Find Amplitude, Period, and Phase Shift y=10sin((6pi)/4(x-pi/2))+25

$y = 10 \sin (\frac{6 π}{4} (x - \frac{π}{2})) + 25$

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

$a = 10$

$b = \frac{3 π}{2}$

$c = \frac{3 π^{2}}{4}$

$d = 25$

Find the amplitude $| a |$ .

Amplitude: $10$

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

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Find the period of $10 \sin (\frac{3 π x}{2} - \frac{3 π^{2}}{4})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{3 π}{2}$ in the formula for period.

$\frac{2 π}{| \frac{3 π}{2} |}$

$\frac{3 π}{2}$ is approximately $4.71238898$ which is positive so remove the absolute value

$\frac{2 π}{\frac{3 π}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{3 π}$

Cancel the common factor of $π$ .

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Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{3 π}$

Factor $π$ out of $3 π$ .

$π \cdot 2 \frac{2}{π \cdot 3}$

Cancel the common factor.

$π \cdot 2 \frac{2}{π \cdot 3}$

Rewrite the expression.

$2 (\frac{2}{3})$

Combine $2$ and $\frac{2}{3}$ .

$\frac{2 \cdot 2}{3}$

Multiply $2$ by $2$ .

$\frac{4}{3}$

Find the period of $25$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{3 π}{2}$ in the formula for period.

$\frac{2 π}{| \frac{3 π}{2} |}$

$\frac{3 π}{2}$ is approximately $4.71238898$ which is positive so remove the absolute value

$\frac{2 π}{\frac{3 π}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{3 π}$

Cancel the common factor of $π$ .

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Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{3 π}$

Factor $π$ out of $3 π$ .

$π \cdot 2 \frac{2}{π \cdot 3}$

Cancel the common factor.

$π \cdot 2 \frac{2}{π \cdot 3}$

Rewrite the expression.

$2 (\frac{2}{3})$

Combine $2$ and $\frac{2}{3}$ .

$\frac{2 \cdot 2}{3}$

Multiply $2$ by $2$ .

$\frac{4}{3}$

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$\frac{4}{3}$

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: $\frac{\frac{3 π^{2}}{4}}{\frac{3 π}{2}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{3 π^{2}}{4} \cdot \frac{2}{3 π}$

Combine.

Phase Shift: $\frac{3 π^{2} \cdot 2}{4 (3 π)}$

Cancel the common factor of $3$ .

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Cancel the common factor.

Phase Shift: $\frac{3 π^{2} \cdot 2}{4 (3 π)}$

Rewrite the expression.

Phase Shift: $\frac{π^{2} \cdot 2}{4 (π)}$

Cancel the common factor of $π^{2}$ and $π$ .

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Factor $π$ out of $π^{2} \cdot 2$ .

Phase Shift: $\frac{π (π \cdot 2)}{4 (π)}$

Cancel the common factors.

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Factor $π$ out of $4 (π)$ .

Phase Shift: $\frac{π (π \cdot 2)}{π \cdot 4}$

Cancel the common factor.

Phase Shift: $\frac{π (π \cdot 2)}{π \cdot 4}$

Rewrite the expression.

Phase Shift: $\frac{π \cdot 2}{4}$

Cancel the common factor of $2$ and $4$ .

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Factor $2$ out of $π \cdot 2$ .

Phase Shift: $\frac{2 \cdot π}{4}$

Cancel the common factors.

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Factor $2$ out of $4$ .

Phase Shift: $\frac{2 \cdot π}{2 \cdot 2}$

Cancel the common factor.

Phase Shift: $\frac{2 \cdot π}{2 \cdot 2}$

Rewrite the expression.

Phase Shift: $\frac{π}{2}$

List the properties of the trigonometric function.

Amplitude: $10$

Period: $\frac{4}{3}$

Phase Shift: $\frac{π}{2}$ ( $\frac{π}{2}$ to the right)

Vertical Shift: $25$

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Name

Name	one hundred fifty-four million three hundred nine thousand ten

Interesting facts

154309010 has 16 divisors, whose sum is 286716672
The reverse of 154309010 is 010903451
Previous prime number is 31

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 23
Digital Root 5

Name

Name	one billion one hundred seventy-four million five hundred fifteen thousand four hundred eighty

Interesting facts

1174515480 has 128 divisors, whose sum is 5652840960
The reverse of 1174515480 is 0845154711
Previous prime number is 389

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 36
Digital Root 9

Name

Name	one billion one hundred sixty-three million one hundred sixty-one thousand nine hundred fifty-three

Interesting facts

1163161953 has 8 divisors, whose sum is 2067843488
The reverse of 1163161953 is 3591613611
Previous prime number is 3

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 36
Digital Root 9

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