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blagie [28]
3 years ago
15

Write an equation of the translated function of the parent function f(x)=|x| with the following transformations:

Mathematics
1 answer:
MrRa [10]3 years ago
8 0

Answer:

Option b

Step-by-step explanation:

To write the searched equation we must modify the function f (x) = | x | in the following way:

1. Do y = f(x + 4)

This operation horizontally shifts the function f(x) = | x | by a factor of 4 units to the left on the x axis.

y = | x +4 |

2. Do y = f (\frac{1}{4}x)

This operation horizontally expands the function f (x) = | x | in a factor of 4 units. y = |\frac{1}{4}x + 1|

3. Do y = f(x)-4

This operation vertically shifts the function f (x) = | x | by a factor of 4 units down on the y-axis.

y = |\frac{1}{4}x +1| -4

4. After these transformations the function f(x) = | x | it looks like:

f(x) = |\frac{1}{4}x +1| -4

Therefore the correct option is option b. You can verify that your vertex is at point (-4, -4) by making f (-4)

f(-4) = |\frac{1}{4}(-4) +1| -4 = -4

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Assume f(x) = g(x). Which of the following pairsof functions may be used to represent theequation 3^x+^2 = 7x + 6?
Aloiza [94]
Functions and equations

We have that:

3^{x+2}^{}=7x+6

Let's name each side of it with f(x) and g(x):

Then, we have that:

\begin{gathered} f(x)=3^{x+2} \\ \text{and} \\ g\mleft(x\mright)=7x+6 \end{gathered}

Then, the answer is C

<h2>Answer: C</h2>

4 0
1 year ago
Sail with the equation by completing the square x^2+8x=-3
maxonik [38]

Answer:

The solutions are x=-4(+/-)\sqrt{13}

Step-by-step explanation:

we have

x^{2} +8x=-3

Complete the square. Remember to balance the equation by adding the same constants to each side

x^{2} +8x+16=-3+16

x^{2} +8x+16=13

Rewrite as perfect squares

(x+4)^{2}=13

square root both sides

x+4=(+/-)\sqrt{13}

x=-4(+/-)\sqrt{13}

5 0
3 years ago
HELPPPP. Need help asap.
mamaluj [8]

Answer:

Just learned this-

They are both 1 because of it being to the power of zero and if you type the both in like...

Step-by-step explanation:

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6 0
3 years ago
What is the sum of 9+10 (;
garri49 [273]

Answer:

19 or 21.

Step-by-step explanation:

I swear.

5 + 5 = 10

10 + 5 = 15

15 + 4 = 19

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9 + 9 = 18

18 + 10 = 28

28 - 5 = 23

There are two nines so

23 - 2 = 21

6 0
3 years ago
Using Laplace transforms, solve x" + 4x' + 6x = 1- e^t with the following initial conditions: x(0) = x'(0) = 1.
professor190 [17]

Answer:

The solution to the differential equation is

X(s)=\cfrac 1{6}  -\cfrac {1}{11}e^{t}+\cfrac {61}{66}e^{-2t}\cos(\sqrt 2t)+\cfrac {97}{66}\sqrt 2 e^{-2t}\sin(\sqrt 2t)

Step-by-step explanation:

Applying Laplace Transform will help us solve differential equations in Algebraic ways to find particular  solutions, thus after applying Laplace transform and evaluating at the initial conditions we need to solve and apply Inverse Laplace transform to find the final answer.

Applying Laplace Transform

We can start applying Laplace at the given ODE

x''(t)+4x'(t)+6x(t)=1-e^t

So we will get

s^2 X(s)-sx(0)-x'(0)+4(sX(s)-x(0))+6X(s)=\cfrac 1s -\cfrac1{s-1}

Applying initial conditions and solving for X(s).

If we apply the initial conditions we get

s^2 X(s)-s-1+4(sX(s)-1)+6X(s)=\cfrac 1s -\cfrac1{s-1}

Simplifying

s^2 X(s)-s-1+4sX(s)-4+6X(s)=\cfrac 1s -\cfrac1{s-1}

s^2 X(s)-s-5+4sX(s)+6X(s)=\cfrac 1s -\cfrac1{s-1}

Moving all terms that do not have X(s) to the other side

s^2 X(s)+4sX(s)+6X(s)=\cfrac 1s -\cfrac1{s-1}+s+5

Factoring X(s) and moving the rest to the other side.

X(s)(s^2 +4s+6)=\cfrac 1s -\cfrac1{s-1}+s+5

X(s)=\cfrac 1{s(s^2 +4s+6)} -\cfrac1{(s-1)(s^2 +4s+6)}+\cfrac {s+5}{s^2 +4s+6}

Partial fraction decomposition method.

In order to apply Inverse Laplace Transform, we need to separate the fractions into the simplest form, so we can apply partial fraction decomposition to the first 2 fractions. For the first one we have

\cfrac 1{s(s^2 +4s+6)}=\cfrac As + \cfrac {Bs+C}{s^2+4s+6}

So if we multiply both sides by the entire denominator we get

1=A(s^2+4s+6) +  (Bs+C)s

At this point we can find the value of A fast if we plug s = 0, so we get

1=A(6)+0

So the value of A is

A = \cfrac 16

We can replace that on the previous equation and multiply all terms by 6

1=\cfrac 16(s^2+4s+6) +  (Bs+C)s

6=s^2+4s+6 +  6Bs^2+6Cs

We can simplify a bit

-s^2-4s=  6Bs^2+6Cs

And by comparing coefficients we can tell the values of B and C

-1= 6B\\B=-1/6\\-4=6C\\C=-4/6

So the separated fraction will be

\cfrac 1{s(s^2 +4s+6)}=\cfrac 1{6s} +\cfrac {-s/6-4/6}{s^2+4s+6}

We can repeat the process for the second fraction.

\cfrac1{(s-1)(s^2 +4s+6)}=\cfrac A{s-1} + \cfrac {Bs+C}{s^2+4s+6}

Multiplying by the entire denominator give us

1=A(s^2+4s+6) + (Bs+C)(s-1)

We can plug the value of s = 1 to find A fast.

1=A(11) + 0

So we get

A = \cfrac1{11}

We can replace that on the previous equation and multiply all terms by 11

1=\cfrac 1{11}(s^2+4s+6) + (Bs+C)(s-1)

11=s^2+4s+6 + 11Bs^2+11Cs-11Bs-11C

Simplifying

-s^2-4s+5= 11Bs^2+11Cs-11Bs-11C

And by comparing coefficients we can tell the values of B and C.

-s^2-4s+5= 11Bs^2+11Cs-11Bs-11C\\-1=11B\\B=-\cfrac{1}{11}\\5=-11C\\C=-\cfrac{5}{11}

So the separated fraction will be

\cfrac1{(s-1)(s^2 +4s+6)}=\cfrac {1/11}{s-1} + \cfrac {-s/11-5/11}{s^2+4s+6}

So far replacing both expanded fractions on the solution

X(s)=\cfrac 1{6s} +\cfrac {-s/6-4/6}{s^2+4s+6} -\cfrac {1/11}{s-1} -\cfrac {-s/11-5/11}{s^2+4s+6}+\cfrac {s+5}{s^2 +4s+6}

We can combine the fractions with the same denominator

X(s)=\cfrac 1{6s}  -\cfrac {1/11}{s-1}+\cfrac {-s/6-4/6+s/11+5/11+s+5}{s^2 +4s+6}

Simplifying give us

X(s)=\cfrac 1{6s}  -\cfrac {1/11}{s-1}+\cfrac {61s/66+158/33}{s^2 +4s+6}

Completing the square

One last step before applying the Inverse Laplace transform is to factor the denominators using completing the square procedure for this case, so we will have

s^2+4s+6 = s^2 +4s+4-4+6

We are adding half of the middle term but squared, so the first 3 terms become the perfect  square, that is

=(s+2)^2+2

So we get

X(s)=\cfrac 1{6s}  -\cfrac {1/11}{s-1}+\cfrac {61s/66+158/33}{(s+2)^2 +(\sqrt 2)^2}

Notice that the denominator has (s+2) inside a square we need to match that on the numerator so we can add and subtract 2

X(s)=\cfrac 1{6s}  -\cfrac {1/11}{s-1}+\cfrac {61(s+2-2)/66+316 /66}{(s+2)^2 +(\sqrt 2)^2}\\X(s)=\cfrac 1{6s}  -\cfrac {1/11}{s-1}+\cfrac {61(s+2)/66+194 /66}{(s+2)^2 +(\sqrt 2)^2}

Lastly we can split the fraction one more

X(s)=\cfrac 1{6s}  -\cfrac {1/11}{s-1}+\cfrac {61(s+2)/66}{(s+2)^2 +(\sqrt 2)^2}+\cfrac {194 /66}{(s+2)^2 +(\sqrt 2)^2}

Applying Inverse Laplace Transform.

Since all terms are ready we can apply Inverse Laplace transform directly to each term and we will get

\boxed{X(s)=\cfrac 1{6}  -\cfrac {1}{11}e^{t}+\cfrac {61}{66}e^{-2t}\cos(\sqrt 2t)+\cfrac {97}{66}\sqrt 2 e^{-2t}\sin(\sqrt 2t)}

6 0
3 years ago
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