Math / Trigonometry

QuestionThe cosine graph shown has a range $[-3,9]$ . The graph has an equation in the form $y=a \cos [b(x-c)]+d, a>0$ . Determine the equation if the graph has a minimum possible phase shift. max $=9$ min $=-5$ $\begin{array}{l} y=a \cos [b(x-c)]+d \\ a=\frac{\max -\min }{2} \quad d=\frac{m_{a x}+\min }{2} \\ a=\frac{4-(-3)}{2} \quad d=\frac{9+(-3)}{2} \\ a=6 \\ a=3 \end{array}$ $y=6 \cos [b(x-c)]+3$

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a cosine graph that goes between $-3$ and $9$ , with the smallest possible shift to the right. Watch out! Remember that cosine normally starts at its maximum.
If it starts at a minimum, that's a shift!

STEP 2

1. Find the amplitude
2. Find the vertical shift
3. Find the period
4. Find the phase shift

STEP 3

The amplitude is half the distance between the highest and lowest points.
It's like finding the radius of a Ferris wheel!

STEP 4

$a = \frac{\textbf{max} - \textbf{min}}{2} = \frac{9 - (-3)}{2} = \frac{12}{2} = 6$ So our amplitude is $\textbf{6}$ !

STEP 5

The vertical shift is the middle value between the highest and lowest points.
It's where the center of the Ferris wheel sits.

STEP 6

$d = \frac{\textbf{max} + \textbf{min}}{2} = \frac{9 + (-3)}{2} = \frac{6}{2} = 3$ Our vertical shift is $\textbf{3}$ !

STEP 7

The graph completes a full cycle between $x = 0$ and $x = \pi$ .
This means the period is $\pi$ .

STEP 8

Since the standard period of cosine is $2\pi$ , we can find $b$ using the equation $2\pi/b = \textbf{period}$ .

STEP 9

$\frac{2\pi}{b} = \pi$ $b = \frac{2\pi}{\pi} = 2$ So, $b = \textbf{2}$ .

STEP 10

A regular cosine graph starts at its maximum.
This graph starts at its minimum.
That's a shift of half a period to the right!

STEP 11

Since our period is $\pi$ , half a period is $\pi/2$ .

STEP 12

So our phase shift $c$ is $\pi/2$ .
We're shifting to the right, so it's positive.

STEP 13

Putting it all together, our equation is: $y = 6 \cos[2(x - \pi/2)] + 3$

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Studdy Solution

STEP 1

What is this asking? We need to find the equation of a cosine graph that goes between $-3$ and $9$ , with the smallest possible shift to the right. Watch out! Remember that cosine normally starts at its maximum.
If it starts at a minimum, that's a shift!

STEP 2

1. Find the amplitude
2. Find the vertical shift
3. Find the period
4. Find the phase shift

STEP 3

The amplitude is half the distance between the highest and lowest points.
It's like finding the radius of a Ferris wheel!

STEP 4

$a = \frac{\textbf{max} - \textbf{min}}{2} = \frac{9 - (-3)}{2} = \frac{12}{2} = 6$ So our amplitude is $\textbf{6}$ !

STEP 5

The vertical shift is the middle value between the highest and lowest points.
It's where the center of the Ferris wheel sits.

STEP 6

$d = \frac{\textbf{max} + \textbf{min}}{2} = \frac{9 + (-3)}{2} = \frac{6}{2} = 3$ Our vertical shift is $\textbf{3}$ !

STEP 7

The graph completes a full cycle between $x = 0$ and $x = \pi$ .
This means the period is $\pi$ .

STEP 8

Since the standard period of cosine is $2\pi$ , we can find $b$ using the equation $2\pi/b = \textbf{period}$ .

STEP 9

$\frac{2\pi}{b} = \pi$ $b = \frac{2\pi}{\pi} = 2$ So, $b = \textbf{2}$ .

STEP 10

A regular cosine graph starts at its maximum.
This graph starts at its minimum.
That's a shift of half a period to the right!

STEP 11

Since our period is $\pi$ , half a period is $\pi/2$ .

STEP 12

So our phase shift $c$ is $\pi/2$ .
We're shifting to the right, so it's positive.

STEP 13

Putting it all together, our equation is: $y = 6 \cos[2(x - \pi/2)] + 3$

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Studdy Solution

STEP 1

What is this asking? We need to find the equation of a cosine graph that goes between −3-3−3 and 999, with the smallest possible shift to the right. Watch out! Remember that cosine normally starts at its maximum.If it starts at a minimum, that's a shift!

STEP 2

1. Find the amplitude2. Find the vertical shift3. Find the period4. Find the phase shift

STEP 3

The **amplitude** is half the distance between the highest and lowest points.It's like finding the radius of a Ferris wheel!

STEP 4

a=max−min2=9−(−3)2=122=6 a = \frac{\textbf{max} - \textbf{min}}{2} = \frac{9 - (-3)}{2} = \frac{12}{2} = 6 a=2max−min​=29−(−3)​=212​=6 So our **amplitude** is 6\textbf{6}6!

STEP 5

The **vertical shift** is the middle value between the highest and lowest points.It's where the center of the Ferris wheel sits.

STEP 6

d=max+min2=9+(−3)2=62=3 d = \frac{\textbf{max} + \textbf{min}}{2} = \frac{9 + (-3)}{2} = \frac{6}{2} = 3 d=2max+min​=29+(−3)​=26​=3 Our **vertical shift** is 3\textbf{3}3!

STEP 7

The graph completes a full cycle between x=0x = 0x=0 and x=πx = \pix=π.This means the **period** is π\piπ.

STEP 8

Since the standard period of cosine is 2π2\pi2π, we can find bbb using the equation 2π/b=period2\pi/b = \textbf{period}2π/b=period.

STEP 9

2πb=π \frac{2\pi}{b} = \pi b2π​=π b=2ππ=2 b = \frac{2\pi}{\pi} = 2 b=π2π​=2 So, b=2b = \textbf{2}b=2.

STEP 10

A regular cosine graph starts at its maximum.This graph starts at its minimum.That's a shift of half a period to the right!

STEP 11

Since our period is π\piπ, half a period is π/2\pi/2π/2.

STEP 12

So our **phase shift** ccc is π/2\pi/2π/2.We're shifting to the *right*, so it's *positive*.

STEP 13

Putting it all together, our equation is: y=6cos⁡[2(x−π/2)]+3 y = 6 \cos[2(x - \pi/2)] + 3 y=6cos[2(x−π/2)]+3

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? We need to find the equation of a cosine graph that goes between $-3$ and $9$ , with the smallest possible shift to the right. Watch out! Remember that cosine normally starts at its maximum.
If it starts at a minimum, that's a shift!

1. Find the amplitude
2. Find the vertical shift
3. Find the period
4. Find the phase shift

The amplitude is half the distance between the highest and lowest points.
It's like finding the radius of a Ferris wheel!

$a = \frac{\textbf{max} - \textbf{min}}{2} = \frac{9 - (-3)}{2} = \frac{12}{2} = 6$ So our amplitude is $\textbf{6}$ !

The vertical shift is the middle value between the highest and lowest points.
It's where the center of the Ferris wheel sits.

$d = \frac{\textbf{max} + \textbf{min}}{2} = \frac{9 + (-3)}{2} = \frac{6}{2} = 3$ Our vertical shift is $\textbf{3}$ !

The graph completes a full cycle between $x = 0$ and $x = \pi$ .
This means the period is $\pi$ .

Since the standard period of cosine is $2\pi$ , we can find $b$ using the equation $2\pi/b = \textbf{period}$ .

$\frac{2\pi}{b} = \pi$ $b = \frac{2\pi}{\pi} = 2$ So, $b = \textbf{2}$ .

A regular cosine graph starts at its maximum.
This graph starts at its minimum.
That's a shift of half a period to the right!

Since our period is $\pi$ , half a period is $\pi/2$ .

So our phase shift $c$ is $\pi/2$ .
We're shifting to the right, so it's positive.

Putting it all together, our equation is: $y = 6 \cos[2(x - \pi/2)] + 3$