Find Amplitude, Period, and Phase Shift f(t)=25sin(160pit)+115

$f (t) = 25 \sin (160 π t) + 115$

Use the form $a \sin (b t - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = 25$

$b = 160 π$

$c = 0$

$d = 115$

Find the amplitude $| a |$ .

Amplitude: $25$

Find the period using the formula $\frac{2 π}{| b |}$ .

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The period of the function can be calculated using $\frac{2 π}{| b |}$ .

Period: $\frac{2 π}{| b |}$

Replace $b$ with $160 π$ in the formula for period.

Period: $\frac{2 π}{| 160 π |}$

Solve the equation.

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$160 π$ is approximately $502.65482457$ which is positive so remove the absolute value

Period: $\frac{2 π}{160 π}$

Cancel the common factor of $2$ and $160$ .

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Factor $2$ out of $2 π$ .

Period: $\frac{2 (π)}{160 π}$

Cancel the common factors.

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Factor $2$ out of $160 π$ .

Period: $\frac{2 (π)}{2 (80 π)}$

Cancel the common factor.

Period: $\frac{2 π}{2 (80 π)}$

Rewrite the expression.

Period: $\frac{π}{80 π}$

Cancel the common factor of $π$ .

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Cancel the common factor.

Period: $\frac{π}{80 π}$

Rewrite the expression.

Period: $\frac{1}{80}$

Find the phase shift using the formula $\frac{c}{b}$ .

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The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{0}{160 π}$

Cancel the common factor of $0$ and $160$ .

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Factor $160$ out of $0$ .

Phase Shift: $\frac{160 (0)}{160 π}$

Cancel the common factors.

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Factor $160$ out of $160 π$ .

Phase Shift: $\frac{160 (0)}{160 (π)}$

Cancel the common factor.

Phase Shift: $\frac{160 \cdot 0}{160 π}$

Rewrite the expression.

Phase Shift: $\frac{0}{π}$

Divide $0$ by $π$ .

Phase Shift: $0$

Find the vertical shift $d$ .

Vertical Shift: $115$

List the properties of the trigonometric function.

Amplitude: $25$

Period: $\frac{1}{80}$

Phase Shift: $0$ ( $0$ to the right)

Vertical Shift: $115$

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Name

Name	one billion two hundred million three hundred thirty-three thousand forty

Interesting facts

1200333040 has 64 divisors, whose sum is 7292023704
The reverse of 1200333040 is 0403330021
Previous prime number is 5

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 16
Digital Root 7

Name

Name	one billion two hundred seventy-seven million one hundred thirty-nine thousand nine hundred fifty-five

Interesting facts

1277139955 has 32 divisors, whose sum is 1591783424
The reverse of 1277139955 is 5599317721
Previous prime number is 223

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 49
Digital Root 4

Name

Name	seven million seven hundred forty-two thousand six hundred twenty-four

Interesting facts

7742624 has 128 divisors, whose sum is 59276448
The reverse of 7742624 is 4262477
Previous prime number is 131

Basic properties

Is Prime? no
Number parity even
Number length 7
Sum of Digits 32
Digital Root 5

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