If f(x) = (x-3)^2+1, what is (6)

Answer:

Step-by-step explanation:

The domain of all polynomials is all real numbers.  To find the range, let's solve that quadratic for its vertex.  We will do this by completing the square.  To begin, set the quadratic equal to 0 and then move the -10 over by addition. The first rule is that the leading coefficient has to be a 1; ours is a 2 so we factor it out.  That gives us:

[tex]2(x^2-2x)=10[/tex]

The second rule is to take half the linear term, square it, and add it to both sides.  Our linear term is 2 (from the -2x).  Half of 2 is 1, and 1 squared is 1.  So we add 1 into the parenthesis on the left.  BUT we cannot ignore the 2 sitting out front of the parenthesis.  It is a multiplier.  That means that we didn't just add in a 1, we added in a 2 * 1 = 2.  So we add 2 to the right as well, giving us now:

[tex]2(x^2-2x+1)=10+2[/tex]

The reason we complete the square (other than as a means of factoring) is to get a quadratic into vertex form.  Completing the square gives us a perfect square binomial on the left.

[tex]x^2-2x+1=(x-1)^2[/tex] and on the right we will just add 10 and 2:

[tex]2(x-1)^2=12[/tex]

Now we move the 12 back over by subtracting and set the quadratic back to equal y:

[tex]2(x-1)^2-12=y[/tex]

From this vertex form we can see that the vertex of the parabola sits at (1,-12).  This tells us that the absolute lowest point of the parabola (since it is positive it opens upwards) is -12.  Therefore, the range is R={y|y ≥ -12}

Answer:

The signa notation to represent the first five ten f(x) is given by [tex] f(x)=\sum\limits^{5}_{n=1}a_n[/tex]

Step-by-step explanation:

Given sequence is -5, -9, 13...

Let f(x) be the given sequence and is denoted by

[tex]f(x)=\{-5,-9,-13,...\}[/tex]

Let the first term be [tex]a_1, 2^{\textrm{nd}}[/tex]  term be [tex]a_3,...[/tex]

ie, [tex]a_1=-5,a_2=-9, a_3=-13,...[/tex]

To find the common difference d:

[tex]d=a_2-a_1[/tex]

[tex]=-9-(-5)[/tex]

[tex]=-9-(+5)[/tex]

[tex]d=-4[/tex]

[tex]d=a_3-a_2[/tex]

[tex]=-13-(-9)[/tex]

[tex]=-13+9[/tex]

[tex]d=-4[/tex]

Therefore the common difference d is -4 for given sequence f(x) with [tex]a_1=-5[/tex] and d=-4, the seqence f(x) is an arithmetic sequence

By defintion of arithmetic sequence

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

Now to find [tex]a_4, a_5[/tex]:

put n=4 in equation (1)

[tex]a_4=a+(4-1)d[/tex]

[tex]a_4=-5+(3)(-4)[/tex]        [since a=-5, d=-4]

[tex]=-5-12[/tex]

[tex]a_4=-17[/tex]

in equation (1)

[tex]a_5=a+(5-1)d[/tex]

[tex]a_5=-5+(4)(-4)[/tex]       [since a=-5, d=-4]

[tex]=-5-16[/tex]

[tex]a_5=-21[/tex]

Therefore [tex]a_4=-17[/tex] and [tex]a_5=-21[/tex]

Therefore [tex]f(x)=\{-5,-9,-13,-17,-21,...\}[/tex]

Now to represent the sum of the first five terms of f(x) using sigma notation as below

[tex]f(x)=\sum_{n=1}^5 a_n[/tex]

where [tex]\sum_{n=1}^5 a_n=a_1+a_2+a_3+a_4+a_5[/tex]

           [tex]-5-9-13-17-21[/tex]

[tex]\sum_{n=1}^5 a_n=-65[/tex]

The length of concrete patio is 6.5 feet which is less than given 7 foot picnic table. So the concrete slab is not enough to fit a 7 foot picnic table

Solution:

Given that concrete patio is [tex]5\frac{2}{3}[/tex] feet wide

Area of concrete patio = [tex]36\frac{2}{3}[/tex] square feet

Therefore,

[tex]\texttt{Width of concrete patio = }5\frac{2}{3}=\frac{5\times 3+2}{3}=\frac{17}{3}feet\\\\\text{Area of concrete patio = }36\frac{2}{3}=\frac{36\times 3+2}{3}=\frac{110}{3}feet^2[/tex]

Let us find the length of concrete patio

Concrete patio is usually of shape rectangle

Area of concrete patio = length x width

[tex]\frac{110}{3} = length \times \frac{17}{3}\\\\length = \frac{110}{3} \times \frac{3}{17}\\\\length = \frac{110}{17} = 6.4705 \approx 6.5[/tex]

Therefore length of concrete patio is 6.5 feet

So it cannot fit the 7 foot picnic table. Since the length of concrete patio is 6.5 feet which is less than given 7 foot picnic table

Answer:

86400 seconds

Step-by-step explanation:

1 year consists of 365 days. 1 day has 24 hours, each hour has 60 minutes and each minute has 60 seconds. 1 day = (24 hours/day) × (60 minutes/hour)(60 seconds/minute) = 86400 seconds

HOPE THIS HELPED ;3

Answer:

-3 2/4

Step-by-step explanation:

I calculated so I am correct and it's not -32

its 3  2/4

Answer:

5/14

Step-by-step explanation:

-5*-1/7*4/8=5/7*4/8=20/56=5/14

Answer:

25%.

Step-by-step explanation:

As a fraction it will be (50 - 40)/ 40 = 10/40 = 1/4.

% change  1/4 * 100 = 25%.

Answer:

[tex]\large\boxed{\dfrac{x+1}{x}=1+\dfrac{1}{x}}[/tex]

Step-by-step explanation:

[tex]\dfrac{x^2-1}{x^2-x}=(*)\\\\x^2-1=x^2-1^2\qquad\text{use}\ a^2-b^2=(a-b)(a+b)\\=(x-1)(x+1)\\\\x^2-x=(x)(x)-(x)(1)\qquad\text{use the distributive property}\\=x(x-1)\\\\(*)=\dfrac{(x-1)(x+1)}{x(x-1)}\qquad\text{cancel}\ (x-1)\\\\=\dfrac{x+1}{x}=\dfrac{x}{x}+\dfrac{1}{x}=1+\dfrac{1}{x}[/tex]

The expression x^2-1 / x^2-x in its simplest form is 1 + 1/x.

The given expression can be simplified by eliminating the common terms from the denominator and numerator. By doing this, the expression can be simplified.

The given expression is:

[tex]\frac{x^{2}-1 }{x^{2}-x}[/tex]

It can be simplified as shown below:

[tex]\frac{x^{2}-1 }{x^{2}-x} = \frac{(x+1)(x-1 )}{x(x-1)}[/tex]

The numerator is split using the identity :

[tex]a^{2} -b^{2} =(a+b)(a-b)[/tex]

It can be further simplified as follows:

[tex]\frac{x^{2}-1 }{x^{2}-x} = \frac{(x+1)(x-1 )}{x(x-1)}\\= \frac{x+1}{x} \\=\frac{x}{x} +\frac{1}{x} \\= 1+\frac{1}{x}[/tex]

We have simplified the given expression into 1 + 1/x.

Thus, the expression x^2-1 / x^2-x in its simplest form is 1 + 1/x.

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Answer:

Explanation:

1) Find the greatest number that divides exactly both 75 (candy canes) and 45 (holiday cookies).

That is, find the GCF (greatest common factor) of 75 and 45.

Then, the greatest number of gift bags, so that each bag has an equal number of candy canes and same number of cookies that Aliyah can make is 15.

2) How many cookies will be in each bag?

To find the number of cookies that will be in each bag, you divide the number of holiday cookies by the number of gift bags:

Thus, each bag will contain 3 holiday cookies.

[tex]\frac{2 \tan x}{1-\tan ^{2} x}+\frac{1}{2 \cos ^{2} x-1}=\frac{\cos x+\sin x}{\cos x-\sin x}[/tex] is proved

Solution:

Given that,

[tex]\frac{2 \tan x}{1-\tan ^{2} x}+\frac{1}{2 \cos ^{2} x-1}=\frac{\cos x+\sin x}{\cos x-\sin x}[/tex]

Let us first solve the L.H.S

[tex]\text { L. H.S }=\frac{2 \tan x}{1-\tan ^{2} x}+\frac{1}{2 \cos ^{2} x-1}[/tex]  --- (1)

By trignometric identities,

[tex]\tan 2 x=\frac{2 \tan x}{1-\tan ^{2} x}[/tex]

[tex]\cos 2 x=2 \cos ^{2} x-1=\cos ^{2} x-\sin ^{2} x[/tex]

By using these in (1) we get,

[tex]\text { L. H.S }=\tan 2 x+\frac{1}{\cos 2 x}[/tex]

By definition of tan,

[tex]tan x = \frac{sinx}{cosx}[/tex]

Therefore,

[tex]L. H.S =\frac{\sin 2 x}{\cos 2 x}+\frac{1}{\cos 2 x}\\\\L. H . S=\frac{1+\sin 2 x}{\cos 2 x}$[/tex]  --- (ii)

By trignometric identities,

[tex]\cos 2 x=2 \cos ^{2} x-1=\cos ^{2} x-\sin ^{2} x[/tex]

[tex]\begin{aligned}&\sin 2 x=2 \sin x \cos x\\\\&\cos ^{2} x+\sin ^{2} x=1\end{aligned}[/tex]

By using these in (ii)

[tex]\text { L. H.S }=\frac{\cos ^{2} x+\sin ^{2} x+2 \sin x \cos x}{\cos ^{2} x-\sin ^{2} x}[/tex]  ----- (iii)

We know that,

[tex](a+b)^2 = a^2 + 2ab + b^2[/tex]

Similarly,

[tex](cosx + sinx)^2 = cos^2x + 2cosxsinx + sin^2x[/tex]

Also,

[tex]a^2 - b^2 = (a+b)(a-b)[/tex]

Similarly,

[tex]cos^2x - sin^2x = (cosx+sinx)(cosx-sinx)[/tex]

Therefore apply these in (iii)

[tex]\begin{aligned}&\text { L. } H . S=\frac{(\cos x+\sin x)^{2}}{\cos ^{2} x-\sin ^{2} x}\\\\&\text { L. H.S }=\frac{(\cos x+\sin x)(\cos x+\sin x)}{(\cos x+\sin x)(\cos x-\sin x)}\end{aligned}[/tex]

Cancel out (cos x + sin x) on numerator and denominator

[tex]\text { L. H.S }=\frac{\cos x+\sin x}{\cos x-\sin x}[/tex]

[tex]\frac{2 \tan x}{1-\tan ^{2} x}+\frac{1}{2 \cos ^{2} x-1}=\frac{\cos x+\sin x}{\cos x-\sin x}[/tex]

Thus L.H.S = R.H.S

Thus proved

Answer:

y=50x-135

Step-by-step explanation:

If each ticket costs 50 then for three tickets, Priya needs 3*50=150

However, since she earned 135, then she's less 150-135=15

The best way to present the money she borrowed is

y=50x-135

Where x is the number of tickets bought

Given the info, Priya needs to borrow $15 from the bank to afford all 3 tickets, so after purchasing, she's out of her own money and in debt to the bank.

Essentially the question asks us how much money Priya has left after buying three concert tickets. Each ticket is priced at $50. She has earned $135 herself, so let's subtract the cost of the tickets from what she has earned.

We find that Priya needs to borrow $15 from the bank in order to purchase all three tickets (because $50 * 3 - $135 = $15).

So after buying the tickets, she would be out of her own money and $15 in debt to the bank.

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Answer:

5 1/4

Step-by-step explanation:

add 1/4 to 5 and you get 5 1/4

Show My Work:

1/4 + 5 =

5 1/4

Answer:

1/4(d+20)

Step-by-step explanation:

1/4d+5=1/4(d+20)

Answer:

Store A:  

$1.98 per loaf

Store B:  

$1.95 per loaf

Store C:  

$2.01 per loaf

Store D:  

$2.02 per loaf

Step-by-step explanation:

5.94 divided by 3 = 1.98

7.80 divided by 4 = 1.95

10.05 divided by 5 = 2.01

12.12 divided by 6 = 2.02

Answer:

Store A:  

$1.98 per loaf

Store B:  

$1.95 per loaf

Store C:  

$2.01 per loaf

Store D:  

$2.02 per loaf

Step-by-step explanation:

:)

Answer:

4! = 24

Step-by-step explanation:

The "!" in math is a factorial.

A product in a four factorial is the product of all the numbers from one to 4.

[tex]4!= 1*2*3*4=24[/tex]

Answer: 1080

Step-by-step explanation: pls mark

brainliest

Answer:

The y-intercept is the point (0,5)

Step-by-step explanation:

we know that

The y-intercept is the value of y when the value of x is equal to zero

In this problem the value of y is a constant for all values of x

so

For x=0

The value of y=5

therefore

The y-intercept is the point (0,5)

Answer:

Step-by-step explanation:

5 gallons to 20 qts = x gallons to 18 qts

 5          x

------ = ------ cross multiply

20        18

(20)(x) = (5)(18)

20x = 90

x = 90/20

x = 4.5 <==== 18 qts is equal to 4.5 gallons

====================

20 qts to 5 gallons = 20/5 = 4 qts per gallon <==

Answer: The unit is “4” quarts per gallon

Step-by-step explanation:

Answer:

3x-5y+11

Step-by-step explanation:

4x-7-4y+18-x-y

4x-x-4y-y-7+18

3x-5y+11

It will take 2.2 hours for the trains to meet.

Step-by-step explanation:

Given,

Distance between two trains = 396 miles

Speed of one train = 95 mph

Speed of other train = 85 mph

Combined speed of trains = 95+85 = 180 mph

Distance = Speed * Time

Time = [tex]\frac{Distance}{Speed}[/tex]

Time = [tex]\frac{396}{180}[/tex]

Time = 2.2 hours

It will take 2.2 hours for the trains to meet.

Keywords: speed, distance

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The two trains will take approximately 2.2 hours to meet.

To find the time it takes for the two trains to meet, we can use the formula:

Time = Distance/Speed

In this case, the distance between the two trains is 396 miles. The combined speed of the two trains is 95 mph + 85 mph = 180 mph. Substituting these values into the formula, we get:

Time = 396/180 = 2.2 (rounded to the nearest tenth)

So it will take approximately "2.2 hours" for the two trains to meet.

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Answer:

V = 314 mm³

Step-by-step explanation:

The volume (V) of a cylinder is calculated as

V = πr²h ( r is the radius and h the height )

Here diameter = 10, hence r = 10 ÷ 2 = 5 and h = 4, thus

V = π × 5² × 4

   = 3.14 × 25 × 4 = 3.14 × 100 = 314 mm³

Answer:

58.8- -7.2= 51.6

Step-by-step explanation:

Because it asked for how much greater, I subtracted.

Answer:

7.2-58.8=51.6

Step-by-step explanation:

Answer:

0.011 (taking till hundredth decimal value)

Step-by-step explanation:

Given: 8 divided by 709.

Lets solve it by using long division.

[tex]8\div 709[/tex]

Divisor is 709 and dividend is 8.

Now, while solving we will get quotient and remainder, if any.

As in this case 709 (divisor) is smaller number than 8 (dividend).

So, we need to take decimal in quotient to add zero with the dividend, which is again smaller than divisor, therefore we will take 0 also in quotient after decimal to add more zero with dividend, now dividend have value "800".

∴[tex]800\div 709= 0.0[/tex]

In the next step, we will divide 800 by 709.

As we know, [tex]709\times 1= 709\\709\times 2= 1418[/tex]

As 709 is lesser than 800 and we don´t have value equal to 800 in the multiple of 709, therefor we will quotient as "1" .

∴ [tex]800\div 709=0.01[/tex]

Now, we have remainder [tex]800-709= 91[/tex]

we cannot divide 91 by 709, so we add zero to the dividend as we have already taken decimal.

∴ with remainder 91 and adding zero to it, make it [tex]910\div 709[/tex]

Now, we have [tex]910\div 709= 0.011[/tex]

As we know, [tex]709\times 1= 709\\709\times 2= 1418[/tex]

As 709 is lesser than 910 and we don´t have value equal to 9100 in the multiple of 709, therefor we will quotient as "1" .

∴ We have result as 0.011 (taking till hundredth decimal value)

Answer:

Subtotal = $145.48

Step-by-step explanation:

Products purchased:

Shoes, Shirt and bag.

Cost of shoes = $64.99

Cost of shirt = $34.99

Cost of bag = $45.50

The subtotal of the the products purchased before taxes can be calculated by finding the sum of all the costs.

[tex]Subtotal = \textrm{Cost of shoes + Cost of shirt + Cost of bag}[/tex]

[tex]Subtotal =\$64.99+\$34.99+\$45.50 [/tex]

[tex]Subtotal = \$145.48[/tex] (Answer)

Answer:

x=-54

Step-by-step explanation:

-4(2x+9)+3x=6-4(x-3)

Reorder the terms in parentheses

+(-8x-36)+3x=6-4*(x-3)

Remove unnecessary parentheses

-8x-36+3x=+6-4+*(+x-3+)

Reorder the terms in parentheses

-8x-36+3x=6+(-4x+12)

Remove unnecessary parentheses

-8x-36+3x=+6-4x+12

move all terms containing x to the left and all other terms to the right.

-8x+3x+4x=+6+12+36

simplify left and right side of the equation.

-1x=+54

finally, divide both sides of the equation by -1 to get x.

x=-54

-4 (2x+9) + 3x = 6 - 4 (x-3)

Simplifying

-4(2x + 9) + 3x = 6 + -4(x + -3)

Reorder the terms:

-4(9 + 2x) + 3x = 6 + -4(x + -3)

(9 * -4 + 2x * -4) + 3x = 6 + -4(x + -3)

(-36 + -8x) + 3x = 6 + -4(x + -3)

Combine like terms: -8x + 3x = -5x

-36 + -5x = 6 + -4(x + -3)

Reorder the terms:

-36 + -5x = 6 + -4(-3 + x)

-36 + -5x = 6 + (-3 * -4 + x * -4)

-36 + -5x = 6 + (12 + -4x)

Combine like terms: 6 + 12 = 18

-36 + -5x = 18 + -4x

Solving

-36 + -5x = 18 + -4x

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '4x' to each side of the equation.

-36 + -5x + 4x = 18 + -4x + 4x

Combine like terms: -5x + 4x = -1x

-36 + -1x = 18 + -4x + 4x

Combine like terms: -4x + 4x = 0

-36 + -1x = 18 + 0

-36 + -1x = 18

Add '36' to each side of the equation.

-36 + 36 + -1x = 18 + 36

Combine like terms: -36 + 36 = 0

0 + -1x = 18 + 36

-1x = 18 + 36

Combine like terms: 18 + 36 = 54

-1x = 54

Divide each side by '-1'.

x = -54

Simplifying

x = -54

Answer:

the answer is 12

Step-by-step explanation:

You just find the total weight and divide it by the number of weights

hope this helped ;3

Answer:

12 g

Step-by-step explanation:

Recall that

Average Weight = Sum of All weights  ÷ Number of Weights

in the first case we are given that average of 5 weights is 13g

i.e Average weight (of 5 weights) = 13g and number of weights = 5

or, using the formula above,

13 = sum of 5 weights ÷ 5   (rearranging)

sum of 5 weights = 13 x 5 = 65 g

we are subsequently told that one more weight of 7 g is added,

hence, new number of weights = 6

and,

sum of 6 weights = sum of original 5 weights + sum of 6th weight

= 65 g + 7 g = 72 g

once again using the first formula

Average Weight (of 6 weights) = Sum of All weights  ÷ Number of Weights

= 72  ÷ 6

= 12g

Answer:

16327.1 m^3

Step-by-step explanation:

V=13pi*r^2*h

V=13pi*(6.2)^2*(10.4)

V=16327.1

Answer:

-6, 6

Step-by-step explanation:

-6+6=0

Answer:

A

Step-by-step explanation:

The volume of the triangular prism is

[tex]V=A_{base}\cdot h,[/tex]

where

h = height

Base = triangle.

In your case,

[tex]A_{base}=\dfrac{1}{2}\cdot 4\cdot 4=8\ ft^2\\ \\h=1\ ft,[/tex]

then

[tex]V=8\cdot 1=8\ ft^3[/tex]