Math / Trigonometry

QuestionWhich function has the greatest maximum value? (a) $y=2 \sin 3\left(x+90^{\circ}\right)+5$ b) $y=3 \sin 2\left(x+90^{\circ}\right)-3$ c) $y=\frac{1}{3} \sin 3\left(x+90^{\circ}\right)-1$ d) $y=\sin 0.5\left(x-90^{\circ}\right)$

Studdy Solution

STEP 1

What is this asking? Which of these wavy sine functions reaches the highest point? Watch out! Don't mix up the amplitude and the vertical shift!

STEP 2

1. Understand the Sine Function
2. Find the Maximum of Each Function

STEP 3

Alright, so we've got these sine functions, and they all look like $y = A \sin(B(x + C)) + D$ .
Remember, $A$ is the amplitude, which tells us how tall the wave is. $B$ affects the period (how long it takes for the wave to repeat), but we don't need to worry about that here! $C$ shifts the graph horizontally, but that doesn't change the maximum height! $D$ is the vertical shift, moving the whole graph up or down.

STEP 4

The sine function, $\sin(\text{something})$ , bounces between -1 and 1.
Its maximum value is 1.

STEP 5

So, the maximum value of $A \sin(\text{something}) + D$ is $A \cdot 1 + D = A + D$ .
This is because the sine part reaches a maximum of 1, gets multiplied by $A$ , and then we add $D$ .

STEP 6

For function (a), $y = 2 \sin 3(x + 90^{\circ}) + 5$ , we have $A = \mathbf{2}$ and $D = \mathbf{5}$ .
So, the maximum value is $2 + 5 = \mathbf{7}$ .

STEP 7

For function (b), $y = 3 \sin 2(x + 90^{\circ}) - 3$ , we have $A = \mathbf{3}$ and $D = \mathbf{-3}$ .
The maximum value is $3 + (-3) = \mathbf{0}$ .

STEP 8

For function (c), $y = \frac{1}{3} \sin 3(x + 90^{\circ}) - 1$ , we have $A = \mathbf{\frac{1}{3}}$ and $D = \mathbf{-1}$ .
The maximum value is $\frac{1}{3} + (-1) = \frac{1}{3} - \frac{3}{3} = \mathbf{-\frac{2}{3}}$ .

STEP 9

For function (d), $y = \sin 0.5(x - 90^{\circ})$ , we have $A = \mathbf{1}$ (since there's no number in front of the sine, it's like having a 1) and $D = \mathbf{0}$ (since there's no number added at the end).
The maximum value is $1 + 0 = \mathbf{1}$ .

STEP 10

Function (a) has the greatest maximum value, which is 7.

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QuestionWhich function has the greatest maximum value? (a) $y=2 \sin 3\left(x+90^{\circ}\right)+5$ b) $y=3 \sin 2\left(x+90^{\circ}\right)-3$ c) $y=\frac{1}{3} \sin 3\left(x+90^{\circ}\right)-1$ d) $y=\sin 0.5\left(x-90^{\circ}\right)$

Studdy Solution

STEP 1

What is this asking? Which of these wavy sine functions reaches the highest point? Watch out! Don't mix up the amplitude and the vertical shift!

STEP 2

1. Understand the Sine Function
2. Find the Maximum of Each Function

STEP 3

STEP 4

The sine function, $\sin(\text{something})$ , bounces between -1 and 1.
Its maximum value is 1.

STEP 5

So, the maximum value of $A \sin(\text{something}) + D$ is $A \cdot 1 + D = A + D$ .
This is because the sine part reaches a maximum of 1, gets multiplied by $A$ , and then we add $D$ .

STEP 6

For function (a), $y = 2 \sin 3(x + 90^{\circ}) + 5$ , we have $A = \mathbf{2}$ and $D = \mathbf{5}$ .
So, the maximum value is $2 + 5 = \mathbf{7}$ .

STEP 7

For function (b), $y = 3 \sin 2(x + 90^{\circ}) - 3$ , we have $A = \mathbf{3}$ and $D = \mathbf{-3}$ .
The maximum value is $3 + (-3) = \mathbf{0}$ .

STEP 8

For function (c), $y = \frac{1}{3} \sin 3(x + 90^{\circ}) - 1$ , we have $A = \mathbf{\frac{1}{3}}$ and $D = \mathbf{-1}$ .
The maximum value is $\frac{1}{3} + (-1) = \frac{1}{3} - \frac{3}{3} = \mathbf{-\frac{2}{3}}$ .

STEP 9

For function (d), $y = \sin 0.5(x - 90^{\circ})$ , we have $A = \mathbf{1}$ (since there's no number in front of the sine, it's like having a 1) and $D = \mathbf{0}$ (since there's no number added at the end).
The maximum value is $1 + 0 = \mathbf{1}$ .

STEP 10

Function (a) has the greatest maximum value, which is 7.

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Studdy Solution

STEP 1

What is this asking? Which of these wavy sine functions reaches the highest point? Watch out! Don't mix up the *amplitude* and the *vertical shift*!

STEP 2

1. Understand the Sine Function2. Find the Maximum of Each Function

STEP 3

STEP 4

The sine function, sin⁡(something)\sin(\text{something})sin(something), bounces between -1 and 1.Its maximum value is **1**.

STEP 5

So, the maximum value of Asin⁡(something)+DA \sin(\text{something}) + DAsin(something)+D is A⋅1+D=A+DA \cdot 1 + D = A + DA⋅1+D=A+D.This is because the sine part reaches a maximum of 1, gets multiplied by AAA, and then we add DDD.

STEP 6

For function (a), y=2sin⁡3(x+90∘)+5y = 2 \sin 3(x + 90^{\circ}) + 5y=2sin3(x+90∘)+5, we have A=2A = \mathbf{2}A=2 and D=5D = \mathbf{5}D=5.So, the maximum value is 2+5=72 + 5 = \mathbf{7}2+5=7.

STEP 7

For function (b), y=3sin⁡2(x+90∘)−3y = 3 \sin 2(x + 90^{\circ}) - 3y=3sin2(x+90∘)−3, we have A=3A = \mathbf{3}A=3 and D=−3D = \mathbf{-3}D=−3.The maximum value is 3+(−3)=03 + (-3) = \mathbf{0}3+(−3)=0.

STEP 8

STEP 9

STEP 10

Function (a) has the greatest maximum value, which is **7**.

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What is this asking? Which of these wavy sine functions reaches the highest point? Watch out! Don't mix up the amplitude and the vertical shift!

1. Understand the Sine Function
2. Find the Maximum of Each Function

The sine function, $\sin(\text{something})$ , bounces between -1 and 1.
Its maximum value is 1.

So, the maximum value of $A \sin(\text{something}) + D$ is $A \cdot 1 + D = A + D$ .
This is because the sine part reaches a maximum of 1, gets multiplied by $A$ , and then we add $D$ .

For function (a), $y = 2 \sin 3(x + 90^{\circ}) + 5$ , we have $A = \mathbf{2}$ and $D = \mathbf{5}$ .
So, the maximum value is $2 + 5 = \mathbf{7}$ .

For function (b), $y = 3 \sin 2(x + 90^{\circ}) - 3$ , we have $A = \mathbf{3}$ and $D = \mathbf{-3}$ .
The maximum value is $3 + (-3) = \mathbf{0}$ .

Function (a) has the greatest maximum value, which is 7.