A student is asked to find the vertex of a parabola whose equation is as follows: y = 3 * (x + 4) ^ 2 - 2 The student concludes the vertex is at (4, - 2) . the student correct? If not, what do you think the student did wrong? Explain with as much relevant mathematical detail as possible. Please use at least 3 sentences total in your answer

The polynomial function of least degree possible using the given information.

Real roots: −1 (with multiplicity 2), 1 and (2,

f(2)) = (2, 7) is f(x) = x³ - 3x² + 3x - 1.

Since the polynomial function has real roots at -1 (with multiplicity 2) and 1, we know that the factors of the polynomial are (x + 1)² and (x - 1).

Let the polynomial be of the form f(x) = ax³ + bx² + cx + d. We know that f(2) = 7, so:

a(2³) + b(2²) + c(2) + d = 7

8a + 4b + 2c + d = 7

Now we need to use the fact that the roots are -1 (with multiplicity 2) and 1. Since the polynomial has factors of (x + 1)² and (x - 1), we can write the polynomial as:

f(x) = a(x + 1)²(x - 1)

Expanding this expression, we get:

f(x) = a(x² - 2x + 1)(x - 1)

f(x) = ax³ - 3ax² + 3ax - a

Now we can equate the coefficients of this expression with the coefficients of our assumed polynomial f(x) = ax³ + bx² + cx + d:

a = a

-3a = b

3a = c

-a = d

Therefore, our polynomial function is:

f(x) = x³ - 3x² + 3x - 1

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After answering the provided question, we can conclude that which  slope equals m = 1/2. As a result, the second line's equation is y - 2 = (1/2)(x - 2), or y = (1/2)x + 1.

In mathematics, the slope-intercept form of a linear equation is an equation of the form y = mx + b, where m is the line's gradient and b is the más, which is the point on the line where it intersects the y-axis. Because it allows you to speedily see the line and ascertain its slope and y-intercept, the slope-intercept form is a useful way to represent a line's equation. The slope of the line indicates its steepness, while the esta indicates in which the line intersects the y-axis.

To answer these questions, we must first determine the equation of the line depicted in the graph. Assume the line is in slope-intercept form, y = mx + b, with m representing the slope and b representing the y-intercept.

where m is the slope to be found. Because we want this line to intersect the given line at (4, 3), we can plug those values into both equations and solve for m: Because y = (1/4)x + 1, we get 3 = (1/4)(4) + 1 = 2 + 2m, which equals m = 1/2. As a result, the second line's equation is y - 2 = (1/2)(x - 2), or y = (1/2)x + 1.

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Answer:

Step-by-step explanation:

The cost of each cell phone case is $7.15, and the cost of each screen protector is $2.25.

To solve this problem, we can create a system of equations.

Let x represent the cost of one cell phone case and y represent the cost of one screen protector.

Then, we can write the following equations:

1x + 2y = 18.95     (equation 1)

2x + 5y = 41.40    (equation 2)

We can solve for x and y by multiplying equation 1 by 2 and subtracting it from equation

2.2x + 5y = 41.40(2 x 1x + 2 x 2y = 2 x 18.95)2x + 5y = 41.4002x + 4y = 37.90

Subtracting equation 1 from equation 2 gives us:

2x + 5y = 41.400- (0.2x + 0.4y = 3.790)

1.8x + 4.6y = 37.61

Now we can solve for y.

1.8x + 4.6y = 37.61

=> 46y = 46.29

=> y = 1.05

Then we can solve for x using equation

1.1x + 2y = 18.95

=> 1x + 2(1.05) = 18.95

=> x = 7.15

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Question:

You order a cell phone case and 2 screen protectors online for a total of $18.95. Your friend orders 2 cell phone cases and 5 screen protectors at the same unit prices for a total of $41.40. What is the cost of each cell phone case? Each screen protector?

The blue line is the tangent line that passes through the point (0.75, 0.25). The point on the tangent line is (3, 4). The gradient of the curve is approximately 0.6.

Answer:

gradient = [tex]\frac{1}{4}[/tex]

Step-by-step explanation:

calculate the gradient m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (4, 2) and (x₂, y₂ ) = (8, 3) ← 2 points on the line

m = [tex]\frac{3-2}{8-4}[/tex] = [tex]\frac{1}{4}[/tex]

The value of x in the similar triangles are:

a. x = 80 ft.

b. x = 16 ft.

Similarity among two or more triangle is a common relations when the properties of the triangles are compared. But in this case, the triangle are NOT congruent.

Considering the triangles in the given question,

1. Comparing the properties of the similar triangles, we have;

So that,

30/ x = 50/ 80

50x = 4000

x = 4000/ 50

  = 80

x = 80 ft.

2. Comparing the properties of the two triangles, we have;

4.5/ 12 = 6/ x

4.5x = 72

x = 72/ 4.5

  = 16

x = 16 ft

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Answer:

Step-by-step explanation:

Answer: identical to/same as

Step-by-step explanation:

Answer:

The ≡  sign means identical to, similar but not exactly the same as, equals

Step-by-step explanation:

Hope this helps! =D

Mark me brianliest and Have a good day! =D

The group of friends can consist of up to 10 people if they want to stay within their budget of $152.75 for parking and admission.

Total cost = parking cost + ticket cost

The ticket cost for x people can be expressed as:

Ticket cost = x × $14.25

the cost of each ticket is $14.25.

the total cost for x people can be expressed as:

= $10.25 + x × $14.25

the group has a budget of $152.75 to spend on parking and admission, so we can set up

$152.75 = $10.25 + x × $14.25

Now we can solve for x by first subtracting $10.25

$152.75 - $10.25 = x × $14.25

Simplifying:

$142.50 = x × $14.25

Finally, we can solve for x

x = $142.50 ÷ $14.25

x = 10

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Thus, the most direct path between the bottle caps and Ann is found as:

d = 10.63 units.

The algebraic phrase known as the distance formula, which provides the distances between two places in terms of respective coordinates (in coordinate system).

The distance formulas with points in rectangular coordinates in two- through three-dimensional Euclidean space are based here on Pythagorean theorem.

The formula used to calculate the distance between both the points (x1, y1) and ((x2, y2)

d = √[(x2 - x1)² + (y2 - y1)²]

Coordinates of pile of bottle caps -:(x1, y1) =   (4, -10).

Coordinates of ANN - (x2, y2) =  (-4, 3).

Using the distance formula:

d = √[(x2 - x1)² + (y2 - y1)²]

d = √[(-4 - 4)² + (-3 + 10)²]

d = √[(64) + 49]

d = 10.63 units

Thus, the most direct path between the bottle caps and Ann is found as:

d = 10.63 units.

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complete question

A pile of bottle caps is located at (4, -10). find the length of the most direct path between Ann and the bottle caps?

The graph for the question is given.

The image of (-5,-9) after dilation by a scale factor of 3 centered at the origin is (-15,-27).

To find the image of the point (-5,-9) after dilation by a scale factor of 3 centered at the origin.

We can use the following formula:

(x', y') = (kx, ky)

Where (x', y') are the coordinates of the dilated point, (x, y) are the coordinates of the original point, and k is the scale factor.

In this case, k = 3, and the coordinates of the original point are (-5,-9).

So we have:

x' = 3(-5) = -15

y' = 3(-9) = -27

Therefore, the image after dilation is (-15,-27).

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Answer:

(2a + 3b)² = 121

Step-by-step explanation:

given 4a² + 9b² = 25 and ab = 8

(2a + 3b)² ← expand using FOIL

= 4a² + 6ab + 6ab + 9b²

= 4a² + 9b² + 12ab ← substitute given values from above

= 25 + 12(8)

= 25 + 96

= 121

Answer:

Fold the paper in half lengthwise. Unfold it, then fold it in half again widthwise. Unfold it, then fold it in thirds lengthwise. Unfold it and then fold it in half again widthwise. You should now have a paper with 8 sections, with 1 on top of 2, 2 on top of 3, 3 on top of 4, etc. You can flip the numbers upside down or backwards as desired.

Step-by-step explanation:

Answer: x=1 y=6

Step-by-step explanation:

The measure of angle A is 112 degrees. Hence, option d is correct.

Rectangles are quadrilaterals having four right angles in the Euclidean plane of geometry. Various definitions include an equiangular quadrilateral, A closed, four-sided rectangle is a two-dimensional shape. A rectangle's opposite sides are equal and parallel to one another, and all of its angles are exactly 90 degrees.

Similarly, since the angle of the cut is 36 degrees, the angle opposite it (angle BCD) is also 36 degrees. Since angles BCD and BDC add up to 90 degrees (because triangle BCD is a right triangle), angle BDC measures 54 degrees (90 - 36 = 54).

Now, we can find angle BDA by subtracting angle ADB and angle BDC from 180 degrees:

angle BDA = 180 - angle ADB - angle BDC

= 180 - 58 - 54

= 68

Finally, we can find angle A by subtracting angle BDA from 180 degrees:

angle A = 180 - angle BDA

= 180 - 68

= 112

Therefore, the measure of angle A is 112 degrees. Hence, option d is correct.

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Complete question:

For the first question, the correct graph would be option B (b-a)², as the water level decreases at a constant rate and then remains constant, which would result in a diagonal line followed by a horizontal line on a graph.

An expression is a combination of symbols and/or numbers that can be evaluated or simplified to obtain a numerical or symbolic result. Expressions can be composed of one or more terms, which are separated by arithmetic operations, such as addition, subtraction, multiplication, or division.

For the second question, the area of the smaller square can be calculated using the expression (c - b)² or option C. This is because the length of the side of the larger square is c, and one side of the smaller square is equal to the difference between the side length of the larger square and the length of the smallest side of one of the congruent triangles, which is c - b.

Taking the square of this difference gives the area of the smaller square. Option D (a-b)² and option E b(c-b) are not correct expressions for the area of the smaller square, and option A does not provide enough information to calculate the area of the smaller square. Option (c - b)² is equivalent to option F.

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29. The correct graph would be option B (b-a)², as the water level decreases at a constant rate and then remains constant.

30. The area of the smaller square can be calculated using the expression (c - b)² or option C.

An expression is a combination of symbols and/or numbers that can be evaluated or simplified to obtain a numerical or symbolic result. Expressions can be composed of one or more terms, which are separated by arithmetic operations, such as addition, subtraction, multiplication, or division.

For the first question, the correct graph would be option B (b-a)², as the water level decreases at a constant rate and then remains constant, which would result in a diagonal line followed by a horizontal line on a graph.

For the second question, the area of the smaller square can be calculated using the expression (c - b)² or option C. This is because the length of the side of the larger square is c, and one side of the smaller square is equal to the difference between the side length of the larger square and the length of the smallest side of one of the congruent triangles, which is c - b.

Taking the square of this difference gives the area of the smaller square. Option D (a-b)² and option E b(c-b) are not correct expressions for the area of the smaller square, and option A does not provide enough information to calculate the area of the smaller square. Option (c - b)² is equivalent to option F.

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Answer:(3x−y+5)(3−+5)

Step-by-step explanation:

No; the triangles are not congruent by any of the congruence criteria.

To determine if two triangles are congruent, we must use one of the five congruence criteria: Side-Angle-Side (SAS), Angle-Angle-Side (AAS), Angle-Side-Angle (ASA), Side-Side-Side (SSS), and Hypotenuse-Leg (HL). To be congruent, the triangles must have three pairs of congruent sides and three pairs of congruent angles.

In the case of △ABC and △DEF, we can compare their sides and angles to determine if they are congruent. We can see that the sides of the two triangles are not equal in length, nor do the angles have the same measure. Therefore, △ABC is not congruent to △DEF.

In order to prove that the two triangles are not congruent, we can use the formula for determining the measure of the angles in a triangle, known as the Law of Cosines. This formula states that c² = a² + b² - 2ab cos γ, where c is the length of the longest side of the triangle, a and b are the lengths of the other sides of the triangle, and γ is the measure of the angle opposite the side c.

Using this formula, we can calculate the measure of the angles in both triangles. For △ABC we have angle A = arccos ( (b² + c² - a²) / 2bc ) and for △DEF we have angle D = arccos ( (e² + f² - d²) / 2ef ). The measure of angle A is not equal to the measure of angle D, so we can conclude that △ABC ≠ △DEF

Therefore, based on the calculation of the angles and the lengths of the sides, it can be concluded that △ABC is not congruent to △DEF.

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Answer: The probability of a student taking the SATs is p = 0.61. Let's consider the first student we sample. The probability that this student does not take the SATs is q = 1 - p = 0.39.

If the first student does not take the SATs, we need to sample another student. The probability that the second student does not take the SATs, given that the first student did not take the SATs, is still q = 0.39.

Similarly, if the second student also does not take the SATs, we need to sample a third student. The probability that the third student does not take the SATs, given that the first two students did not take the SATs, is still q = 0.39.

We continue this process until we find the first student who takes the SATs. Let X be the number of students we sample until we find the first student who takes the SATs. Then X can take on the values 1, 2, 3, and so on.

The probability that we find the first student who takes the SATs on the first try is p = 0.61. The probability that we find the first student who takes the SATs on the second try is (q)(p) = (0.39)(0.61) = 0.238.

Similarly, the probability that we find the first student who takes the SATs on the third try is (q)(q)(p) = (0.39)(0.39)(0.61) = 0.147.

In general, the probability that we find the first student who takes the SATs on the kth try is (q)^(k-1)(p).

Therefore, the probability that we find the first student who takes the SATs in exactly k tries is:

P(X=k) = (q)^(k-1)(p)

We want to find the probability of sampling five successive students at random until we find the first student to take the SATs. So we want to find:

P(X=1) ∗ P(X=2) ∗ P(X=3) ∗ P(X=4) ∗ P(X=5)

= p ∗ (q)(p) ∗ (q)(q)(p) ∗ (q)(q)(q)(p) ∗ (q)(q)(q)(q)(p)

= (0.61) ∗ (0.39)(0.61) ∗ (0.39)(0.39)(0.61) ∗ (0.39)(0.39)(0.39)(0.61) ∗ (0.39)(0.39)(0.39)(0.39)(0.61)

= 0.61 ∗ 0.238 ∗ 0.147 ∗ 0.0905 ∗ 0.0554

≈ 0.00028

Therefore, the probability of sampling five successive students at random until you find the first student to take the SATs is approximately 0.00028 or 0.028%.

Step-by-step explanation:

The mean of the data is X = 85. The variance is 187.5. The standard deviation is σ = 13.7

Standard deviation is a measure of the amount of variability or dispersion of a set of data points from the mean value. It is calculated by finding the square root of the variance, which is the average of the squared differences between each data point and the mean. The standard deviation indicates how far the data points are from the mean on average, with larger values indicating greater variability and smaller values indicating less variability. The standard deviation is a commonly used measure of spread in statistics and is often used to describe the distribution of a dataset. It is denoted by the symbol σ (sigma) for a population and s for a sample.

To find the mean of the data, we add up all the values and divide by the total number of values:

X = (90 + 75 + 60 + 85 + 65 + 80 + 70 + 75) / 8

= 680 / 8

= 85

So the mean of the data is X = 85.

To find the variance, we need to first find the deviation of each value from the mean, square each deviation, and then take the average of the squared deviations:

deviation of 90 from mean = 90 - 85 = 5

deviation of 75 from mean = 75 - 85 = -10

deviation of 60 from mean = 60 - 85 = -25

deviation of 85 from mean = 85 - 85 = 0

deviation of 65 from mean = 65 - 85 = -20

deviation of 80 from mean = 80 - 85 = -5

deviation of 70 from mean = 70 - 85 = -15

deviation of 75 from mean = 75 - 85 = -10

Now, we square each deviation and find the average:

variance = [(5)² + (-10)² + (-25)² + (0)² + (-20)² + (-5)² + (-15)² + (-10)²] / 8

= (25 + 100 + 625 + 0 + 400 + 25 + 225 + 100) / 8

= 1500 / 8

= 187.5

So the variance is 187.5.

Finally, the standard deviation is the square root of the variance:

σ = √(187.5)

= 13.7 (rounded to the nearest tenth)

Therefore, the standard deviation is σ = 13.7 (rounded to the nearest tenth).

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The distance north of Helicopter A from helicopter B is 1. 03 km.

The shortest distance between the polar bear and the conservationists is 9.36 km.

To find the distance from Helicopter A to Helicopter B in the north direction, we can use trigonometry. We can create a right triangle with the north direction as one side, the distance between the helicopters as the hypotenuse, and the angle of 122° between them.

Let N be the north distance between A and B. We can use the sine function:

sin(angle) = opposite side / hypotenuse

sin(122°) = N / 1.2 km

Now, solve for N:

N = 1.2 km x sin (122°) = 1.03 km

The polar bear's position is given as 3 km eastward and on a bearing of 104°. To find the shortest distance between the polar bear and the conservationists, we need to consider the right triangle formed by the eastward direction, the north direction, and the distance between the polar bear and the conservationists as the hypotenuse.

Let D be the shortest distance between the polar bear and the conservationists. We can use the cosine function:

cos(angle) = adjacent side / hypotenuse

cos(104°) = 3 km / D

Now, solve for D:

D = 3 km / cos(104°) = 9.36 km

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Answer: The area of the mosaic is 175 cm². To calculate this, multiply the two sides of each triangle together (7 cm * 5 cm = 35 cm²) and then multiply that by the number of tiles (35 cm² * 20 = 175 cm²).

Step-by-step explanation:

Answer:

perpendicular slope = 2

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

y = - [tex]\frac{1}{2}[/tex] x + 7 ← is in slope- intercept form

with slope m = - [tex]\frac{1}{2}[/tex]

given a line with slope m then the slope of a line perpendicular to it is

[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{-\frac{1}{2} }[/tex] = 2

After simplifying this fraction by dividing both the numerator and denominator by their greatest common factor, which is 1 so the answer is 19/15.

Suppose there are two pizza slices on a plate. One slice is divided into 3 equal parts, and the other slice is divided into 5 equal parts. If you eat 2 out of the 3 parts of the first slice and 3 out of the 5 parts of the second slice, what fraction of the total pizza have you eaten?

To solve the problem, we need to add the fractions together. The fraction of the first slice eaten is 2/3, and the fraction of the second slice eaten is 3/5. We must identify a common denominator in order to add these fractions. In this case, the least common multiple of 3 and 5 is 15. So we need to convert both fractions so that they have a denominator of 15.

To convert 2/3 to a fraction with a denominator of 15, we need to multiply both the numerator and denominator by 5. This gives us 10/15. To convert 3/5 to a fraction with a denominator of 15, we need to multiply both the numerator and denominator by 3. This gives us 9/15.

We can now put the two fractions together because they both have the same denominator. This results in:

10/15 + 9/15 = 19/15

By dividing the denominator and numerator by their 1 greatest common factor, we may simplify this fraction. The final response is thus: 19/15

To act out this problem, we can draw a pizza with two slices, one divided into 3 parts and the other divided into 5 parts. We can then shade in the parts that have been eaten (2 out of 3 for the first slice and 3 out of 5 for the second slice), and count the total number of shaded parts. We can then convert this count into a fraction and simplify it if necessary.

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Answer:

25.1

Step-by-step explanation:

formula for circumfrance is 2piR

so (2pi)(4)= 25.1

:)

The shapes that are nets of a pentagonal-based pyramid are shapes A and D.

A net of a pentagonal-based pyramid is a 2-dimensional pattern that can be folded into a 3-dimensional pentagonal-based pyramid. The net of a pentagonal-based pyramid has the following properties:

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Answer:

Step-by-step explanation: The answer is 115+2c+5c.

Friday the reception desk which gives her 115 dollars.

Thursday and Saturday she cut hair for 2 hrs and 5 hrs respectively.so if she earns ‘c’ dollars per hour for cutting hair it is 2c and 5c respectively.

so the total amount of money is 2c+115+5c