π1
Step 4.4
Divide 2π2π by 11.
2π2π
2π2π
Step 5
Step 5.1
The phase shift of the function can be calculated from cbcb.
Phase Shift: cbcb
Step 5.2
Replace the values of cc and bb in the equation for phase shift.
Phase Shift: 0101
Step 5.3
Divide 00 by 11.
Phase Shift: 00
Phase Shift: 00
Step 6
List the properties of the trigonometric function.
Amplitude: None
Period: 2π2π
Phase Shift: None
Vertical Shift: None
Step 7
The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.
Vertical Asymptotes: x=3π2+πnx=3π2+πn for any integer nn
Amplitude: None
Period: 2π2π
Phase Shift: None
Vertical Shift: None
Step 8
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