What are the amplitude, period, and midline of f(x) = 3 cos(4x − π) + 2?

Answers

Answer 1

Amplitude: 3; period: pi/2; midline: y = 2

Answer 2

Answer:

The amplitude of the function is 3, period of the function is [tex]\frac{\pi}{2}[/tex] and midline of the function is 2.

Step-by-step explanation:

The given function is

[tex]f(x)=3\cos(4x-\pi)+2[/tex] .... (1)

The standard form of a Cosine function is

[tex]g(x)=a\cos(bx+c)+d[/tex] ... (2)

where, a is amplitude, period is [tex]\frac{2\pi}{b}[/tex] and d is midline.

From (1) and (2), we get

[tex]a=3,b=4,d=2[/tex]

The amplitude of the function is

[tex]a=3[/tex]

The period of the function is

[tex]\frac{2\pi}{b}=\frac{2\pi}{4}=\frac{\pi}{2}[/tex]

The midline of the function is

[tex]d=2[/tex]

Therefore the amplitude of the function is 3, period of the function is [tex]\frac{\pi}{2}[/tex] and midline of the function is 2.

Related Questions

A cone has a volume of about 28 cubic inches. Which are possible dimensions for the cone?

Answers

The formula for the volume, V of a cone is V = (1/3)*π*(radius)^2 * height.

So, as you see you have two variables, radius and height, which means that there are different combinations of radius and height that lead to the same volume.

That implies that your questions is missing some information.

I searched for it and found that the question has a list of answer choices. This is the list:

a) radius 6 inches, height 3 inches

b) diameter 6 inches, height 3 inches

c) diameter 4 inches, height 6 inches

d) diameter 6 inches, height 6 inches

Now you can probe those options to see which leads to an approximate volume of 28 cubic inches.

a) radius 6 in, height 3 in

=> V = (1/3)*3.14*(6in)^2 * 3in = 113.04 in^3 => not possible

b) diameter 6 in, height 3 inc

diameter 6 in => radius 3 in

=> V = (1/3)*3.14*(3in)^2 * 3in = 28.26 in^3 => this is the answer

The other options lead to values far from 28 in^3.

Answer: diameter 6 inches, height 3 inches.

A quadratic equation has a discriminant of 0. which describes the number and type of solutions of the equation?

Answers

Discriminant 0 in quadratic equation means 1 real solution—a repeated root where parabola touches x-axis once.

When the discriminant of a quadratic equation is 0, it means that the quadratic equation has exactly one real solution. This solution is considered a "double root" or "repeated root," meaning that the parabola defined by the quadratic equation touches the x-axis at exactly one point. Mathematically, this occurs when the quadratic equation has two identical roots.

The general form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex], and the discriminant, denoted by [tex]\(b^2 - 4ac\),[/tex] helps determine the nature of the roots.

When the discriminant is zero [tex](\(b^2 - 4ac = 0\))[/tex], the quadratic equation has one real root. This happens when the parabola defined by the equation just touches the x-axis at one point. The solution is given by:

[tex]\[x = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{{2a}}\][/tex]

The following conditions are:

D < 0 ; there are two non-real or imaginary roots which are complex conjugates

D = 0 ; there is one real root and one imaginary (non-real)

D > 0 ; there are two real distinct roots

Therefore the answer to this question is:

The solution has one real root and one imaginary root.

Find the volume of revolution bounded by the curves y = 4 – x2 , y = x, and x = 0, and is revolved about the vertical axis.

Answers

[tex]4-x^2=x\\ x^2+x-4=0\\ \Delta=1^2-4\cdot1\cdot(-4)=1+16=17\\ x_1=\dfrac{-1-\sqrt{17}}{2}\\ x_2=\dfrac{-1+\sqrt{17}}{2}\\\\ \displaystyle V=\pi\int\limits_0^{\dfrac{-1+\sqrt{17}}{2}}(4-x^2-x)^2\,dx\\ V=\pi\int\limits_0^{\dfrac{-1+\sqrt{17}}{2}}(16-4x^2-4x-4x^2+x^4+x^3-4x+x^3+x^2)\,dx\\ V=\pi\int\limits_0^{\dfrac{-1+\sqrt{17}}{2}}(x^4+2x^3-7x^2-8x+16)\,dx\\ V=\pi \left[\dfrac{x^5}{5}+\dfrac{x^4}{2}-\dfrac{7x^3}{3}-4x^2+16x\right]_0^{\dfrac{-1+\sqrt{17}}{2}}\\ [/tex]

The rest of solution in the attachment.

There's a mistake in the picture
It shoud be
[tex]V=\pi\left(\dfrac{289\sqrt{17}-521}{60}\right)\approx35[/tex]

Find all complex solutions of 3x^2+3x+4=0.

(If there is more than one solution, separate them with commas.)

Answers

Given equation is [tex]3x^2 + 3x + 4 = 0[/tex]

Now we can compare it with general form of quadratic equation ([tex]ax^2 + bx + c = 0[/tex])
a = 3 , b = 3 and c = 4

Now we can apply quadratic formula which is given as
[tex]x =\frac{ -b+/- \sqrt{b^2-4ac} }{2a}[/tex]

Now we can plugin value of a , b or c
[tex]x = \frac{-3+/- \sqrt{(3)^2 - 4*3*4} }{2*3} [/tex]
[tex]= \frac{-3+/- \sqrt{9 - 48} }{6} = \frac{-3+/- \sqrt{-39} }{6} [/tex]
In general we know [tex] \sqrt{-1} = i [/tex]
So we can write [tex] \sqrt{-39 } = \sqrt{-1} * \sqrt{39} = i \sqrt{39} [/tex]
So
[tex]x = \frac{-3+/-i \sqrt{39} }{6} [/tex]
So [tex]x = \frac{-3+i \sqrt{39} }{6} [/tex] or [tex]x = \frac{-3- \sqrt{39} }{6} [/tex]

Final answer:

The complex solutions to the equation[tex]3x^2+3x+4=0 are x = (-3 + i\sqrt{39})/6 and x = (-3 - i\sqrt{39})/6.[/tex]

Explanation:

To find all complex solutions of the quadratic equation [tex]3x^2+3x+4=0[/tex]e the quadratic formula:

[tex]x = \((-b \pm \sqrt{b^2-4ac})/(2a)\).[/tex]

Here, a = 3, b = 3, and c = 4. Plugging these values into the formula, we get:

[tex]x = \((-3 \pm \sqrt{3^2-4 \cdot 3 \cdot 4})/(2 \cdot 3)\).[/tex]

This simplifies to:

[tex]x = \((-3 \pm \sqrt{-39})/6\).[/tex]

Since the discriminant (under the square root sign) is negative, we know that the solutions will be complex. Using i to represent the square root of -1, we can write the solutions as:

[tex]x = \((-3 \pm i\sqrt{39})/6\).[/tex]

So, the complex solutions are [tex]x = (-3 + i\sqrt{39})/6 and x = (-3 - i\sqrt{39})/6.[/tex]

Avery can run at 10 uph. The bank of a river is represented by the line 4x + 3y = 12, and Avery is at (7, 5). How much time does Avery need to reach the river?

Answers

The minimum distance will be along a perpendicular line to the river that passes through the point (7,5)

4x+3y=12

3y=-4x+12

y=-4x/3+12/3

So a line perpendicular to the bank will be:

y=3x/4+b, and we need it to pass through (7,5) so

5=3(7)/4+b

5=21/4+b

20/4-21/4=b

-1/4=b so the perpendicular line is:

y=3x/4-1/4

So now we want to know the point where this perpendicular line meets with the river bank. When it does y=y so we can say:

(3x-1)/4=(-4x+12)/3 cross multiply

3(3x-1)=4(-4x+12)

9x-3=-16x+48

25x=51

x=51/25

x=2.04

y=(3x-1)/4

y=(3*2.04-1)/4

y=1.28

So now that we know the point on the river that is closest to Avery we can calculate his distance from that point...

d^2=(x2-x1)^2+(y2-y1)^2

d^2=(7-2.04)^2+(5-1.28)^2

d^2=38.44

d=√38.44

d=6.2 units

Since he can run at 10 uph...

t=d/v

t=6.2/10

t=0.62 hours (37 min 12 sec)

So it will take him 0.62 hours or 37 minutes and 12 seconds for him to reach the river.

How many three digit numbers can be made from the digits 1,\ldots,9 if repetitions of digits are not allowed?

Answers

9 values for each digit, order matters: 123 is different than 321, and no repetion

9*8*7 = 504

Look at a simpler example, with 1, 2, 3:

123
132
213
231
312
321

6 numbers

3*2*1 = 6

There are 84 three digit numbers can be made from the digits 1, ..., 9

What is Combination?

A combination is a technique to determines the number of possible arrangements in a collection of items where the order of the selection does not matter.

Given that;

The numbers are,

⇒ 1, 2, ..., 9

Now,

All the three digit numbers can be made from the digits 1, ..., 9 are;

⇒ [tex]^{9} C_{3}[/tex]

⇒ 9! / 3! 6!

⇒ 9 × 8 × 7 / 6

⇒ 84

Thus, There are 84 three digit numbers can be made from the digits

1, ..., 9.

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What kind of transformation is illustrated in this figure ?

Answers

Your answer would be translation (brainliest answer please)

Translation it's moving one place to another

Find the probability of at least 2 girls in births. Assume that male and female births are equally likely and that the births are independent events. Round to three decimal places.

Answers

we assume that male and female births are equally likely, it means that the probability of birth of male= probability of birth of female = 100%/2=50% or 0.5We have 2 independent events. So what will be the variants:
Male and FemaleMale and Male Female and Male Female and Female
All four variants are equally likely.Probability of each one is 1/4 = 0.25.So, Result "Female and Female"
are probability = 0.250 --- if round to three decimal places.

Need help with this please

Answers

First, are you really familiar with the graph of y = [x]? If so, your answer choices move that graph up or down.

A. Move the graph of y=[x] UP 3 units
B. Move the graph UP 1 unit
C. Move the graph DOWN 1 unit
D. Move the graph DOWN 3 units

The graph you're given on the problem screen is not a very good one, I think. Would it help to draw y = [x] on a piece of graph paper where one square = 1 unit? The graph you're shown has one square = 2 units, which could lead to some confusion.

So, draw the graph of y = [x], then figure out which way the graph was moved -- and how far -- to get the graph in the problem.

Here is a link to a YouTube video that might help a lot.
https://youtu.be/UQ3a2QH_-GU

What is the factored form of the expression k^2 - 9h^2

Answers

The equation which is supposed to help you is this one:
a^2 - b^2 = (a+b) (a-b)

So, what we need to do here is apply this equation:
k^2 - 9h^2 =
k * k - (3h)^2 =
(k + 3h) (k - 3h)

ANSWER ASAP Find the value of x. A. sqrt 3 B. 3 sqrt 2/2 C. 3 sqrt 2 D. 3 sqrt 3

Answers

Since this is a right triangle we can use the Pythagorean Theorem:

h^2=x^2+y^2, and since x and y are both opposite 45° angles, they are equal to one another, so we have:

h^2=2x^2 since h=3

2x^2=9

x^2=9/2

x=√(9/2)

x=3√(1/2) if we rationalize the denominator by multiplying the fraction by √2/√2 we get

x=(3√2)/2

Answer:

Step-by-step explanation:

Factor completely 36x2− 121.

Answers

The expression 36x² - 121 is in the 'difference of two squares form' which is (a² - b²)

Factorising (a² - b²) gives (a+b)(a-b)

Factorising 36x² - 121 gives (6x+11)(6x-11)

How to find the area of region composed of rectangles and/or right triangles?

Answers

To determine the area of a region that is composed of rectangles and triangles, you should just calculate for the individual area of each of the rectangles and the triangles. Then, after determining the areas, you add them all up. The sum would represent the total area of the region. The area is the number of square units in a shape. It is one way of describing the size of a particular surface. For a rectangle, it is calculated from the product of its length and its width. For a triangle, it obtained from one half the product of its base and the height.

If ON = 9x – 4, LM = 8x + 7, NM = x – 3, and OL = 4y – 8, find the values of x and y for which LMNO must be a parallelogram.

Answers

x=11 y=4 :) i need 20 characters so im randomly typing

A. x = 11, y = 4 If this is the case, then: 8x + 7 = 9x -4 --> x = 11 4y - 8 = x - 3 --> 4y -8 = 11 -3 and 4y =16 and y = 4.

What value of x makes the denominator of the function equal zero? y= 6/4x-40

Answers

To find the value that would make the denominator equal to zero, set the denominator equal to zero and solve for x.
4x - 40 = 0
Add 40 to both sides.
4x = 40
Divide both sides by 4
x = 10

The value of x makes the denominator of the function equal zero is 10

what is an equation?

An equation is a mathematical expression that contains an equals symbol. Equations often contain algebra.

Given that:

y = [tex]\frac{6}{4x-40}[/tex]

Now the denominator is: 4x - 40

So, 4x =40

x =40/4

x =10

So, make the denominator 0, put x= 10.

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If XYZ measures 75, what is the measure of XWZ ? A. 285 B. 210 C. 75 D. 150
(Its a circle, and its saying that the arc is 775 and wants to know what the rest of the circle is)

Answers

Your answer is 285. ADD ME!!!!!!!!!!!

The measure of XWZ is 285

what is arc?

The arc of a circle is defined as the part or segment of the circumference of a circle.

Given:

<XYZ = 75

As, we know central angle of 360

So, arc(XWZ) + <XYZ = 360

arc (XWZ) = 360 - 75

arc(XWZ) = 285

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Jake has proved that a function, f(x), is a geometric sequence. How did he prove that?
A He showed that an explicit formula could be created.
B He showed that a recursive formula could be created.
C He showed that f(n) ÷ f(n − 1) was a constant ratio.
D He showed that f(n) − f(n − 1) was a constant difference.

Answers

The defining characteristic of any geometric sequence is the common ratio, which is the constant found when dividing any term by the term preceding it.

C. He showed that f(n)/(f(n-1) was a constant ratio.

The answer to your question is : C. He showed that f(n) ÷ f(n − 1) was a constant ratio.

Answer please math isn't my forte

Answers

4)
x + 5 < 14

Subtract 5 on both sides
x + 5 - 5 < 14 - 5
x < 14 - 5
x < 9
Answer is D.

5)
< (less than) and > (greater than) have empty circles because the number is not included.
For example, x > 5
5 is not a solution because we are looking for all x that is greater than 5.

≤ (less than or equal to) and ≥ (greater than or equal to) have a closed circle because the number is included.
For example, x ≤ 7
7 is a solution because we are looking for all x that is less than or equal to 7.

Answer is C.

6)
4(x-1) ≥ 8
Divide 4 on both sides
(x-1) ≥ 2
Add 1 on both sides
x ≥ 3

Answer is D.

If a radius of a circle bisects a chord which is not a diameter, then ______

Answers

Frst, some notation. Let AB be a chord on circle O, and let CD be a diameter of O that passes through AB at M.

Answer:

the radius is perpendicular to the chord.

Step-by-step explanation:

The geometry is drawn in the image shown below in which AB is the chorh and O is the centre of the circle. Om is the radius which bisects the chord.(Given) So, AN = NB

From the image, considering ΔAON and ΔBON,

AO = BO (radius of circle)

AN = NB (given)

ON = ON (common)

So,

ΔAON ≅ ΔBON

Hence, ∠ANO = ∠BNO

Also, ∠ANO + ∠BNO = 180° (Linear Pair)

So,

∠ANO = ∠BNO = 90°

Hence, it is perpendicular to the chord.

in a certain county, the number of charter schools is 4 less than twice the number of alternative schools. We know that there are 48 charter schools in the county. How many alternative schools are in the county?

Answers

Answer:

There are [tex]26[/tex] alternative schools in the country

Step-by-step explanation:

Let

x------> the number of charter schools

y-----> the number of alternative schools

we know that

[tex]x=48[/tex]

[tex]x=2y-4[/tex] -----> equation A

substitute the value of x in the equation A and solve for y

[tex]48=2y-4[/tex]

[tex]2y=48+4[/tex]

[tex]2y=52[/tex]

[tex]y=26[/tex]

Solve the following system by graphing.

x - y = 4
x + y = 2

What is the solution of the system?

(3, -1)
(3, 1)
(-1, 3)

Answers

hello :
the solution is : (3, -1)
put x= 3 and y = -1
you have : 3-(-1) = 4 and 3+(-1) = 2 ..... right

The solution for the system of equation x - y = 4 and x + y = 2 is (3, -1).

What is an equation ?

An equation is a combination of different variables, in which two mathematical expressions are equal to each other.

The given pair of equations,

x - y = 4 (1)

And x + y = 2 (2)

To find the solution of the equations,

Add both the equations,

x - y + x + y = 4 + 2

2x = 6

x = 3,

Substitute the value of x = 3 in equation (1),

3 - y = 4

y = -1

The values of x and y are 3 and -1 respectively.

Hence, option (A) is correct.

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Rationalize the denominator

Answers

see attached picture:

Answer:

[tex]\frac{60 - 10\sqrt10-6\sqrt3+\sqrt30}{97}[/tex]

Step-by-step explanation:

Hello!

To rationalize the denominator, we have to remove any root operations from the denominator.

We can do that by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate simply means the same terms with different operations.

Rationalize[tex]\frac{6 - \sqrt10}{10 + \sqrt3}[/tex][tex]\frac{6 - \sqrt10}{10 + \sqrt3} * \frac{10 - \sqrt3}{10 - \sqrt3}[/tex][tex]\frac{(6 - \sqrt10)(10 - \sqrt3)}{100 - 3}[/tex][tex]\frac{60 - 10\sqrt10-6\sqrt3+\sqrt30}{97}[/tex]

The answer is [tex]\frac{60 - 10\sqrt10-6\sqrt3+\sqrt30}{97}[/tex].

How do I solve this

Answers

[tex]\bf \cfrac{x+2}{x^2+6x-7}[/tex] so, that function is "defined", ok, what values of "x" are not in the domain, namely, what values can "x" take on and not make the function "undefined", well, you know, if we end up with a 0 at the denominator, like [tex]\bf \cfrac{x+2}{0}[/tex] then, we'd have an "undefined" expression...so... any values of "x" that make the denominator 0, are not really the ones we want, and thus they'd be excluded from the domain.

so, hmm which are those? let's check, let's set the denominator to 0, and solve for "x".

[tex]\bf x^2+6x-7=0\implies (x+7)(x-1)=0\implies x= \begin{cases} -7\\ 1 \end{cases} \\\\\\ \textit{let's check, } x=-7\quad \cfrac{(-7)+2}{(-7)^2+6(-7)-7}\implies \cfrac{-5}{49-42-7}\implies \cfrac{-5}{0} \\\\\\ x=1\quad \cfrac{(1)+2}{(1)^2+6(1)-7}\implies \cfrac{3}{1+6-7}\implies \cfrac{-3}{0}[/tex]

Find the distance between the points (13, 20) and (18, 8).

Answers

distance = sqrt (13-18)^2 + (20-8)^2 )

= sqrt ( 5^2 + 12^2)

= sqrt 169 = 13

Answer: 13 units

Step-by-step explanation:

The distance formula to calculate distance between two points (a,b) and (c,d) is given by :-

[tex]d=\sqrt{(d-b)^2+(c-a)^2}[/tex]

Given points : (13, 20) and (18, 8)

Now, the distance between the points (13, 20) and (18, 8)is given by ;-

[tex]d=\sqrt{(8-20)^2+(18-13)^2}\\\\\Rightarrow\ d=\sqrt{(-12)^2+(5)^2}\\\\\Rightarrow\ d=\sqrt{144+25}\\\\\Rightarrow\ d=\sqrt{169}\\\\\Rightarrow\ d=13\text{units}[/tex]

Hence, the distance between the points (13, 20) and (18, 8) is 13 units.

During one year about 163 million adults over 18 years old in the United States spent a total of about 93 billion hours online at home. On average, how many hours per day did each adult spent online at home?

1. How do you write each number in scientific notation?
2 How do you convert the units to hours per day.

Answers

one billion is 9 zeros, on million is 6 zeros, thus

[tex]\bf 163,000,000\implies 163\times 10^6 \\\\\\ 93,000,000,000\implies 93\times 10^9\\\\ \cfrac{93\times 10^9}{163\times 10^6}\impliedby \textit{hours per year for one adult} \\\\\\ \cfrac{93\times 10^9\times 10^{-6}}{163}\implie
s \cfrac{93\times 10^{9-6}}{163}\implies \cfrac{93\times 10^3}{163}[/tex]

that's how many hours each adult spent a year... now, how many hours is that per day? well, there are 365 days a year, so, we simply divide it by 365

[tex]\bf \cfrac{\frac{93\times 10^3}{163}}{365}\implies \cfrac{\frac{93\times 10^3}{163}}{\frac{365}{1}}\implies \cfrac{93\times 10^3}{163}\cdot \cfrac{1}{365}\implies \cfrac{93\times 10^3}{163\cdot 365}\quad \cfrac{hours}{day}[/tex]

Final answer:

To write each number in scientific notation, express it as a product between 1 and 10 and a power of 10. Each adult spent an average of 570.55 hours per day online at home.

Explanation:

To write each number in scientific notation, we need to express it as a product of a number between 1 and 10 and a power of 10.

163 million can be written as 1.63 x 10⁸

93 billion can be written as 9.3 x 10¹⁰

To convert the units to hours per day, we need to divide the total number of hours by the number of adults.

So, each adult spent an average of (93 x 10¹⁰) / (163 x 10⁸) = 570.55 hours per day online at home. Scientific notation simplifies large numbers, facilitating computations and providing a concise representation of quantities in mathematical contexts.

Write 11•47 using the distributive property. Then simplify.

Answers

Distributive property :

a (b+c) = ab + bc

Try writing 47 = 50-3 or 47 = 40+7.

11 x 47 = 11 x (50-3) = 11 x 50 - 11 x 3 = 550 - 33 = 517

or

11 x 47 = 11 x (40+7) = 11 x 40 + 11 x 7 = 440 + 77 = 517

Approximately what percentage of scores falls below the mean in a standard normal distribution

Answers

50% of scores will fall below the mean in a standard distribution

Four less than a number is greater than -28

Answers

This question implies that a number which is subtracted by 4 will still be larger than -28.
Any number which is larger than -28 by at least 5 would work.
Examples, -23, 0, 99 etc.

Hope this helps!

Write 5000 = 12 as an order pair

Answers

Hejejdjshdjsjskskskskskskskskskkskskskdkdjdkddkkd

what is the slope of a line that passes through (-4,-13) and (19,11)

Answers

the slope of the line is the gradient, which you can find through rise over run

m (gradient) = (y1 - y2) / (x1 - x2)

where (x1, y1) is the coordinate of the first point, and (x2, x2) is the coordinate of the second point

in your question:
x1 = -4
x2 = 19
y1 = -13
y2 = 11

m = (-13 -11) / (-4 -19) = -24 / -23 = 24/23 or 1.04 (2d.p.)

hope that helps :)

Answer:

24/23

Step-by-step explanation:

- vs - = + after -24,-23 =

Answer = 24, 23

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