Concept explainers
For Exercises 1—16, identify which functions shown here (f, g, h, and so on) have the given characteristics.
f(x)=−3sec(2x+π) g(x)=−3cos(12x−π4) h(x)=3sin(−12x−π4)k(x)=sin(π2x)+3 m(x)=2csc(2x−π4)−3 n(x)=3tan(x−π2)p(x)=−2cot(12x+π) t(x)=−3+2cosx
1. Has an amplitude of 3
Toidentify : the functions that have the given characteristics
Answer to Problem 1PRE
The functions that have amplitude of 3 are: g(x)=−3cos(12x−π4) , & h(x)=3sin(−12x−π4)
Explanation of Solution
Given information:Listed functions in below
f(x)=sin(π2x)+3 | g(x)=−3cos(12x−π4) | h(x)=3sin(−12x−π4) |
k(x)=−3sec(2x+π) | m(x)=2csc(2x−π2)−3 | n(x)=3tan(x−π2) |
p(x)=−2cot(12x+π) | t(x)=−3+2cosx |
Concept Involved:
Properties of the General Sine and Cosine Functions:
Consider the graphs of y=Asin(Bx−C)+D and y=Acos(Bx−C)+D with B>0 .
- The amplitude is |A|.
- The period is 2πB .
- The phase shift is CB .
- The vertical shift is D.
- One full cycle is given on the interval 0≤Bx−C≤2π .
- The domain is −∞<x<∞ .
- The range is −|A|+D≤y≤|A|+D .
Properties of the tangent and cotangent functions y=Atan(Bx−C)+D and y=Acot(Bx−C)+D with B>0 (If B<0 , we can rewrite the equations using the odd function property of the tangent and cotangent functions.)
- The value |A| is the vertical scaling factor.
- If A<0, the graph is reflected across the x-axis
- The period is πB
- The phase shift is CB
- The vertical shift is D
Calculation:
Function | Period | Phase Shift | Vertical shift | Amplitude |
f(x)=sin(π2x)+3 | 2ππ2=4ππ=4 | 0 | 3 | 1 |
g(x)=−3cos(12x−π4) | 2π12=4π1=4π | π4÷12=2π4=π2 | 0 | 3 |
h(x)=3sin(−12x−π4) | 2π12=4π1=4π | −π4÷12=−2π4=−π2 | 0 | 3 |
k(x)=−3sec(2x+π) | 2π2=π | −π2 | 0 | None |
m(x)=2csc(2x−π2)−3 | 2π2=π | π2÷21=π2⋅12=π4 | -3 | None |
n(x)=3tan(x−π2) | π1=π | π2÷1=π2 | 0 | None |
p(x)=−2cot(12x+π) | π12=π1⋅21=2π | −π÷12=−π⋅21=−2π | 0 | None |
t(x)=−3+2cosx | 2π1=2π | 0 | -3 | 2 |
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Chapter 5 Solutions
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