College basketball team has won 14 games and lost 6 games. If they continue to win at this rate, how many games would the team win in a 50 game schedule?

Answer:

  see below

Step-by-step explanation:

The rotation transformations are ...

  90° : (x, y) ⇒ (-y, x)

  180° : (x, y) ⇒ (-x, -y)

  270° : (x, y) ⇒ (y, -x)

Applying these to the given points, you get ...

9) A'(6, 9)

10) A'(15, 11)

11) A'(9, 6)

12) A'(-11, 15)

13) A'(6, -9)

14) A'(15, -11)

Answer:

B.

Step-by-step explanation:

The line is going up 1 and over 2, making the slope 1/2 and the y-intercept is 1. Hope this helped!

Step-by-step explanation:

ok so your gonna need to set up a proportion, specifically

25 is to x as x is to 16

mathematically, it's a cross multiplying problem:

25                X

           =          

X                   16

cross multiply and you get 400 = x²

which when solved is 20.

Have a good night dude.

Answer:

There will be total 2048 parts of the given paper if James if able to fold the paper eleven times.

The needed function is [tex]y = 2 ^n[/tex]

Step-by-step explanation:

Let us assume the piece of paper James decides to fold is a SQUARE.

Now, let us assume:

n : the number of times the paper is folded.

y : The number of parts obtained after folds.

Now, if the paper if folded ONCE ⇒  n = 1

Also, when the pap er is folded once, the parts obtained are TWO equal parts.

⇒  for n = 1 , y = 2       ..... (1)

Similarly, if the paper if folded TWICE  ⇒  n = 2

Also, when the paper is folded twice, the parts obtained are FOUR equal parts.

⇒  for n = 2 , y = 4       ..... (2)

⇒[tex]y = 2^2 = 2^n[/tex]

Continuing the same way, if the paper is folded SEVEN times  ⇒  n = 7

So, [tex]y = 2^ n = 2^7 = 128[/tex]

⇒  There are total 128 equal parts.

Lastly,  if the paper is folded ELEVEN  times  ⇒  n = 11

So, [tex]y = 2^ n = 2^{11} = 2048[/tex]

⇒  There are total 2048 equal parts.

Hence, there will be total 2048 parts of the given paper if James if able to fold the paper eleven times.

And the needed function is [tex]y = 2 ^n[/tex]

Answer:

(option C/3) 12^3 + 6x^2

Step-by-step explanation:

Took test and got it right.

The product of 2x^2 and (6x + 3) is calculated using the distributive property of multiplication over addition, resulting in 12x^3 + 6x^2.

The question requires us to find the product of the polynomial 2x^2 with the binomial (6x + 3). We use the distributive property of multiplication over addition to multiply 2x^2 by each term in the binomial.

Step 1: Multiply 2x^2 by 6x: 2x^2 * 6x = 12x^3

Step 2: Multiply 2x^2 by 3: 2x^2 * 3 = 6x^2

Thus, the product of 2x^2(6x + 3) = 12x^3 + 6x^2. So the correct answer is D. 12x^3 + 6x^2.

Learn more about Polynomial multiplication here:

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Answer:

It'll take the robot 8 minutes to complete one task

7.5 tasks will be completed in one hour

Step-by-step explanation:

Total time to complete 5 tasks is 2/3hr (40 minutes)

Time it takes to complete one task = 40 ÷ 5 = 8 minutes

Since the robot completes one task in 8 minutes, x tasks will be completed in 60 minutes.

x = 60 ÷ 8 =  7.5 tasks

Answer:

A. 8 minutes; B. 7.5 tasks in one hour; C. It takes the robot about [tex] \\ \frac{2}{15}\;hour[/tex] or about 0.1333 hour to complete one task or 13.33% of one hour.

Step-by-step explanation:

Part A

Two thirds hours is

[tex] \\ \frac{2}{3}*60 = 40\;min[/tex]  

We know that each task takes the same amount of time. So, 40min can be divided by 5:

[tex] \\ \frac{40}{5} = 8\;min[/tex]

Thus, each task takes 8 min to be completed. Then, it takes the robot 8 minutes to complete one task.

Part B

The robot can complete 5 tasks in 40 minutes, how many tasks can the robot complete in 60 minutes or one hour?

There are 20 minutes ahead to complete one hour. In the next 8 minutes, the robot can complete one task. There are still 12 minutes ahead. In the next 8 minutes, the robot completes another task. There is still 4 minutes ahead to complete the hour, but in 4 minutes the robot can complete half of the task because it takes 8 minutes for a complete task. Therefore, the robot can complete 5 tasks + 2 tasks + 0.5 task = 7.5 tasks in one hour or 60 minutes.

We can obtain the same answer using proportions. That is, if 5 tasks are completed in 40 minutes, how many of them will be completed in one hour or 60 minutes.  

Then

[tex] \\ \frac{5\;tasks}{40\;min} = \frac{x}{60\;min}[/tex]

[tex] \\ \frac{5\;tasks}{40\;min}*60\;min = x[/tex]

[tex] \\ x = \frac{5\;tasks*60\;min}{40\;min}[/tex]

[tex] \\ x = \frac{300\;tasks}{40} = 7.5\;tasks[/tex]

Part C (A)

From part A, we already know that the robot can complete a task in 8 minutes, which is a fraction of one hour. What is this fraction? In one hour we have 60 minutes, then

[tex] \\ 8\;min*\frac{1\;hour}{60\;min} = 1\;hour*\frac{8}{60} = 1\;hour*\frac{4}{30} = 1\;hour*\frac{2}{15} = 0.1333333....\;hours \approx 0.1333\;hours[/tex].

Therefore, it takes the robot about [tex] \\ \frac{2}{15}\;hour[/tex] or 0.1333 hour to complete one task (rounding to four decimal places) or 13.33% of one hour.

Answer:

um in the first place why the hell did they steal a car? and paint the dmn car?

Step-by-step explanation:

Answer:

F(x)=-1/5x-7

Step-by-step explanation:

You have to switch signs and flip the orgininal slope to get a perpendicular one.

5 will become - 5

Then it will become -1/5 because 5 on its own is equivalent to -5/1

Answer: the slope is - 1/5

Step-by-step explanation:

The equation of a straight line can be represented in the slope-intercept form which is expressed as

y = mx + c

Where

m represents the slope

c represents the y intercept

Comparing with the given equation,

Slope, m = 5

If two lines are perpendicular, it means that the slope of one line is the negative reciprocal of the slope of the other line. Therefore, the slope of the line perpendicular to

F(x)=5x-7 is - 1/5

Answer:

2.7%.

Step-by-step explanation:.

Given:

Net profit margin ( profitability rate) =  15%

Total sales = $155 million

Total assets = $312 million

Total equity = $223 million.

Dividend rate = 10%

Question asked:

What is the firm's sustainable growth rate ?

First of all we will find these thing.

1. Asset utilization rate = [tex]\frac{Total \ sales}{Total \ assets}[/tex]

                                     = [tex]\frac{155}{312} = 0.496\ million= 0.5\%[/tex]

2. Financial utilization rate   =   [tex]\frac{Total\ debt}{Total\ equity} \\[/tex]

  Total debt = Total asset - Total equity

                   = $312 million -   $223 million = $89 million

                                           = [tex]\frac{89}{223} = 0.4\%[/tex]

3. Return on equity rate = Asset utilization rate [tex]\times[/tex] profitability rate

Return on equity rate = [tex]0.5\times15\times0.4=3\%[/tex]

4. Business retention rate = 100 - Dividend rate

                                           = 100 - 10 = 90%

Now, finally we will calculate sustainable growth rate :

Sustainable growth rate =  Return on equity rate [tex]\times[/tex]  Business retention rate

                                        = [tex]3\%\times90\%=2.7\%[/tex]

Therefore, firm's sustainable growth rate is 2.7%

Answer:

Adding a constant to every score increases the mean by the same constant amount. Thus, μ

= 50+10= 60.

Adding a constant to every score has no effect on the standard deviation. σ = 5

Step-by-step explanation:

If a constant value is added to every score in a distribution, the same constant will be added to the mean. similarly, if you subtract a constant from every score, the same constant will be subtracted from the mean.

A;so recall the definition of standard deviation, it measures how each observation is far from its center on average, so if you shift your data by A then, also every observation is shifted by A and then standard deviations stays the same. also think standard deviation as measure of spread not a measure of scale.

When 10 points are added to every score in a population with a mean of 50 and standard deviation of 5, the new mean will be 60 but the standard deviation does not change and remains 5.

In the given scenario, the population has a mean of 50 (µ = 50) and a standard deviation of 5 (σ = 5). When you add 10 points to every score in the population, the mean of the population, which is the average of all scores, will increase by 10, resulting in a new mean of 60.

The standard deviation, which measures the dispersion or spread of scores from the mean, will not change. This happens because adding a constant to every score only shifts the entire distribution of scores, but does not increase or decrease the spread, or 'standard deviation,' among them. In other words, the new value of the standard deviation will remain as 5.

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Answer:

Both apply

Step-by-step explanation:

SSS can be used because sides are already known to be equal

SAS can also be used because when you have the 3 sides, you can use the cosine law to find any angle

Answer:

After 12 years height of both the trees would be same.

Step-by-step explanation:

Given,

Height of tree type A = 5 ft

Height of tree type B = 8 ft

We need to find after how many years both the trees will be of same height.

Solution,

Firstly we will convert the height of both plants into inches.

Since we know that 1 feet is equal to 12 inches.

So height of tree type A =[tex]5\ ft=5\times12=60\ in[/tex]

Similarly, height of tree type B =[tex]8\ ft=8\times12=96\ in[/tex]

Also given that;

Rate of growth of tree type A = 9 in/year

and rate of growth of tree type A = 6 in/year

Let the number of years be 'x'.

So according to question after 'x' years the height of both trees type A and type B will be same.

Now we can frame the equation as;

[tex]60+9x=96+6x[/tex]

Combining the like terms, we get;

[tex]9x-6x=96-60\\\\3x=36[/tex]

On dividing both side by '3' using division property, we get;

[tex]\frac{3x}{3}=\frac{36}{3}\\\\x=12[/tex]

Hence after 12 years height of both the trees would be same.

Answer:

(a)∴A=$6753.71.

(b)∴A=$1480.24

(c) ∴P=$7424.70

(d)∴P=$49759.62

(e)∴P=$5040.99

(f) ∴P=$2496.11

Step-by-step explanation:

We use the following formula

[tex]A=P(1+\frac rn)^{nt}[/tex]

A=amount in dollar

P=principal

r=rate of interest

(a)

P=$2,500, r=5%=0.05,t =20 years , n= 4

[tex]A=\$2500(1+\frac{0.05}{4})^{(20\times 4)[/tex]

   =$6753.71

∴A=$6753.71.

(b)

P=$1,000, r=8% =0.08,t =5 years , n= 2

[tex]A=\$1000(1+\frac{0.08}{2})^{(5\times 2)[/tex]

   =$1480.24

∴A=$1480.24

(c)

A=$10,000, r=6% =0.06,t =5 years , n= 4

[tex]10000=P(1+\frac{0.06}{4})^{(5\times 4)}[/tex]

[tex]\Rightarrow P=\frac{10000}{(1+0.015)^{20}}[/tex]

[tex]\Rightarrow P=7424.70[/tex]

∴P=$7424.70

(d)

A=$100,000, r=6% =0.06,t =10 years , n= 12

[tex]100000=P(1+\frac{0.07}{12})^{(10\times 12)}[/tex]

[tex]\Rightarrow P=\frac{100000}{(1+\frac{0.07}{12})^{120}}[/tex]

[tex]\Rightarrow P=49759.62[/tex]

∴P=$49759.62

(e)

A=$100,000, r=10% =0.10,t =30 years , n= 12

[tex]100000=P(1+\frac{0.10}{12})^{(30\times 12)}[/tex]

[tex]\Rightarrow P=\frac{100000}{(1+\frac{0.10}{12})^{360}}[/tex]

[tex]\Rightarrow P=5040.99[/tex]

∴P=$5040.99

(f)

A=$100,000, r=7% =0.07,t =20 years , n= 4

[tex]100000=P(1+\frac{0.07}{4})^{(20\times 4)}[/tex]

[tex]\Rightarrow P=\frac{100000}{(1+\frac{0.07}{4})^{80}}[/tex]

[tex]\Rightarrow P=2496.11[/tex]

∴P=$2496.11

To find the balance A, we use the formula A = P(1 + r/n)^(nt). To find the principal P, we rearrange the formula to P = A / ((1 + r/n)^(nt)).

(a) To find the balance A, we can use the formula A = P(1 + r/n)^(nt). Plugging in the given values, we have:

A = $2,500(1 + 0.05/4)^(4*20) = $2,500(1.0125)^80 ≈ $9,005.29

(b) Using the same formula, we can calculate:

A = $1,000(1 + 0.08/2)^(2*5) = $1,000(1.04)^10 ≈ $1,483.11

(c) To find the principal P, we rearrange the formula: P = A / ((1 + r/n)^(nt)). Plugging in the given values, we get:

P = $10,000 / ((1 + 0.06/4)^(4*5)) ≈ $7,772.22

(d) Using the rearranged formula, we can calculate:

P = $50,000 / ((1 + 0.07/12)^(12*10)) ≈ $23,022.34

(e) Since the compounding is monthly, we need to calculate the value of r/n first: r/n = 0.10/12 ≈ 0.0083. Plugging in the values, we have:

P = $100,000 / ((1 + 0.0083)^(12*30)) ≈ $3,791.61

(f) Similarly, we calculate:

P = $100,000 / ((1 + 0.07/4)^(4*20)) ≈ $24,084.48

Option B:

The relationship between angle j and angle k is j° + k° = 180°.

Solution:

Given JKLM is a parallelogram.

angle J and angle K are consecutive angles.

To find the relationship between angle j and angle k:

In parallelogram, the sum of the consecutive angles is 180°.

⇒ m∠J + m∠K = 180°

⇒ j° + k° = 180°

Hence the relationship between angle j and angle k is j° + k° = 180°.

Option B is the correct answer.

In parallelogram JKLM, angle J and angle K are congruent or have the same measure.

In parallelogram JKLM, angle J and angle K are congruent. This means that they have the same measure.

A parallelogram is a quadrilateral with opposite sides that are parallel. Since the opposite sides of a parallelogram are parallel, the opposite angles are also congruent.

Therefore, angle J and angle K in parallelogram JKLM have the same measure.

See the attached picture:

Answer:

  see below

Step-by-step explanation:

Rotation problems can be worked fairly easily if you have tracing paper or a transparency. Overlay the (semi-)transparent material on the given graph and trace the axes and figure. Then rotate the material according to the directions and copy the new position back to the graph.

(I find this much easier than trying to figure the coordinates.)

The quadratic function in vertex form to model the height of the flea compared to the horizontal distance travelled is h = -2/15*(d-15)² + 30 with the maximum point at (15, 30) and passing through the origin.

The problem here can be diagnosed using concepts of

quadratic functions

and

vertex form

. In a real world scenario, the motion of a projectile like the flea jumping can be modeled using a downward opening parabola represented by a quadratic function. In this case, we are asked to find the quadratic function in vertex form, which is given by

h = a(d - h1)² + k

where (h1,k) is the vertex of the parabola. In the given scenario, the maximum height attained by the flea is 30 cm which is at a horizontal distance of 15 cm from the starting point, thus the vertex of the parabola is (15, 30). From the information given, we know that the flea starts from the ground, so at the origin, height h = 0. Substituting these values, we get the equation of the parabola as

h = -a(d-15)^2 + 30

.

To find the value of 'a'

, we can use the information that the parabola passes through the origin (0,0). Substituting these values in the equation, we get a = -30/225 = -2/15. Therefore, the quadratic function in vertex form to model the height of the flea compared to the horizontal distance travelled becomes

h = -2/15*(d-15)² + 30

.

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Answer: she bought 5 greeting cards at $2.50 each.

Step-by-step explanation:

Let x represent the number of greeting cards that she bought at $2.50 each.

Let y represent the number of greeting cards that she bought at $4 each.

She buys some greeting cards that cost $2.50 each and some greeting cards that cost $4 each. She buys 12 cards. This means that

x + y = 12

The total amount that she spent in buying the greeting cards is $40.50. It means that

2.5x + 4y = 40.5- - - - - - - - - - 1

Substituting x = 12 - y into equation 1, it becomes

2.5(12 - y) + 4y = 40.5

30 - 2.5y + 4y = 40.5

- 2.5y + 4y = 40.5 - 30

1.5y = 10.5

y = 10.5/1.5

y = 7

x = 12 - y = 12 - 7

x = 5

Maya bought 5 greeting cards that cost $2.50 each.

Step 1

Let's denote the number of greeting cards that cost $2.50 as x, and the number of greeting cards that cost $4 as y . We know two things from the problem statement:

1. Maya buys a total of 12 cards: x + y = 12 .

2. The total cost of the cards is $40.50: 2.50x + 4y = 40.50.

Now, we'll solve these equations simultaneously to find x .

From equation (1):

x + y = 12

y = 12 - x

Step 2

Substitute y = 12 - x into equation (2):

[tex]\[ 2.50x + 4(12 - x) = 40.50 \][/tex]

Expand and simplify:

[tex]\[ 2.50x + 48 - 4x = 40.50 \][/tex]

[tex]\[ -1.50x + 48 = 40.50 \][/tex]

Subtract 48 from both sides:

[tex]\[ -1.50x = 40.50 - 48 \][/tex]

[tex]\[ -1.50x = -7.50 \][/tex]

Step 3

Divide both sides by -1.50 to solve for x :

[tex]\[ x = \frac{-7.50}{-1.50} \][/tex]

[tex]\[ x = 5 \][/tex]

So, Maya bought x = 5 greeting cards that cost $2.50 each.

Verification:

Now, substitute x = 5 back into the equation y = 12 - x  to find y :

y = 12 - 5

y = 7

Check the total cost:

[tex]\[ 2.50 \cdot 5 + 4 \cdot 7 = 12.50 + 28 = 40.50 \][/tex]

Everything checks out correctly. Therefore, Maya bought 5 greeting cards that cost $2.50 each.

Answer:

4457400

Step-by-step explanation:

25C14 = 4457400

Not practical, these are too many

Answer:

4,457,400  combinations.

Step-by-step explanation:

The number of combinations of 14 from 25 is a very large number :

25C14 =   25! / 14! 11!

=  4,457,400.

With so many possible combinations it would not be practical to make a Cd for all these possibilities.

Answer:

Step-by-step explanation:

it is 11423 ft3

Final answer:

The difference in volume of the weather balloon from launch to burst is calculated using the formula for the volume of a sphere, considering the change in diameters from 5 ft to 28 ft. The resulting difference is approximately 11,428.89 cubic feet.

Explanation:

The question asks for the difference in volume of a weather balloon when it burst compared to at launch. To solve this, we use the formula for the volume of a sphere, which is V = \(\frac{4}{3}\)\(\pi\)r^3, where r is the radius of the sphere. Given that the diameter at launch was 5 ft and at burst was 28 ft, the radiuses would be 2.5 ft and 14 ft, respectively.

Volume at launch: V1 = \(\frac{4}{3}\)\(\pi\)(2.5)^3 \approx 65.45 cubic feet. Volume at burst: V2 = \(\frac{4}{3}\)\(\pi\)(14)^3 \approx 11,494.34 cubic feet. The difference in volume: V2 - V1 \approx 11,494.34 - 65.45 \approx 11,428.89 cubic feet.

Therefore, the difference in volume of the balloon when it burst compared to at launch is approximately 11,428.89 cubic feet.

*100% CORRECT ANSWERS

Question 1

Alan is writing out the steps using the "shortest Route Algorithm". On the second step, he just circled the route ABD as the shortest route from A to D. What should he cross out next?  

AD; 6  

Question 2

Beth is writing out the steps using the "Shortest Route Algorithm". She just finished writing out all the routes for the third step. What route should she circle next?  

ACE; 6  

Question 3

If you start at vertex A and use the "shortest route" algorithm, what would be the second path to be selected/highlighted?  

ACF  

(SEE ATTACHMENTS BELOW)

Answer: ACF

Step-by-step explanation: Starting at vertex A and using shortest routes algorithm the secondary route to be selected would be ACF = 1+2 = 3

The first route would be AC = 1. A vertex is a point where two straight lines meet or join, they are usually found in angles.

Answer:

a) For Bella, x has to be a positive even values

For Tito, x has to be a negative even values

b) For Bella, x = 4

For Tito, x = -4

Answer: Xn = n(n+1)/2

Step-by-step explanation:

Firstly, you work with asingle dot for each and let n= 1,2,3,4....

Now if you double the dots, it will form a rectangle and it is easier to work with many dots i.e just multiple n by n+1

Dots in rectangle= n(n+1)

But remember the number of dots were doubled therefore,

Dots in triangle = n(n+1)/2.

Answer:

See attached picture.

Step-by-step explanation:

See attached picture.

For remaining parts resubmit question.  

Answer: the distance of the boat from the base of the lighthouse is 192.9 m

Step-by-step explanation:

The scenario is represented in the right angle triangle shown in the attached photo.

Looking at triangle ABC, the height of the light house operator above sea level represents the opposite side of the right angle triangle.

Angle A = 10° because it is alternate to the angle of depression.

To determine AB, the distance of the boat from the base of the lighthouse, we would apply

the tangent trigonometric ratio which is expressed as

Tan θ, = opposite side/adjacent side. Therefore,

Tan 10 = 34/AB

AB = 34/Tan 10 = 34/0.1763

AB = 192.9 m

Answer:

= 1

Step-by-step explanation:

The demand for that college will be equal to 1 or it can be said as unit elastic demand over the tuition range. This means that the demand for the college would move proportionately with the tuition range of that college, since the change in revenue is due to the change in tuition only.

Hope this helps.

Good luck and cheers.

Answer:

demand for the college is equal to 1

Step-by-step explanation:

- We know that the change in Total Revenue is only the function of change in tuition fee only.

- The change in tution fee is subjected to 15% last year multiplied by the corresponding change in demand for the college will lead to a change in total revenue.

- The relation can be expressed as:

                                         ΔTR =  Δ P *ΔD

Where,

         ΔTR : Change in Total Revenue

         Δ P : Change in tuition fee

         ΔD : Change in demand.

- For TR to remain unchanged then ΔTR = 0. Hence,

                                         ΔTR = 0 = Δ P *ΔD

- We are given a change in Δ P = 15%, so that means for ΔTR = 0, the change in demand ΔD = 0.

- ΔD = 0, also means that the elasticity of the demand curve is perfectly elastic or in other words the demand for the college is equal to 1.

Step-by-step explanation:

Let us assume the two needed vehicles are A and Q.

Let P(A) be the probability of the vehicle A available when needed.

And, P(Q) be the probability of the vehicle Q available when needed.

Now, P(A) = 90 % = 0.90

⇒ P (not A) =  1 - P(A)  

                    =  1- 0.9 =  0.1

⇒ P (not A) = 0.1

Similarly,  P(Q) = 90 % = 0.90

⇒ P (not Q) =  1 - P(Q)  

                    =  1- 0.9 =  0.1

⇒ P (not Q) = 0.1

So, the probability that both the vehicles are NOT available when needed  

= P(not A) x P(not Q)  

= 0.1 x 0.1 = 0.01

Hence, the probability that neither vehicle is available at a given time is 0.01

Answer:

  see below

Step-by-step explanation:

In the attachment, the points are listed in the order given in the problem statement. (They are listed to the right of the "rotation matrix", with x-coordinates above y-coordinates.)

__

I really don't like to do repetitive calculations, so I try to use a graphing calculator or spreadsheet whenever possible. Angles are measured CCW.

As always, the rotation transformations are ...

  180° — (x, y) ⇒ (-x, -y)

  270° — (x, y) ⇒ (y, -x)

Answer:

y=2x+7

Step-by-step explanation:

2x−y+4=0

y=2x+4

m=2

Line are parallel, so their slope is the same.

A(-1,5)... x1 =-1,y1 =5

y-y1 =m(x-x1)

y1 - 5=2(x-(-1))

y1 - 5=2(x+1)

y1-5=2x+2

y=2x+2+5

y=2x+7

Answer: y = 2x + 7

Step-by-step explanation:

The equation of a straight line can be represented in the slope intercept form as

y = mx + c

Where

c represents y intercept

m represents the slope of the line.

The equation of the given line is

2x - y + 4 = 0

y = 2x + 4

Comparing with the slope intercept form, slope = 2

If two lines are parallel, it means that they have the same slope. Therefore, the slope of the line passing through (- 1, 5) is 2

To determine the y intercept, we would substitute m = 2, x = - 1 and y = 5 into y = mx + c. It becomes

5 = 2 × - 1 + c

5 = - 2 + c

c = 5 + 2 = 7

The equation becomes

y = 2x + 7

Answer:

[tex]f(x)=x^{4}+x^{3}-10x^{2} +8x[/tex]

Step-by-step explanation:

A number is a factor of f(x) if and only if f(x) is zero for that value/number.

For the factors of a function we write the factors as x-a where a is the zero of function i.e. value at which f(x) is zero.

To write the polynomial function of minimum degree with real coefficients whose zeros include 2, -4, and 1, 3i, we find the f(x) is the product of all factors i.e x-a where a will represent the given zeros.

[tex]f(x)=(x-2)(x-(-4))(x-1)(x-3i)\\f(x)=(x-2)(x+4))(x-1)(x-3i)\\f(x)=(x^{2} +4x-2x-8)(x^{2} -3xi-x+3i )\\f(x)=(x^{2} +2x-8)(x^{2} -x-(3x+3)i)\\[/tex]

As it is stated that polynomial should have real coefficients so skipping the terms with 'i' we get

[tex]f(x)=(x^{2} +2x-8)(x^{2} -x)\\f(x)=x^{4}-x^{3}+2x^{3}-2x^{2} -8x^{2} +8x\\f(x)=x^{4}+x^{3}-10x^{2} +8x[/tex]

Answer:

f(x) = x4 - 2x2 + 36x - 80

Step-by-step explanation:

Answer:

F=3

Step-by-step explanation:

Due to the difficulty of visualizing the graph of the function in degrees (graph 1), we will graph it in radians (graph 2)

f(x)=−sin(3x)−1 ≡ y=−sin(3x)−1

To graph y=−sin(3x)−1

y=a.sin(bx+c)+d, where

a=-1, b=3, c=0, d=-1 and the period (T) of the function  is:

[tex]T=\frac{2\pi }{b}=\frac{2\pi }{3}[/tex]

On the graph 2 we place the original function y=sin(x) to compare

We watch that y=−sin(3x)−1 moves 1 down (-), but amplitud is the same (1)

Frequency is the number of repetitions (3x) of a function in a given interval, so

F=3

The frequency of the function f(x) = -sin(3x) - 1 is 3/(2π), determined by the coefficient of x inside the sine function.

The frequency of the function f(x) = -sin(3x) - 1 can be determined by examining the coefficient of x within the sine function. The standard form for a sine function is f(x) = sin(Bx), and the frequency f is given by f = B/(2π). In this case, the coefficient B is 3, so the frequency of the function is 3/(2π), which is already in simplest fraction form.