convert from standard form to vertex form. y=x^2 -6x +8​

a)2

b)4

c)6

d)x2

e)-2a2b

For questions 7, 8, and 10:

The cube root of 7 is 2, the cube root of 40 is 4, and the cube root of 162 is 6. The cube root of an expression with an exponent can be found by dividing the exponent by 3; for example, the cube root of x8 is x2.

For question 11:

The cube root of -16a5b is -2a2b.

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The area of the sector outlined by the given arc is 32π square units.

What is the area of the sector?

The area of a sector of a circle with radius r and central angle θ (in radians) is given by the formula:

sector area = (1/2) x r² x θ

The length of an arc of a circle with radius r and central angle θ (in radians) is given by the formula:

arc length = r x θ

In this case, we know that the radius is 8 units and the arc length is 4π units. So we can set up an equation:

4π = 8θ

Solving for θ:

θ = (4π)/8 = π/2

So the central angle is π/2 radians.

The area of a sector of a circle with radius r and central angle θ (in radians) is given by the formula:

sector area = (1/2) x r² x θ

Plugging in the values we know:

sector area = (1/2) x 8² x π/2

sector area = 32π

Therefore, the area of the sector outlined by the given arc is 32π square units.

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The simplest form of the expression (x^(2)+8x+16)/(x^(2)-4x-32) is (x+4)/(x-8) with restrictions x≠8 and x≠-4.

The expression (x^(2)+8x+16)/(x^(2)-4x-32) can be simplified by factoring the numerator and denominator.

First, we can factor the numerator:
(x^(2)+8x+16) = (x+4)(x+4)

Next, we can factor the denominator:
(x^(2)-4x-32) = (x-8)(x+4)

Now, we can simplify the expression by canceling out the common factor of (x+4):
(x+4)(x+4)/(x-8)(x+4) = (x+4)/(x-8)

Therefore, the simplest form of the expression is (x+4)/(x-8).

However, there are restrictions on the variable x. The denominator of the expression cannot equal zero, so we must find the values of x that make the denominator zero and exclude them from the domain of the expression.

To find the restrictions, we set the denominator equal to zero and solve for x:
(x-8)(x+4) = 0

This gives us two solutions:
x = 8
x = -4

Therefore, the restrictions on the variable are x≠8 and x≠-4.

In conclusion, the simplest form of the expression (x^(2)+8x+16)/(x^(2)-4x-32) is (x+4)/(x-8) with restrictions x≠8 and x≠-4.

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The fraction of total distance that was travelled on Sunday is 3/8.

Let total distance to Fernando's grandmother's house be = D;

On Saturday, they traveled 3/8 of D,

So, the remaining distance is 5/8 of D.

On Sunday, they traveled 3/5 of the remaining distance, which is:

⇒ (3/5) × (5/8) × D = 3/8 × D,

So, on Sunday, they traveled 3/8 of D.

To find the fraction of the total distance traveled on Sunday, we divide the distance traveled on Sunday by the total distance, which is;

⇒ (3D/8)/D = 3/8

Therefore, Fernando's family traveled 3/8 of the total distance on Sunday.

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The given question is incomplete, the complete question is

Fernando's family traveled 3/8 of the distance to his grandmother’s house on Saturday. They traveled 3/5 of the remaining distance on Sunday.

What fraction of the total distance to his grandmother’s house was traveled on Sunday?

The measures of angles 4 and 5 are:

∠5 = 104°

∠4 =76°

On the image we can see that angles 1 and 2 are next to eachother, that means that the sum of their measures must be a plane angle, that is an angle of 180°.

Then we can write the sum:

∠1 + ∠2 = 180°

(3x + 5) + (2x + 10) = 180

5x + 15 = 180

5x = 180 - 15

x = 165/5 = 33

the measure of angle 5 is the same one of angle 1 then:

∠5 = (3*33 + 5)° = 104°

And angle 4 is supplementary of angle 1, then:

∠4 + 104° = 180°

∠4 = 180 - 104= 76°

These are the measures

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Equation Slope-Intercept Form Slope y-Intercept

y = 2x + 5 y = 2x + 5 2 5

4x - 3y = 12 y = (4/3)x - 4 4/3 -4

x = 7 Does not exist Does not exist Does not exist

3y = 9 y = 3 0 3

For the first equation, y = 2x + 5, it is already in slope-intercept form. The slope is the coefficient of x, which is 2, and the y-intercept is the constant term, which is 5.

For the second equation, 4x - 3y = 12, we can rearrange it to get it in slope-intercept form by isolating y on one side of the equation. We can do this by subtracting 4x from both sides and then dividing by -3: 4x - 3y = 12 -3y = -4x + 12 y = (4/3)x - 4

The slope is the coefficient of x, which is 4/3, and the y-intercept is the constant term, which is -4. For the third equation, x = 7, there is no y term, so it cannot be written in slope-intercept form and the slope and y-intercept do not exist.

For the fourth equation, 3y = 9, we can rearrange it to get it in slope-intercept form by dividing both sides by 3: 3y = 9 y = 3 The slope is 0, since there is no x term, and the y-intercept is the constant term, which is 3.

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Answer:

35.13

Rounded: 35.1

Step-by-step explanation:

The figure will look like the attached image.

(My guess, based on hint)

We will first find the area of the circle.

Area of circle is radius squared times Pi

3x3x3.14=28.26

It is half circle so 28.26/2=14.13

The question didn't state the height, but the base is 6 and we'll assume it is 7 because the question has 7 in the third to last row. 7x6/2=21

14.13+21=35.13

the factored form of the expression 9m^(2)-49 is (3m+7)(3m-7).

To factor the expression 9m^(2)-49, we need to use the difference of squares formula.

The difference of squares formula states that a^(2)-b^(2)=(a+b)(a-b).

In this case, we can see that 9m^(2) is a perfect square and 49 is also a perfect square. So, we can write the expression as (3m)^(2)-(7)^(2).

Using the difference of squares formula, we get:

(3m)^(2)-(7)^(2)=(3m+7)(3m-7)

So, the factored form of the expression 9m^(2)-49 is (3m+7)(3m-7).

Therefore, the answer is (3m+7)(3m-7).

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For part (a), the answer is f(x+h) = x² + 2xh + h² - 4x - 4h. for par (b) h = -x ± sqrt(x² + 6), the value of h depends on the value of x

(a) To find f(x+h), we substitute x+h for x in the expression for f(x):

f(x+h) = (x+h)² - 4(x+h)

Expanding the square and simplifying, we get:

f(x+h) = x² + 2xh + h² - 4x - 4h

(b) To find h, we start with the expression for f(x+h) that we found in part (a):

f(x+h) = x² + 2xh + h² - 4x - 4h

We want to simplify this expression so that we can identify h. To do this, we start by subtracting f(x) from both sides:

f(x+h) - f(x) = (x² + 2xh + h² - 4x - 4h) - (x² - 4x)

Simplifying, we get:

f(x+h) - f(x) = 2xh + h² - 4h

Now we can identify h by setting this expression equal to some value and solving for h. For example, if we set f(x+h) - f(x) equal to 5, we get:

2xh + h² - 4h = 5

Simplifying and rearranging, we get a quadratic equation in h:

h² + 2xh - 4h - 5 = 0

We can solve this using the quadratic formula:

h = (-2x ± √(4x² + 24))/2

Simplifying, we get:

h = -x ± √(x² + 6)

So the value of h depends on the value of x

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This is the vertex form equation of the parabola with the given information.

From the information provided, we can write the vertex form equation of the parabola as follows:

First, we need to determine the value of "a" in the equation y = a(x - h)^2 + k, where (h,k) is the vertex and "a" determines the width and direction of the parabola.

The distance between the vertex and the focus is equal to 1/4a, so we can use this to find "a":

1/4a = (41/4 - 10)

1/4a = 1/4

a = 1

Now we can plug in the values for the vertex and "a" into the equation:

y = 1(x - 10)^2 - 8

This is the vertex form equation of the parabola with the given information.

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The correct option representing possible sum for the long chain is B. 13 3 4 + 15 2 4 = 29 1 4.

Firstly let us convert the mixed fraction to fraction. So, the length of first chain = (13×4)+3/4

Length of first chain = 55/4 inches

Length of first chain = 13.75 inches

Length of second chain = (15×4)+2/4

Length of second chain = 62/4

Second chain length = 15.5 inches

So, total length = length of first chain + length of second chain

Total length = 13.75 + 15.5

Total length = 29.25 inches

Converting the decimal back to mixed fraction -

Total length = 29 1/4 inches

Thus, the correct answer B. 13 3 4 + 15 2 4 = 29 1 4.

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Answer: y = 4 + 3x

Step-by-step explanation:

Input (x) = any number

Output (y) = 4 more than 3 times x

more: +

times: ×

The linear equation y = 3x - 7 has a data that is correlated because it has a linear relationship and we cannot determine if there's a causation between x and y

A. y causes a. is not applicable in this case, as there is no variable named "a" in the equation.

B. The equation y = 3x - 7 represents a linear relationship between two variables, x and y. As the equation has a slope of 3, it indicates that as the value of x increases by 1, the value of y increases by 3. Therefore, the data are correlated positively.

C. This is incorrect. As mentioned in B, the equation represents a linear relationship between two variables, x and y, which are positively correlated.

D. This is correct. Although the equation represents a linear relationship between x and y, it does not imply causation. It is possible that x causes y, y causes x, or some other variable or variables may be influencing both x and y. Therefore, we cannot determine if there is causation between x and y based on the equation y = 3x - 7 alone

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The graph of the linear equation, y = 5x + 2, is shown below in the attachment.

To graph the linear equation, y = 5x + 2, first plot the y-intercept at (0, 2) on the coordinate plane. Then, use the slope of 5 to move up 5 units and right 1 unit from the y-intercept to find another point on the line.

Continue this process to plot several more points, and then connect the points with a straight line to complete the graph of the equation. The resulting graph will be a straight line with a positive slope of 5 and y-intercept of 2.

See the attachment below for the graph.

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Answer:

C but it's a guess........

The next term in the sequence would be -31.

What is Sequence ?

A sequence is a list of numbers or objects that follow a specific pattern or rule. Each number in the sequence is called a term, and the terms can be finite or infinite.

A. {31, 26, 21, 16, 11,...}

To find the common difference (d) between consecutive terms in the arithmetic sequence, we can subtract any term from the term that comes after it. For example, we can subtract 26 from 31 to get:

31 - 26 = 5

Similarly, we can subtract 21 from 26, 16 from 21, and so on:

26 - 21 = 5

21 - 16 = 5

16 - 11 = 5

Since the differences between consecutive terms are all the same, we can conclude that the common difference is d = 5.

To find the next term in the sequence, we can add the common difference (d = 5) to the last term (11):

11 + 5 = 16

Therefore, the next term in the sequence would be 16.

B. {-45, -38, -31, -24, . . .}

To find the common difference (d) between consecutive terms in the arithmetic sequence, we can subtract any term from the term that comes after it. For example, we can subtract -38 from -45 to get:

-45 - (-38) = -7

Similarly, we can subtract -31 from -38, -24 from -31, and so on:

-38 - (-31) = -7

-31 - (-24) = -7

Since the differences between consecutive terms are all the same, we can conclude that the common difference is d = -7.

To find the next term in the sequence, we can add the common difference (d = -7) to the last term (-24):

-24 - 7 = -31

Therefore, the next term in the sequence would be -31.

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Answer:

4:3

Step-by-step explanation:

16:12=8:6=4:3, that is simplest ratio

(a) Possible number(s) of positive real zeros: 0, 1, or 2
(b) Possible number(s) of negative real zeros: 0, 1, or 2


To determine the possible number of positive and negative real zeros of a polynomial function, we can use the Descartes' Rule of Signs.

This rule states that the number of positive real zeros of a polynomial function is equal to the number of sign changes in the coefficients of the polynomial, or less than that by an even number. Similarly, the number of negative real zeros of a polynomial function is equal to the number of sign changes in the coefficients of the polynomial when the variable x is replaced with -x, or less than that by an even number.

For the given polynomial function Q(x) = -x⁴ - 6x³ - 8x² - 5x + 1, the number of sign changes in the coefficients is 1 (from -5x to +1). Therefore, the possible number of positive real zeros is 1 or 0 (1-2).

To find the possible number of negative real zeros, we replace x with -x and simplify the polynomial:

Q(-x) = -(-x)⁴ - 6(-x)³ - 8(-x)² - 5(-x) + 1
= -x⁴ + 6x³ - 8x² + 5x + 1

The number of sign changes in the coefficients of this polynomial is 2 (from -x⁴ to +6x³ and from -8x² to +5x). Therefore, the possible number of negative real zeros is 2 or 0 (2-2).

So, the possible number of positive real zeros is 0, 1, or 2, and the possible number of negative real zeros is 0, 1, or 2.

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Answer: Your welcome!

Step-by-step explanation:

The students can decorate the circles for placemats in a variety of ways. They can use paint, markers, fabric, or any other creative material of their choice. They can also add images, shapes, and words to the circles. They could even attach ribbons or other decorations to the circles to create a unique design. The possibilities are endless!

Answer:

he walks 300 meters

Step-by-step explanation:

nbsjdjsj d f f f f f f f. f f f f f f f f f f

Answer: surface area is _______ square centimeters.

To answer this question, I would need to know the dimensions of the rectangular prism, as both the total surface area and the lateral surface area depend on the dimensions of the prism.

Assuming the rectangular prism has length (l), width (w), and height (h), the formulas for the total surface area and the lateral surface area are:

Total surface area = 2lw + 2lh + 2wh

Lateral surface area = 2lh + 2wh

Therefore, to find the total surface area and lateral surface area of a specific rectangular prism, I would need to know its dimensions (length, width, and height) and plug them into these formulas.

Step-by-step explanation:

Answer:

5

Step-by-step explanation:

The sum of angle in a triangle is 180°

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted \triangle ABC.

The sum of angle A, B and C is 180 i.e A+B +C = 180°

Also a triangle is a 3 sided polygon. The sum of of angle in a polygon is( n-2)180

How do we prove that the sum of angle in a triangle is 180°?

Since triangle is 3 sided, n= 3, because n denote the number if sides

therefore the sum of angle = (n-2) 180 = (3-2)×180

= 180°

therefore the sum of angle In a triangle is 180°

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These statistics can help us better understand the data and make comparisons between the two classes.

The given data shows the results of two classes, Class A and Class B.

Class A: 34, 31, 29, 24, 27, 37, 26, 24, 20, 37, 22, 18, 36, 26, 28, 23, 23, 27, 33, 25, 36, 32, 33, 25, 28, 35

Class B: (no data given)

In order to analyze the data, we can use statistics. Statistics is the science of collecting, analyzing, and interpreting data. One way to analyze the data is to find the mean, median, and mode of the data set.

Mean: The mean is the average of the data set. To find the mean, add up all the numbers and divide by the total number of data points.

Mean of Class A = (34+31+29+24+27+37+26+24+20+37+22+18+36+26+28+23+23+27+33+25+36+32+33+25+28+35) / 26 = 28.42

Median: The median is the middle number in the data set. To find the median, sort the data set in ascending or descending order and find the middle number. If there are an even number of data points, the median is the average of the two middle numbers.

Sorted Class A: 18, 20, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 31, 32, 33, 33, 34, 35, 36, 36, 37, 37

Median of Class A = (26+27) / 2 = 26.5

Mode: The mode is the number that appears most frequently in the data set.

Mode of Class A = 24, 26, 27, 33, 36, 37 (all appear twice)

These statistics can help us better understand the data and make comparisons between the two classes.

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The given polynomial g(x) is g(x)=x^(3)+(p-4)x^(2)+(p-9)x-4.

To show that the polynomial g(x) is divisible by (x+1) for all values of p, we must prove that g(x) has a factor of (x+1). This can be done by expanding the given polynomial and rewriting it in a form that contains the factor (x+1).

First, we expand the given polynomial g(x):

g(x)=(x^3 + (p-4)x^2 + (p-9)x - 4)

= x^3 + px^2 - 4x^2 + px - 9x - 4

= x^3 + (p-4)x^2 + (px - 9x) - 4

= x^3 + (p-4)x^2 + (p - 9)x - 4

Now we can factor out the term (x+1) from the expression:

= (x^3 + (p-4)x^2 + (p-9)x - 4)

= (x+1)(x^2 + (p-5)x + (p-9)) - 4

Finally, we can rewrite the expression as:

g(x)=(x+1)(x^2 + (p-5)x + (p-9)) - 4

This shows that the polynomial g(x) is divisible by (x+1) for all values of p, as there is a factor of (x+1) in the expression.

Therefore, we have successfully proved that the polynomial g(x) is divisible by (x+1) for all values of p.

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a) The inverse sine of 1 is π/2 radians, which is equivalent to 90 degrees.

b) The inverse sine of √2 / 2 is π/4 radians, which is equivalent to 45 degrees.
c) The inverse sine of - √2 / 2 is -π/4 radians, which is equivalent to -45 degrees.

Evaluate the following expressions. Your answer will be an angle in radians and in the interval [-π/2, π/2]
(a) sin^-1 (1) = π/2
(b) sin^-1 (√2 / 2)= π/4
(c) sin^-1 (- √2 / 2)= -π/4


(a) sin^-1 (1) = π/2
The inverse sine of 1 is π/2 radians, which is equivalent to 90 degrees.

(b) sin^-1 (√2 / 2)= π/4
The inverse sine of √2 / 2 is π/4 radians, which is equivalent to 45 degrees.

(c) sin^-1 (- √2 / 2)= -π/4
The inverse sine of - √2 / 2 is -π/4 radians, which is equivalent to -45 degrees.

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Yes, the pair (x, y) exists and satisfies the given equation.

Yes, there exists a pair of integers (x, y) such that the greatest common divisor (d) of a and b can be expressed as a linear combination of the two numbers, i.e. d = ax + by.

To prove this, first note that the greatest common divisor (d) of a and b divides both a and b. Therefore, by the Division Algorithm, there exist integers q and r such that a = dq + r and 0 ≤ r < d. Similarly, there exist integers p and s such that b = dp + s and 0 ≤ s < d.

Subtracting these two equations yields d = (a - dq) + (b - dp) = (r - q) + (s - p). Therefore, if we let x = r - q and y = s - p, then d = ax + by, where x and y are both integers. Thus, the pair (x, y) exists and satisfies the given equation.

The integral of f(x) from a=-1 to b=4 is \(\frac{21}{64}\). The lower and upper limits of integration are the same, the integral is 0.

a) To evaluate the integral of \(f(x)=\frac{1}{x^4}\) from a=-1 to b=4, we need to first find the antiderivative of f(x). The antiderivative of \(\frac{1}{x^4}\) is \(-\frac{1}{3x^3}\). Now we can use the Fundamental Theorem of Calculus to evaluate the integral:

Integral [a=-1,b=4] f(x) dx = \(-\frac{1}{3(4)^3} - (-\frac{1}{3(-1)^3})\)

= \(-\frac{1}{192} + \frac{1}{3}\)

= \(\frac{63}{192}\)

= \(\frac{21}{64}\)

So the integral of f(x) from a=-1 to b=4 is \(\frac{21}{64}\).

b) The integral of f(x) from a=4 to b=4 is 0. This is because the integral of a function over an interval of length 0 is always 0. In other words, if the lower and upper limits of integration are the same, the integral is 0.

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The rate of change is 961.37 gallons/hour. This means that the water level decreases by 961.37 gallons every hour.

In mathematics, comparing two related quantities stated in different units is known as a rate. A rate is the ratio of two distinct values that are gauged in various ways. It is not the same as a unit rate to compare a certain number of units of the first quantity to one unit of the second. In general, the rate can be expressed as the ratio of two quantities with distinct units. With the help of rate, we can create a connection between two otherwise unconnected components. By doing this, we can gain a deeper understanding of a scenario.

Given,

The total amount of water the pool can hold =  9613.7 gallons

The time taken to drain the pool completely = 10 hours

We are asked to find the change in water level per hour of the pool.

This can be called the rate of change.

Rate of change = Total amount of water/ Time taken to drain

                       = 9613.7 / 10 = 961.37 gallons / hour

Therefore the rate of change is 961.37 gallons/hour. This means that the water level decreases by 961.37 gallons every hour.

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