question down below about linear equations and grapghing.

The correctly matched quadratic functions are given as follows:

Quadratics are polynomial equations of the second degree, which means that they contain at least one squared term.

Quadratic equations are another name for it. The quadratic equation has the following general form: ax2 + bx + c = 0.

Recall that:
f(x) - a (x -c) (x - d)

where

Part A) First graph

We are given to know

The solutions or zeros of the first graph are

x=-1 and x=3

The parabola open up, so the leading coefficient a is positive, the function therefore, is equal to

The vertex is equal to the point (1, -4)

Substitute and solve for a

-4 = a(1-3)(1+1)

-4 = a(-2)(2)

a = 1

Hence, the function is equal to f(x) = (x-3) (x + 1)

Part B) Second Graph

We are also given to know that

The solutions or zeros of the first graph are

x=-3 and x=1

The parabola open down, so the leading coefficient a is negative

The function is equal to f(x) = a (x-1)(x+3)

Find the value of the coefficient a

The vertex is equal to the point (-1,8)

to solve for a, we must substitute:

8 = a(-1-1) (-1+3)

8 = a(-2)(2)

a = -2

Hence, the function is equal to: f(x) = -2 (x - 1) ( x + 3)

Part C)

We are given to know that the solutions or zeros of the first graph are

x=-2 and x=6

The parabola open up, so the leading coefficient a is positive

The function is equal to f(x)  = a(x-6) (x+2).

Find the value of the coefficient a

The vertex is equal to the point (2,-8)

We substitute and solve for a:

-8 = a(2-6) (2+2)

-8 = a(-4)(4)

a = 0.5

Hence, the function is equal to f(x) = 0.5(x - 6) (x + 2)

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The surface area of this pyramid with a regular pentagon as its base is equal to 851.09 square units.

In Mathematics, a complete circle has a magnitude of 360° and when it is divided by the 10 triangles, the angle formed at the center of the pentagon would always be equal to 36°.

Also, the length of the apothem of a pentagon can be calculated by using this formula:

a = s/2tan(36)

Where:

Substituting the given parameters into the formula, we have;

a = 10/2tan(36)

a = 10/1.45

a = 6.90 units.

Next, we would determine the slant height of this pyramid by applying Pythagorean theorem:

c² = a² + b²

c² = 6.9² + 13²

c² = 47.61 + 169

c² = 216.1

c = √216.1

c = 14.72 units.

For the surface area, we have:

Mathematically, the surface area of a pyramid with a regular pentagon as its base can be calculated by using this formula:

SA = 1.72l² × (5/2 × l × h)

SA = 1.72(14.72)² × (5/2 × 14.72 × 13)

SA = 372.69 + 478.4

SA = 851.09 square units.

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#43

So

200/2=100 pairs of 201

Sum

#44

400/2=200 pairs of 401

Sum

#45

800/2=400 pairs of 801

Sum

#46

2000/2=1000 pairs of 2001

Sum

Answer:

43.  20,100

44.  80,200

45.  320,400

46.  2,001,000

Step-by-step explanation:

Gauss's method

General formula for the sum of the first n integers:

[tex]1+2+3+ \dots +n=\dfrac{1}{2}n(n+1)[/tex]

Question 43

Given sum:  

Therefore, n = 200.

Substitute the value of n into the formula:

[tex]\implies \dfrac{1}{2}(200)(200+1)=100 \cdot 201=20100[/tex]

Question 44

Given sum:  

Therefore, n = 400.

Substitute the value of n into the formula:

[tex]\implies \dfrac{1}{2}(400)(400+1)=200 \cdot 401=80200[/tex]

Question 45

Given sum:  

Therefore, n = 800.

Substitute the value of n into the formula:

[tex]\implies \dfrac{1}{2}(800)(800+1)=400 \cdot 801 =320400[/tex]

Question 46

Given sum:  

Therefore, n = 2000.

Substitute the value of n into the formula:

[tex]\implies \dfrac{1}{2}(2000)(2000+1)=1000 \cdot 2001=2001000[/tex]

[tex]\frac{-2x^{2} +25 }{(x-2) (x-5)} ,x\neq 2,x\neq 5[/tex]  is the difference.

What is an expression?

⇒ An expression is a combination of numbers, variables, and functions (such as addition, subtraction, multiplication or division, etc.)

Calculation :

⇒   [tex]\frac{3x^{2} }{(x^{2} -7x+10)}- \frac{5x} {x-2}[/tex]

⇒   [tex]\frac{3x^{2} }{(x-2)(x-5)}- \frac{5x} {x-2}[/tex]

⇒1/(x-2) [(3x²-5x{x-5})/(x-5)]

⇒1/(x-2) [(3x²-5x²+25x)/(x-5)]

⇒1/(x-2) [ (-2x²+25x)/(x-5)]

⇒[tex]\frac{-2x^{2} +25x }{(x-2)(x-5)}[/tex]

at x=2 and x=5 the denominator becomes zero If the denominator of a fraction is zero, the expression is not a legal fraction because its overall value is undefined.

⇒   [tex]\frac{-2x^{2} +25 }{(x-2) (x-5)} ,x\neq 2,x\neq 5[/tex] is the difference

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Answer:

y = - 5x² - 10x + 2

Step-by-step explanation:

y = - 5(x + 1)² + 7 ← expand (x + 1)² using FOIL

  = - 5(x² + 2x + 1) + 7 ← distribute parenthesis by - 5

 = - 5x² - 10x - 5 + 7 ← collect like terms

 = -5x² - 10x + 2 ← in standard form

Answer: y = -5x² - 10x + 2

Step-by-step explanation:

Given quadratic function

y = -5 (x + 1)² + 7

Given requirement

Quadratic Standard Form: y = ax² + bx + c

Simplify the exponents

y = -5 (x + 1) (x + 1) + 7

y = -5 (x² + x + x + 1) + 7

y = -5 (x² + 2x + 1) + 7

Expand the parenthesis by distributive property

y = (-5) · x² + (-5) · 2x + (-5) · 1 + 7

y = -5x² + (-10x) + (-5) + 7

y = -5x² - 10x - 5 + 7

Combine like terms

[tex]\Large\boxed{y=-5x^2-10x+2}[/tex]

Hope this helps!! :)

Please let me know if you have any questions

Answer:

Step-by-step explanation:

y=-1/2x+6

The amount saved in 10 years is = $1,04,461.13

The correct option is B.

An annuity is a series of payments that are distributed evenly over time.

One way to express the future value of an annuity of payment P for n years is,

Total salary is $65000

The interest rate is 5.32%

Trent save 8%  of his salary every year.

Company puts  4.6% of salary every year.

[tex]\text { Future Value }=\mathrm{P} \times \frac{\left(1+\frac{i}{n}\right)^{n t}}{\frac{i}{n}}[/tex]

Given:

Total salary is $65000

The interest rate is 5.32%

Trent save 8% of his salary every year.

Company puts  4.6%of salary every year.

The 8% of salary is calculated as:

[tex]\text { Amount } &=\frac{8}{100} \times 65000\\[/tex]

               [tex]&=\$ 5200[/tex]

The following information is available on the company's put:

[tex]\text { Amount } &=\frac{4.6}{100} \times 65000\\[/tex]

                [tex]&=\$ 2990[/tex]

Total amount as follows:

[tex]\begin{aligned}\text { Amount } &=5200+2990 \\&=\$ 8190\end{aligned}[/tex]

The future value can be obtained as follows

[tex]\begin{aligned}\mathrm{FV} &=8190 \times \frac{(1+0.053)^{10}-1}{0.053} \\&=8190 \times 12.75 \\&=1,04,461.13\ \\\end{aligned}[/tex]

Hence, the amount saved in 10 years is = $1,04,461.13

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By substitution, the value of x and y are 1 and 1 respectively.

3 / 8 x + 1 /3  y = 17 / 24

x + 7y = 8

x = 8 - 7y

Let's substitute the value of x in equation(ii)

3 / 8(8 - 7y) + 1 / 3 y = 17 / 24

3 - 21 / 8 y + 1 / 3 y = 17 / 24

1 / 3 y - 21 / 8 y = 17 / 24 - 3

8y - 63y/24 =  17 - 72/ 24

-55y / 24 = -55 / 24

cross multiply

-1320y = -1320

divide both sides by -1320

y = -1320 / -1320

y = 1

Hence,

x + 7(1) = 8

x = 8 - 7

x = 1

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The vertex of the function is a minimum and the coordinate of the vertex of the function is (0.2, -2.4)

The function is given as:

f(x) = 5x^2 + 2x - 3

Expand the function

f(x) = 5x^2 + 5x - 3x - 3

Factorize the function

f(x) = (5x - 3)(x + 1)

Set the function to 0

(5x - 3)(x + 1) = 0

Solve for x

x = 3/5 and x =-1

Hence, the x-intercept is 3/5 and -1

The vertex of the function is a minimum.

This is so because the leading coefficient of the function is positive

Here, we have:

f(x) = 5x^2 + 2x - 3

Differentiate and set to 0

10x + 2 = 0

Solve for x

x = -0.2

Substitute x = -0.2 in f(x) = 5x^2 + 2x - 3

f(0.2) = 5(0.2)^2 + 2(0.2) - 3

Evaluate

f(0.2) = -2.4

Hence, the vertex of the function is (0.2, -2.4)

To do this, we simply plot the x-intercept and the vertex.

And then connect the points

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Based on the calculations, the sum of this geometric series is equal to 9,990.

Mathematically, the standard form of a geometric series can be represented by the following expression:

[tex]\sum^{n-1}_{k=0}a_1(r)^k[/tex]

Where:

Also, the sum of a geometric series is given by this mathematical expression:

[tex]S=\frac{a_1(1-r^n)}{1-r}[/tex]

Given the following data:

Substituting the given parameters into the formula, we have;

[tex]S=\frac{9000(1-0.1^3)}{1-0.1}\\\\S=\frac{9000(1-0.001)}{1-0.1}[/tex]

S = 9000(0.999)/0.9

S = 8,991/0.9

S = 9,990.

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The value of integration of y=16-[tex]x^{2}[/tex] from x=-1 to x=1 is 94/3.

Given the equation y=16-[tex]x^{2}[/tex] and the limit of the integral be x=-1,x=1.

We are required to find the value of integration of y=16-[tex]x^{2}[/tex] from x=-1 to x=1.

Equation is relationship between two or more variables that are expressed in equal to form.Equation of two variables look like ax+by=c.It may be linear equation, quadratic equation, or many more depending on the power of variable.

Integration is basically opposite of differentiation.

y=16-[tex]x^{2}[/tex]

Find the integration of 16-[tex]x^{2}[/tex].

=16x-[tex]x^{3}/3[/tex]

Now find the value of integration from x=-1 to x=1.

=16(1)-[tex](1)^{3}/3[/tex]-16(-1)-[tex](-1)^{3}/3[/tex]

=16(1)-1/3+16-1/3

=32-2/3

=(96-2)/3

=94/3

Hence the value of integration of y=16-[tex]x^{2}[/tex] from x=-1 to x=1 is 94/3.

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Answer:

  24 m depth

Step-by-step explanation:

The quadratic function modeling depth will have zeros at x=1 and x=12, the distances in meters from the boat where the diver is at the surface.

The factored form of a quadratic function can be written as ...

  y = a(x -p)(x -q)

where p and q are zeros of the function. The value 'a' is a vertical scale factor.

Here, we are given y=0 at the surface, at points where x=1 and x=12. This means the function can be written as ...

  y = a(x -1)(x -12)

To find the value of 'a', we can use the maximum depth value. That depth will be halfway between the function zeros, at x = (1+12)/2 = 6.5

  -30.25 = a(6.5 -1)(6.5 -12)

  30.25 = 5.5²·a = 30.25a   ⇒   a=1

Then the function modeling the diver's depth in meters is ...

  y = (x -1)(x -12)

And the depth at x=4 will be ...

  y = (4 -1)(4 -12) = 3(-8) = -24 . . . . meters

The diver's depth is 24 meters when she is 4 m away from the boat.

Answer:

[tex]\textsf{19.} \quad S_{20}=1490[/tex]

33.  None of these are correct.

Step-by-step explanation:

General form of an arithmetic sequence:

[tex]a_n=a+(n-1)d[/tex]

where:

Given values:

Substitute the given values into the formula and solve for a:

[tex]\implies 27=a+(20-1)(-5)[/tex]

[tex]\implies 27=a+(19)(-5)[/tex]

[tex]\implies 27=a-95[/tex]

[tex]\implies a=27+95[/tex]

[tex]\implies a=122[/tex]

Sum of the first n terms of an arithmetic series:

[tex]S_n=\dfrac12n[2a+(n-1)d][/tex]

where:

Given values:

Substitute the given values into the formula and solve:

[tex]\implies S_{20}=\dfrac{1}{2}(20)[2(122)+(20-1)(-5)][/tex]

[tex]\implies S_{20}=10[244-95][/tex]

[tex]\implies S_{20}=10[149][/tex]

[tex]\implies S_{20}=1490[/tex]

Sum of the first n terms of a geometric series:

[tex]S_n=\dfrac{a(1-r^n)}{1-r}[/tex]

where:

Given values:

Substitute the given values into the formula and solve:

[tex]\implies S_6=\dfrac{1458\left(1-\frac{1}{3}^6\right)}{1-\frac{1}{3}}[/tex]

[tex]\implies S_6=\dfrac{1456}{\frac{2}{3}}[/tex]

[tex]\implies S_6=\dfrac{1456 \cdot 3}{2}[/tex]

[tex]\implies S_6=2184[/tex]

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Answer:

[tex]z=\frac{t^{2}(m-3) }{4}[/tex]    or    [tex]z=\frac{1}{4}t^{2} (m-3)[/tex]

Step-by-step explanation:

[tex]t=\sqrt{\frac{4z}{m-3} }[/tex]

[tex]t^{2} =\frac{4z}{m-3}[/tex]

[tex]4z=t^{2} (m-3)[/tex]

[tex]z=\frac{t^{2}(m-3) }{4}[/tex]

Hope this helps

Based on the information given about the temperature, Daniela made her error on step 3 because it says: "Find the ratio of the differences of the means compared to the mean absolute deviation.

From the information given, it van be seen that Daniela recorded the low temperatures during the school day last week and this week and her results are shown in the table.

Here, she didn't find the difference between the today and last week means. She just put the mean as a whole.

Therefore, Daniela made her error on step 3 because it says: "Find the ratio of the differences of the means compared to the mean absolute deviation. This was the error.

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Given the equation written by Fiona and Henry, If both linear equations have the same solutions, Henry's equation is x - 5/2y = 25/2.

Hence, option D is the correct answer.

This question is incomplete, the missing answer choices are;

A. x- 5/4y =25/4

B. x-5/2y=25/4

C. x-5/4y =25/4

D. x- 5/2y=25/2

Given the linear equation written by Fiona; y = 2/5x - 5.

From the answer choices provided, they are in the form of x - (m)y = b.

We will transform Fiona's equation into that form.

y = 2/5x - 5

Divide each term by the coefficient of x.

y(5/2) = (5/2)2/5x - (5/2)5

5/2y = x - 25/2

25/2 = x - 5/2y

x - 5/2y = 25/2

Given the equation written by Fiona and Henry, If both linear equations have the same solutions, Henry's equation is x - 5/2y = 25/2.

Hence, option D is the correct answer.

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x=-2

Step-by-step explanation:

Solution Given:

3(x-3) = 5(2x + 1)

Opening bracket we get

3x-9=10x+5

keeping common value in same side

-9-5=10x-3x

-14=7x

x=-14/7

x= -2

Answer:

-2 =x

Step-by-step explanation:

3(x-3) = 5(2x + 1)

To solve for x

Distribute

3x -9 = 10x +5

Subtract 3x from each side

3x-9-3x = 10x -3x+5

-9 = 7x+5

Subtract 5 from each side

-9-5 = 7x+5-5

-14 = 7x

Divide by 7

-14/7 = 7x/7

-2 =x

The height of the cone in terms of y is h = 12x⁴y

The cylinder and the cone have the same volume.

Volume of a cylinder = πr²h

where

Therefore,

Volume of a cylinder = πx²y

volume of a cone = 1 / 3 πr²h

where

Therefore,

πx²y = 1 / 3 × π × (1 / 2x)² × h

πx²y = πh / 12x²

πx²y × 12x² / π = h

h = 12x⁴y

Therefore, the height of the cone in terms of y is h = 12x⁴y

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Using it's concept, the probability that two red marbles are chosen from the vase is of 0%.

A probability is given by the number of desired outcomes divided by the number of total outcomes.

In this problem, we have that there are no red marbles, hence the number of desired outcomes is of 0, and there is a 0% probability that two red marbles are chosen from the vase.

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Answer:

93.32 %

Step-by-step explanation:

33.2   is    1.2   away from 32     this is  1.2 / .8 = 1.5 standard deviations ABOVE the mean     z -score = + 1.5

from z-score table this corresponds to .9332     this is  93.32 %

The probability that a randomly selected passenger car gets more than 33.2 mpg is 0.14.

The p-value, used in null-hypothesis significance testing, represents the likelihood that the test findings will be at least as extreme as the result actually observed, assuming that the null hypothesis is true.

Let the random variable X represent the miles-per-gallon rating of passenger cars. Compute the probability that a randomly selected passenger car gets more than 33.2 mpg as follows:

The probability will be calculated as below;-

[tex]P(x > 33.2) = p(\dfrac{x-\mu}{\sigma} > \dfrac{33.2 - 32}{0.8})[/tex]

P( x > 33.2) = 1 - P ( z < 1 )

P( x > 33.2) = 1 - 0.86

P( x > 33.2) = 0.14

Thus, the probability that a randomly selected passenger car gets more than 33.2 mpg is 0.14.

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Answer:

Step-by-step explanation:

A square is made up of 4 sides that are equal in length. They can all be called "x" or "s" or whatever variable you want. The formula for the area of a square (and also the area of a rectangle) is Area = side × side.

If you called your sides "x", then the formula is A = x × x or A = x². Another way to note the area of either a square or a rectangle is to call the "bottom" the base and one of the sides the height and then the area formula looks like this:

A = b × h

or

A = length × width

They are all the same thing.

Answer:

There are two solutions:  
x = 0, y = -3  Point (0, -3)
x = 3, y = -6  Point (3, 6)

The solutions are at the intersections of the two individual graphs corresponding to each of the two equations (see attached figure)

Step-by-step explanation:

Use a graphing calculator or tool to plot each equation. The point of intersection of the two is the solution to the system of equations. See attached image

Algebraically

The first equation must be equal to the second equation for a unique value of y and x

[tex]y = x^2 - 3 = 3x -3\\x^2 - 3 = 3x - 3[/tex]

Subtract 3x - 3 from both sides giving

[tex]x^2 - 3 - (3x -3) = 0\\x^2 -3 -3x + 3 = 0\\x^2 - 3x = 0\\[/tex]

Factoring out x we get
[tex]x(x-3) = 0[/tex]

This means that x = 0 and also
x -3 = 0, or x = 3

Substituting for x = 0 in y = 3x-3 gives
y = 3.0 - 3 = -3
So x= 0, y = -3 is one solution (0,-3)

Substituting for x = 3 in the same equation gives
y = 3(3) - 3 = 6
So x = 3, y = 6 is another solution (3, 6)

5. Alternate interior angles: ∠1 ≅ ∠5; ∠5 ≅ ∠7 and ∠4 ≅ ∠6

6. Alternate exterior angles: ∠3 ≅ ∠9 and ∠2 ≅ ∠8

7. Corresponding angles: ∠2 ≅ ∠6; ∠3 ≅ ∠7; ∠5 ≅ ∠9; ∠4 ≅ ∠8 and ∠1 ≅ ∠3

8. Vertical angles: ∠2 ≅ ∠4; ∠3 ≅ ∠5; ∠7 ≅ ∠9; and ∠6 ≅ ∠8 .

9.  ∠1 and ∠7.

If two lines that are cut across by a transversal are parallel, the following special angles are formed:

Given the image given, we can identify the following special angles:

5. Alternate interior angles: ∠1 ≅ ∠5; ∠5 ≅ ∠7 and ∠4 ≅ ∠6

6. Alternate exterior angles: ∠3 ≅ ∠9 and ∠2 ≅ ∠8

7. Corresponding angles: ∠2 ≅ ∠6; ∠3 ≅ ∠7; ∠5 ≅ ∠9; ∠4 ≅ ∠8 and ∠1 ≅ ∠3

8. Vertical angles: ∠2 ≅ ∠4; ∠3 ≅ ∠5; ∠7 ≅ ∠9; and ∠6 ≅ ∠8 .

9.  ∠1 ≅ ∠3 and ∠3 ≅ ∠7. So the two angles would be ∠1 ≅ ∠7.

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The equation represents a population of 370 animals that decreases at an annual rate of 11% is [tex]p=370(0.89)^{t}[/tex].

It is defined as the function that rapidly decreases and the value of the exponential function is always positive. It denotes with exponent y = a^x

where a is a constant and a>1.

Given that,

A population of 370 animals that decreases at an annual rate of 11%

a = 370, r = 11%

p = [tex]a(1-r)^{t}[/tex]

 =  [tex]370(1-0.11)^{t}[/tex]

p  =  [tex]370(0.89)^{t}[/tex]

Hence, The equation represents a population of 370 animals that decreases at an annual rate of 11% is [tex]p=370(0.89)^{t}[/tex].

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Answer:

[tex]\Large\boxed{1)~6x+7}[/tex]

[tex]\Large\boxed{2)~-7x+5y+8}[/tex]

Step-by-step explanation:

Given expression

4x + 7 + 2x

Move like terms together

= 4x + 2x + 7

= (4x + 2x) + 7

Combine like terms

[tex]\Large\boxed{=6x+7}[/tex]

Given expression

-4x + 3y - 3x + 8 + 2y

Move like terms together

= -4x - 3x + 3y + 2y + 8

= (-4x - 3x) + (3y + 2y) + 8

Combine like terms

[tex]\Large\boxed{=-7x+5y+8}[/tex]

Hope this helps!! :)

Please let me know if you have any questions

Answer:

24.315°

Step-by-step explanation:

First use the pythagorean theorem to get the missing side since this is a right triangle.

a^2 + b^2 = c^2 →

b^2 = c^2 – a^2 →

b = (c^2 – a^2)^½

b = (17^2 – 7^2)^½

b = (289 – 49)^½

b = √240.

Then use the law of cosines to find the angle in between.

b^2 = a^2 + c^2 - 2ac * cos(B) →

b^2 - a^2 - c^2 = -2ac * cos(B) →

a^2 + c^2 – b^2 / 2ac = cos(B) →

cos^-1(a^2 + c ^2 – b^2 / 2ac) = C →

B = cos^-1(a^2 + c ^2 – b^2 / 2ac)

B = cos^-1(49 + 289 – √240 / 238)

B ≈ 75.685°.

A = 180 – B

A = 180 – 75.685

A = 24.315°

Answer:

x = 7; y = 4 1/3

Step-by-step explanation:

The solution is the point of intersection of the lines.

Read the point of intersection.

You can also solve the system using algebra.

y = 1/3 x + 2

y = 4/3 x - 5

1/3 x + 2 = 4/3 x - 5

x + 6 = 4x - 15

-3x = -21

x = 7

y = 1/3 x + 2

y = 1/3 × 7 + 2

y = 2 1/3 + 2

y = 4 1/3

Answer: x = 7; y = 4 1/3

There will be a vertical stretch with a factor of 3.