Mathematics : asked on isabellamason1499

29.05.2023

F is a trigonometric function of the form f(x) = a cos( bx + c) + d

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09.07.2023, solved by verified expert

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The function is f(x) = 3cos(2πx)—6 if the function has a maximum
point at (-2, -3) and a minimum point at (–2.5, -9).
What is trigonometry?
Trigonometry is a branch of mathematics that deals with the relationship between sides and angles of a right-angle triangle.
We have a function:
f(x) = a cos(bx + c) + d
Here a is the amplitude:
Midline line y = -6
a = |-6-(-3)| = 3
d = -6 (midline)
Difference = |-3 + 2| = 1
b = 2π/1 = 2π
By plugging points in the equation, we get
c = 0
The equation become:

Thus, the function is f(x) = 3cos(2πx)—6 if the function has a maximum
point at (-2, -3) and a minimum point at (–2.5, -9).

F is a trigonometric function of the form f(x), №18011287, 29.05.2023 18:16

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Mathematics

f is a trigonometric function of the form of a sin(bx+c) + d. Below is the graph f(x). The function intersects its midline at (3, -6.5) and has a maximum point at (4, - 2). Find a formula for f(x). Give an exact expression.

Step-by-step answer

P Answered by Master

See below for answer and explanation (and also a graph)

Step-by-step explanation:

For a sine function f(x) = a * sin(bx + c) + d:

Period: 2π/|b|

Phase shift: -c/b

Midline: d

Amplitude: |a|

The amplitude is the distance between the maximum and the midline, which is a = |-6.5-(-2)| = |-6.5+2| = |-4.5| = 4.5

The midline is already given to us as d=-6.5

The period would be the length of a cycle, which one finished cycle is defined at (4,-2) where the function must start at the maximum (0,-2) which is the distance of b, making the period of the function 2π/|4| = 2π/4 = π/2.

The last variable c can be found by plugging in a given point and solving for c. Plugging (4,-2) into the equation, you have -2=4.5sin(π/2(4)+c)-6.5 which gets you c=π/2+2πn for any integer n but we are only taking the first solution c=π/2

Therefore, your final equation is f(x) = 4.5sin(π/2x+π/2)-6.5

f is a trigonometric function of the form of a sin(bx+c) + d. Below is the graph f(x). The function

Mathematics

F is a trigonometric function of the form f(x)=a sin(bx+c)+d Below is the graph f(x). The function intersects its midline at (3,-6.5) and has a maximum point at (4,-2) Find a formula for f(x). Give an exact expression.

Step-by-step answer

P Answered by Specialist

Step-by-step explanation:no thank for the points

no thank for the points

$F is a trigonometric function of the form f(x)=a\sin(bx+c)+d Below is the graph f(x). The function$
$F is a trigonometric function of the form f(x)=a\sin(bx+c)+d Below is the graph f(x). The function$
$F is a trigonometric function of the form f(x)=a\sin(bx+c)+d Below is the graph f(x). The function$
$F is a trigonometric function of the form f(x)=a\sin(bx+c)+d Below is the graph f(x). The function$
$F is a trigonometric function of the form f(x)=a\sin(bx+c)+d Below is the graph f(x). The function$

Mathematics

g is a trigonometric function of the form g(x) = a sin(bx + c) + d. Below is the graph g(x). The function intersects its midline at 3/4 *pi, 3) and has a maximum point at (pi,7) Find a formula for g(x). Give an exact expression.

Step-by-step answer

P Answered by Specialist

$g(x)=4sin(2x+\frac{\pi}{2})+3$

Step-by-step explanation:

Because the function intersects its midline at $(\frac{3\pi}{4},3)$ , then the midline is d=3 .

Additionally, the amplitude is just the positive distance between the maximum/minimum and the midline, so the amplitude is a=7-3=4 .

Also, given that period is $\frac{2\pi}{b}$ and the fact that the period is $\pi-0=\pi$ from our given maximum, we have the equation $\frac{2\pi}{b}=\pi$ where b=2 .

Lastly, to find , we know that the phase shift, $-\frac{c}{b}$ , is $-\frac{\pi}{4}$ (or $\frac{\pi}{4}$ to the left) since $\pi-\frac{3\pi}{4}=-\frac{\pi}{4}$ . Therefore, we have the equation $-\frac{c}{2}=-\frac{\pi}{4}$ where $c=\frac{\pi}{2}$ .

Putting it all together, our final equation is $g(x)=4sin(2x+\frac{\pi}{2})+3$

g is a trigonometric function of the form g(x) = a sin(bx + c) + d. Below is the graph g(x). The fun

Mathematics

G is a trigonometric function of the form g(x)=acos(bx+c)+d Below is the graph of g(x). The function has a maximum point at (3.5,-4) and a minimum point at (-1,-5) Find a formula for g(x). Give an exact expression.

Step-by-step answer

P Answered by Specialist

9514 1404 393

a = 0.5b = π/4.5c = 7π/9d = -4.5

Step-by-step explanation:

The value of d is the average of the maximum and the minimum.

d = (-4 +-5)/2 = -9/2 = -4.5

The value of 'a' is the difference between the maximum and d.

a = -4 -(-4.5) = 0.5

The value of b is 2π divided by the period. Here, the half-period is 3.5 -(-1) = 4.5, so ...

b = 2π/9 = π/4.5

The value of c is the value that makes the cosine argument zero at x=3.5.

(π/4.5)(3.5) +c = 0

c = -7π/9

G is a trigonometric function of the form g(x)=acos(bx+c)+d

Below is the graph of g(x). The functio

Mathematics

G is a trigonometric function of the form g(x)=a sin (bx+c)+d. The function intersects its midline at (-1 , 6) and has a minimum point at (-3.5 , 3) Find a formula for g (x). Give an exact expression.

Step-by-step answer

P Answered by Specialist

Formula for g(x) is $g(x) = 3 sin(\frac{\pi }{5}x + \frac{\pi }{5} ) + 6$

Step-by-step explanation:

Given - g is a trigonometric function of the form g(x)=a sin (bx+c)+d. The function intersects its midline at (-1 , 6) and has a minimum point at (-3.5 , 3)

To find - Find a formula for g (x). Give an exact expression.

Proof -

Given that,

g(x)=a sin (bx+c)+d

We know that, Midline is present in between maximum and minimum

Here given that, minimum is present is 3 and midline is present at 6

So, Maximum occurs at 9.

Now,

We know that,

Standard form of sine function is - g(x) = Asin(B(x-C)) + D

Where

A = Amplitude

and Amplitude = (Maximum - minimum) / 2

= (9 - 3)/ 2

= 6/2 = 3

⇒A = 3

Now,

Period = $\frac{2\pi }{B}$

⇒B = (2π) / Period

= (2π) / 10

= π/5

⇒B = π/5

Now,

Phase Shift : C = -1 ( i.e. 1 to the left)

Vertical Shift : D = 6

So,

We get

g(x) = Asin(B(x-C)) + D

= $3 sin(\frac{\pi }{5}(x -(- 1))) + 6$

⇒ $g(x) = 3 sin(\frac{\pi }{5}x + \frac{\pi }{5} ) + 6$

∴ we get

Formula for g(x) is $g(x) = 3 sin(\frac{\pi }{5}x + \frac{\pi }{5} ) + 6$

G is a trigonometric function of the form g(x)=a sin (bx+c)+d.

The function intersects its midline

Mathematics

Step-by-step answer

P Answered by Specialist

Step-by-step explanation:no thank for the points

no thank for the points

Mathematics

h is a trigonometric function of the form h(x)= a cos(bx + c)+d. Below is the graph of h(x). The function has a maximum point at (-pi/2,5) and a minimum point at (pi/4,-4). Find a formula for h(x). Give an expression. h(x)=

Step-by-step answer

P Answered by PhD

h(x)= a cos(bx + c)+d

h(x) is maximum when a cos(bx + c)+d is maximum

and the maximum value of a cos(bx + c)+d = a + d when cos(bx + c) = 1

and cos(bx + c ) = 1, when bx + c = 0

as per the given question function has a maximum point at (-pi/2,5)

This means a + d = 5 and x = -c/b = -pi/2...........(1)

similarly for minimum value

-a + d = -4 and bx + c = pi

-a + d = -4 and x = (-pi -c)/b = pi/4 ..........(2)

solving (1) and (2)

a = 9/2

d= 1 /2

c = -2pi/3

b = -4/3

Therefore the function h(x) = 9/2 cos(-4x/3 -2pi/3) +1/2

h(x) = 9/2 cos(4x/3 + 2pi/3) + 1/2

Mathematics

h is a trigonometric function of the form h(x)= a cos(bx + c)+d. Below is the graph of h(x). The function has a maximum point at (-2pi,3) and a minimum point at (-pi/2,-4). Find a formula for h(x). Give an expression. h(x)=

Step-by-step answer

P Answered by PhD

h(x)= a cos(bx + c)+d

h(x) is maximum when a cos(bx + c)+d is maximum

and the maximum value of a cos(bx + c)+d = a + d when cos(bx + c) = 1

and cos(bx + c ) = 1, when bx + c = 0

as per the given question function has a maximum point at (-2pi,3)

This means a + d = 3 and x = -c/b = -2pi .............(1)

similarly for minimum value

-a + d = -4 and bx + c = pi

-a + d = -4 and x = (-pi -c)/b = -pi/2 ..........(2)

solving (1) and (2)

a = 7/2

d= -1 /2

c = -4pi/3

b = -2/3

Therefore the function h(x) = 7/2 cos(-2x/3 -4pi/3) - 1/2

h(x) = 7/2 cos(2x/3 + 4pi/3) - 1/2

Mathematics

F is a trigonometric function of the form f(x) = a cos( bx + c) + d

Step-by-step answer

P Answered by Specialist

F is a trigonometric function of the form f(x) = a cos( bx + c) + d

Mathematics

Fit a trigonometric function of the form f(t)=c0+c1sin(t)+c2cos(t)f(t)=c0+c1sin⁡(t)+c2cos⁡(t) to the data points (0,5.5)(0,5.5), (π2,0.5)(π2,0.5), (π,−2.5)(π,−2.5), (3π2,−7.5)(3π2,−7.5), using least squares.

Step-by-step answer

P Answered by PhD

Answer

f(t)=-0.2+4.1sin(t)+4cos(t)

Step-By-Step Explanation

Given the function f(t)=c_0+c_1sin(t)+c_2cos(t) .

For each pair (t, f(t)) in the data points (0,5.5), (π/2,0.5), (π,−2.5), (3π/2,−7.5)

f(0)=c_0+0c_1+c_2=5.5 .

$f(\pi /2)=c_0+c_1+0c_2=0.5$ .

$f(\pi)=c_0+0sin(t)-c_2=-2.5$ .

$f(3\pi /2)=c_0-c_1+0c_2=-7.5$ .

Expressing this as a system of linear equations in matrix form AX=B

$\left(\begin{array}{ccc} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & -1 \\ 0 & -1 & 0 \end{array} \right)\left( \begin{array}{c} c_{0} \\ c_{1} \\ c_{2}\\ \end{array} \right)=\left(\begin{array}{c} 5.5 \\ 0.5 \\ -2.5 \\ -7.5 \end{array} \right)$

Where

$A=\left(\begin{array}{ccc} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & -1 \\ 0 & -1 & 0 \end{array} \right)$ ,

$B=\left(\begin{array}{c}5.5\\0.5\\-2.5\\-7.5\end{array} \right)$

$X=\left(\begin{array}{c}c_0\\c_1\\c_2\end{array}\right)$

To determine the values of X, we use the expression

$X=(A^{T}A)^{-1}A^{T}B$

$A^{T}A= \left(\begin{array}{ccc} 3 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 2 \end{array} \right)$

$(A^{T}A)^{-1}= \left(\begin{array}{ccc} 0.4 & -0.2 & 0 \\ -0.2 & 0.6 & 0 \\ 0 & 0 & 0.5 \end{array} \right)$

$A^{T}B=\left(\begin{array}{c} 3.5 \\ 8 \\ 8 \end{array} \right)$

Therefore:

$X=\left(\begin{array}{ccc} 0.4 & -0.2 & 0 \\ -0.2 & 0.6 & 0 \\ 0 & 0 & 0.5 \end{array} \right)\left( \begin{array}{c} 3.5 \\ 8 \\ 8 \end{array} \right)$