For the function that is described by the provided characteristics, write an equation in terms of x and y. A sine curve with a period of&nbsp;π, an amplitude of 3, a right phase shift of&nbsp;π2, and a vertical translation up 2 units.

The graph of y=asin⁡b(x−d)+c or y=acos⁡b(x−d)+c, has: amplitude ∣a∣, period 2πb, vertical translation c units up if c&gt;0 or ∣c∣ units down if c&lt;0, and phase shift dd units to the right if d&gt;0 or ∣d∣ units to the left if d&lt;0. Use the sine function: y=asin⁡b(x−d)+c The period is ππ so 2πb=π→2. The amplitude is 3 so a=∣3∣=3 The phase shift is π2 right so d=π2. The vertical translation is 2 units so d=2. So, the equation is: y=3sin⁡2(x−π2)+2 or y=3sin⁡(2x−π)+2

Question: "For the function that is described by the..."

College
Algebra
College algebra
Transformations of functions

Daniaal Sanchez

Answered question

2021-07-04

For the function that is described by the provided characteristics, write an equation in terms of x and y. A sine curve with a period of $π$ , an amplitude of 3, a right phase shift of $\frac{π}{2}$ , and a vertical translation up 2 units.

Answer & Explanation

wheezym

Skilled2021-07-05Added 103 answers

The graph of $y = a \sin b (x - d) + c o r y = a \cos b (x - d) + c$ , has: amplitude ∣a∣, period $\frac{2 π}{b}$ , vertical translation c units up if $c > 0$ or $∣ c ∣$ units down if $c < 0$ , and phase shift dd units to the right if $d > 0$ or $∣ d ∣$ units to the left if $d < 0$ .

Use the sine function:

$y = a \sin b (x - d) + c$

The period is $π π$ so $\frac{2 π}{b} = π \to 2$ . The amplitude is 3 so $a =∣ 3 ∣= 3$

The phase shift is $\frac{π}{2}$ right so $d = π 2$ . The vertical translation is 2 units so $d = 2$ . So, the equation is:

$y = 3 \sin 2 (x - \frac{π}{2}) + 2$

$y = 3 \sin (2 x - π) + 2$

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