Question

Difficulty: EasyTrigonometric Graphs and Simple Equations

What is the maximum value of the trigonometric function $y = 3\sin x + 2$ ?

Answer: 5

The maximum value of the function is 5.

The basic sine function

\sin x

reaches a maximum value of

1

. Substituting

\sin x = 1

into

y = 3\sin x + 2

yields

y = 3(1) + 2 = 5

Identify the maximum value of the sine term

The range of

\sin x

[-1, 1]

, so its maximum value is

1

The sine function oscillates between a minimum of

-1

and a maximum of

1

for all real numbers

x

Calculate the maximum value of the transformed function

y_{\text{max}} = 3(1) + 2 = 5

Multiplying by the positive amplitude coefficient

3

scales the peak to

3

, and adding the vertical shift of

2

raises the peak to

5

Maximum and Minimum Values of Trigonometric Functions

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