Explain all the basic trigonometric functions

Answer: 2(27+16)

Step-by-step explanation:

The professional set of drawing pencils costs $18.51 more than the basic set. This is a significant difference in price and may be worth the additional cost depending on the user's needs.

First, we need to calculate the difference in price between the professional set and the basic set of drawing pencils. To do this, we will subtract the cost of the basic set ($8.98) from the cost of the professional set ($27.49).

We can write this difference as an equation:

Difference = Professional Set Cost - Basic Set Cost

Difference = 27.49 - 8.98

Next, we can solve for the difference in price by subtracting the two values.

Difference = 18.51

Therefore, the professional set of drawing pencils costs $18.51 more than the basic set. This is a significant difference in price and may be worth the additional cost depending on the user's needs.

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The slope of the line is 2/5 and to represent the "rise" and "run" on the line, a vertical line and a horizontal line are drawn to show the change in y and x, respectively.

To find the slope of the line, we can choose two points on the line and use the formula:

slope = (change in y)/(change in x)

Let's choose the points (-2, 3) and (3, 5) on the line. The "rise" is the change in y and the "run" is the change in x between these two points:

Rise = change in y = 5 - 3 = 2

Run = change in x = 3 - (-2) = 5

So the slope of the line is:

slope = (change in y)/(change in x) = 2/5

Therefore, the slope of the line in simplest form is 2/5.

To represent the "rise and run" on the line, we can draw a vertical line going from the point (-2, 3) to the point (-2, 5) to represent the "rise" of 2 units, and a horizontal line going from the point (-2, 5) to the point (3, 5) to represent the "run" of 5 units.

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The HCF is 10, and the LCM is 2100

The scientific nοtatiοn fοr 36,000,000 is 3.6 107, where 3.6 is the number between 1 and 10, and 7 is the pοwer οf 10, which gives the number οf digits in 36,000,000.

Scientific nοtatiοn is a way οf representing very large οr very small numbers in a mοre cοmpact and manageable fοrm. It invοlves expressing a number as a prοduct οf a cοefficient and a pοwer οf 10.

Scientific nοtatiοn is a way οf expressing numbers that are very large οr very small using a pοwer οf 10.

The general fοrm οf a number in scientific nοtatiοn is a × 10ⁿ, where a is a number between 1 and 10 (but nοt equal tο 10), and n is an integer.

Hence, The number 36,000,000 can be written in scientific nοtatiοn as 3.6 × 10⁷, where 3.6 is between 1 and 10, and 7 is the pοwer οf 10 that gives the number οf zerοs in 36,000,000.

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Complete Question:

Shantel read that the distance between the sun and Mercury is about 36,000,000 miles. Which of the following correctly represents this distance in scientific notation?

A. 36 x 10⁸

B. 3.6 x 10⁷

C. 3.6 x 10⁻⁷

D. 3.6 x 10⁶

In an inverse variation  as one value increases, the other decreases. Option D

Inverse variation is a mathematical relationship between two variables in which the product of the two variables is a constant. In other words, if the value of one variable increases, then the value of the other variable decreases in such a way that their product remains constant.

For example, if y is inversely proportional to x, then we can write it as y = k/x, where k is the constant of variation. As x increases, y must decrease to keep the product of x and y constant.

So, the correct answer is D, as one value increases, the other decreases. As one value decreases, the other increases to maintain the constant product. Inverse variation is also called inverse proportionality.

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Answer:

b) [tex]\sqrt{208}[/tex]

Step-by-step explanation:

The distance from B to D splits the rectangle into two right triangles with the same dimensions. This distance also equals the hypotenuse of either triangle so pick one triangle to work off of. I will work with triangle BCD.

To solve this problem, you must use the Pythagorean Theorem:

[tex]a^{2} + b^{2} = c^{2}[/tex]

where "a" is one leg of the triangle, "b" is the other leg, and "c" is the hypotenuse.

Leg BC = 8 meters and leg DC = 12 meters.

Substitute "a" as leg BC so [tex]a = 8[/tex].

Substitute "b" as leg DC so [tex]b = 12[/tex]

Now we just plug in our values to find "c", the hypotenuse.

[tex]a^{2} + b^{2} = c^{2} \\8^{2} + 12^{2} = c^{2}\\64 + 144 = c^{2}\\208 = c^{2}\\c = \sqrt{208}[/tex]

Therefore, the answer is b) [tex]\sqrt{208}[/tex]

Answer: its 7

Step-by-step explanation:

its 7

Answer:      from year 3 to 4, percentage is 32%

Step-by-step explanation:       the growth is th largest, showing a difference of eight. and 8% is so little it would not make sense

The arithmetic sequence is:

3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43,....

Common difference = 4

First term = 3

The formula for the sum of the first n terms of an arithmetic sequence is:

Sₙ = ⁿ/₂[2a + (n − 1)d]

where:

n = the number of terms to be added.

a = the first term in the sequence.

d = common difference

We are told that sum of the first 9 term is 171. Thus:

(9/2) [2a + (9 − 1)d]  = 171

(2a + 8d) = 38   ----(1)

Sum of the next 5 term is 235.

Thus, sum of the first 14 terms = 235 + 171 = 406

(14/2) [2a + (14 − 1)d]  = 406

7(2a + 13d) = 406

2a + 13d = 58    -----(2)

Subtract eq 1 from eq 2 to get:

5d = 20

d = 4

2a + 13(4) = 58

2a = 58 - 52

a = 3

Sequence is:

3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43,....

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The function to represent the average cost of each satellite television subscription per month would be:

f(x) = (140 + 30x) / 60 for the Orbit company

f(x) = (20 + 35x) / 60 for the Vision company

What is time complexity?

Time complexity refers to the amount of time an algorithm or program takes to run as the input size increases. It is a measure of the efficiency of an algorithm, and it is usually expressed in terms of the "big O" notation.

Cost of Orbit company subscription

= $140 + ($30/month * 12 months * 5 years) = $2140

Cost of Vision company subscription

= $20 + ($35/month * 12 months * 5 years) = $2120

Therefore, the average cost of each satellite television subscription per month for the given period is:

(Cost of Orbit subscription + Cost of Vision subscription) / (12 months * 5 years) = ($2140 + $2120) / (12 * 5) = $35.67 per month

Time complexity O(1), and space complexity V(1) because the calculation only involves basic arithmetic operations and a fixed number of variables.

So, the function to represent the average cost of each satellite television subscription per month would be:

f(x) = (140 + 30x) / 60 for the Orbit company

f(x) = (20 + 35x) / 60 for the Vision company

where x represents the number of years Federico plans to stay in his house.

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Answer:

1/2

Step-by-step explanation:

'%' literally means 'out of 100'. So if it is 50 out of 100, we can write it as 50/100. This can then be simplified down to 1/2.

Answer:

50% can be written as 50/100 or can be simplified and written as 25/50 or 5/25 or 1/5.

If you want to write it as a whole number, you can divide the fraction of 50% ex:- 1/5. So, the whole number you get is 1/5 = 0.2.

(a)the Jordan canonical form of A is:

[tex]\left[\begin{array}{ccc}-1&1&0\\0&3&0\\0&0&3\end{array}\right][/tex]

(b)the Jordan canonical form of A is:

[tex]\left[\begin{array}{ccc}4&0\\0&1\end{array}\right][/tex]

(a) Matrix A = (-1 3)

To find the Jordan canonical form of A, we first need to find the eigenvalues of A. The characteristic polynomial of A is given by:

p(λ) = det(A - λI) = det([-1-λ, 3; 0, 3-λ]) = (λ+1)(λ-3)

So the eigenvalues of A are  λ1= -1 and λ2 = 3.

Next, we need to find a basis for each generalized eigenspace of A. For λ1 = -1, we have:

(A - λ1I) = [tex]\left[\begin{array}{ccc}0&3\\0&4\end{array}\right][/tex]with rank 1

So the dimension of the generalized eigenspace for λ1 is 2 - 1 = 1. We need to find a vector x such that (A - λ1I)x = 0, but (A - λ1I)^2x ≠ 0. In this case, we can take x = (3,0). Then we have:

(A - λ1I)x = [tex]\left[\begin{array}{ccc}1&3\\0&4\end{array}\right][/tex] [tex]\left[\begin{array}{ccc}3\\0\end{array}\right][/tex] = [tex]\left[\begin{array}{ccc}0\\0\end{array}\right][/tex]

(A - λ1I)^2x = [tex]\left[\begin{array}{ccc}1&3\\0&4\end{array}\right][/tex][tex]\left[\begin{array}{ccc}1&3\\0&4\end{array}\right][/tex] [tex]\left[\begin{array}{ccc}3\\0\end{array}\right][/tex]= [tex]\left[\begin{array}{ccc}0\\0\end{array}\right][/tex]

So the generalized eigenspace for λ1 is spanned by the vector x = (3,0).

For λ2 = 3, we have:

(A - λ2I) = [tex]\left[\begin{array}{ccc}-4&3\\0&0\end{array}\right][/tex] with rank 1

So the dimension of the generalized eigenspace for λ2 is 2 - 1 = 1. We need to find a vector x such that (A - λ2I)x = 0, but (A - λ2I)^2x ≠ 0. In this case, we can take x = (3,4). Then we have:

(A - λ2I)x = [tex]\left[\begin{array}{ccc}-4&3\\0&0\end{array}\right][/tex] [tex]\left[\begin{array}{ccc}3\\4\end{array}\right][/tex] = [tex]\left[\begin{array}{ccc}0\\0\end{array}\right][/tex]

(A - λ2I)^2x =[tex]\left[\begin{array}{ccc}-4&3\\0&0\end{array}\right][/tex]  [tex]\left[\begin{array}{ccc}-4&3\\0&0\end{array}\right][/tex] [tex]\left[\begin{array}{ccc}3\\4\end{array}\right][/tex] = [tex]\left[\begin{array}{ccc}0\\0\end{array}\right][/tex]

So the generalized eigenspace for λ2 is spanned by the vector x = (3,4).

Now we can form the Jordan canonical form of A using the basis vectors we found for the generalized eigenspaces:

J = [(c1, 1, 0), (0, λ2, 0), (0, 0, λ2)] = [tex]\left[\begin{array}{ccc}-1&1&0\\0&3&0\\0&0&3\end{array}\right][/tex]

(b) Matrix A = [tex]\left[\begin{array}{ccc}1&2\\3&2\end{array}\right][/tex]

To find the Jordan canonical form of A, we first need to find the eigenvalues of A. The characteristic polynomial of A is given by:

p(λ) = det(A - λI) = det([(1-λ), 2; 3, (2-λ)]) = λ^2 - 3λ - 8 = (λ-4)(λ+1)

So the eigenvalues of A are λ1 = 4 and λ2 = 1.

Next, we need to find a basis for each generalized eigenspace of A. For λ1 = 4, we have:

(A - λ1I) = [tex]\left[\begin{array}{ccc}-3&2\\3&-2\end{array}\right][/tex] with rank 1

For λ2 = 1, we have A - λ2I =

[tex]\left[\begin{array}{ccc}0&2\\3&1\end{array}\right][/tex]

and the corresponding generalized eigenspace is the span of the vectors {(2,-3), (1,0)}. To find a basis for the cycle, we need to find a generalized eigenvector v such that (A - λI)v = (A - I)v = u, where u is the original eigenvector. Solving this equation gives v = (-1,3), so a basis for the cycle is {(2,-3), (1,0), (-1,3)}.

Therefore, the Jordan canonical form of A is:

[tex]\left[\begin{array}{ccc}4&0\\0&1\end{array}\right][/tex]

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The values of a and b in the expression are a = 1 and b = 7

From the question, we have the following parameters that can be used in our computation:

The expression

When the expression on the left hand side is simplified, we have

xy^7

So, we have

xy^7 = x^a * y^b

By comparison, we have

a = 1 and b = 7

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Step-by-step explanation:

vertex             vertical angles

Answer:

4992/640

Step-by-step explanation:

As there are decimals in both, you can move the decimal point by 2!

49.92 will become 4992,

and 6.4 will become 640!

:)

The proportionality constant, based on the information provided, will be 3/7.

The graph, which is based on the information provided, depicts the relationship between the usage rates of blue and orange fabrics.

As a result, the proportionality constant will be determined as follows:

y = kx

0.085 = 0.2k

k = 0.085/0.2

k = 0.425

k = 3/7

Hence, 3/7 is the right answer.

If the corresponding elements of two sequences of numbers, frequently experimental data, have a constant ratio, known as the coefficient of proportionality or proportionality constant, then the two sequences of numbers are proportional or directly proportional.

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Full Question:

Maya uses blue and orange fabric to make identical wall decorations. The graph below shows the relationship between the amounts of blue and orange fabrics used.

The center of the circle is (7, -4). The new center of the new circle is (9, 0). The new equation of the circle is:

(x + 9)² + y² = 6²

Given the circle's center and radius, the equation of a circle offers an algebraic approach to describe a circle. The formulae used to determine a circle's area or circumference are different from the equation for a circle. Many coordinate geometry problems involving circles employ this equation. We need the equation of the circle in order to represent a circle on the Cartesian plane. If we are aware of a circle's centre and radius, we can draw it on a sheet of paper. Similar to this, if we know the coordinates of the center and radius of a circle on a Cartesian plane, we can draw it.

The general equation of the circle is given as:

(x - h)² + (y - k)² = r²

Where, h and k are the coordinates of the center.

The given equation is (x + 7)² + (y - 4)² = 6²

The center is (7, -4).

Given that, the circle is shifted by 3 units to the right and 4 down.

Then the x-coordinate of the circle is:

x + (7 + 3) = (x + 9)

and y-coordinate is:

y - (4 - 4) = y

Thus, the new equation of the circle is:

(x + 9)² + y² = 6²

The new center of the new circle is (9, 0).

Hence, the center of the circle is (7, -4). The new center of the new circle is (9, 0). The new equation of the circle is:

(x + 9)² + y² = 6²

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Answer:21

Step-by-step explanation:

there are four numbers in this data set we divide by 4 to get the mean which is 6.5/4 = 1.6251.

What  is mean ?

The mean is generally the average of a given set of numbers. It is the value that gives the central tendency from a given set of observations.

In statistics, three measures define the central values of the observed data they are mean, median, and mode

The mean for a given set of numbers can be computed in more than one way, including the arithmetic mean method, which uses the sum of the numbers in the series

The mean of a data set is calculated by adding up all the numbers in the data set and then dividing by the total number of values in the set.

In this case, we have a data set of

{1/2,1/3,2/3,11/2}.

Adding these numbers together gives us

1/2 + 1/3 + 2/3 + 11/2 = 6.5.

Since there are four numbers in this data set,

we divide by 4 to get the mean which is 6.5/4 = 1.6251.

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The ultralight airplane traveled a total of 1350 miles in September while tracking the monarch butterflies migrating to Mexico.

A mile is unit of distance on land in English-speaking countries equal to 5280 feet, or 1760 yards.

The ultralight airplane tracked monarch butterflies migrating to Mexico for 30 days in September, with a flight mile of 45 miles each day. To find the total miles traveled by the ultralight airplane in September, we can do the following:

Multiply flight mile per day by the number of days

= 45 miles/day x 30 days

= 1,350 miles.

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Alexis can run at a rate οf 10 km/hr

Time: a measure οf the duratiοn between twο events.

Speed: the rate at which an οbject cοvers a distance.

Distance: the amοunt οf space between twο pοints οr οbjects.

We can use the fοrmula:

rate = distance / time

tο calculate the rate at which Alexis runs and rides his bicycle. Then, we can use the rate οf running tο find οut hοw many kilοmeters he can run in 1 hοur.

Fοr running, rate = distance / time = 5 km / 0.5 hr = 10 km/hr

Fοr riding his bicycle, rate = distance / time = 20 km / 0.8 hr = 25 km/hr

Therefοre, Alexis can run at a rate οf 10 km/hr

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256 in²  is the area of the triangle represented.

What is known as a triangle?

A triangle is a three-sided polygon because it has three vertices. The connection of the three sides end to end at a single point creates the angles of the triangle.

                                 The triangle's three points amount to a sum of 180 degrees. This two-dimensional shape has three straight sides. A triangle is included as a polygon with three sides. The sum of the three angles in a triangle is 180 degrees.

Area of a triangle = 1/2 * base * height

for this triangle:       base = 32 in       height = 16 in

 area = 1/2 ( 32)(16)  = 256 in²

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The equation of a polar curve r^2 = a cos 2θ describes a limaçon, which is a type of polar curve with a loop. When a is positive, the limaçon has an outer loop and an inner loop. When a is negative, the limaçon has a dimple instead of an inner loop.

In this case, a = 81, which is positive, so the limaçon will have an outer loop and an inner loop.

To graph the limaçon, we can start by considering the behavior of the curve at θ = 0, π/4, π/2, 3π/4, and π.

At θ = 0, r^2 = 81cos(2θ) = 81cos(0) = 81, so r = ±9. This means that there are two points on the curve at θ = 0, one at (9, 0) and one at (-9, 0).

At θ = π/4 and 3π/4, cos(2θ) = cos(π/2) = 0, so r^2 = 0 and r = 0. This means that there are two points on the curve at θ = π/4 and two points at θ = 3π/4, all located at the origin.

At θ = π/2, r^2 = 81cos(2θ) = 81cos(π) = -81, which is negative. Since r is always non-negative, there are no points on the curve at θ = π/2.

At θ = π, r^2 = 81cos(2θ) = 81cos(2π) = 81, so r = ±9. This means that there are two points on the curve at θ = π, one at (9, π) and one at (-9, π).

Based on this information, we can sketch the graph of the limaçon as follows:

There are two points on the x-axis, one at (9, 0) and one at (-9, 0).

There are four points at the origin.

There are two points on the line θ = π, one at (9, π) and one at (-9, π).

The curve passes through the origin at θ = π/4, π/2, and 3π/4, and has a loop that encloses the origin.

Overall, the limaçon has a symmetric, flower-like shape with a loop that encloses the origin.

Answer:

an infinity sign

Step-by-step explanation:

We reject the null hypothesis that a sample of 400 observations with mean 95 and standard deviation 12 was randomly drawn from a population with mean 98 at a 5% level of significance.

To test whether a sample of 400 observations has been randomly drawn from a population with a mean of 98, we can use a one-sample t-test. The null hypothesis for this test is that the sample was drawn from a population with a mean of 98, while the alternative hypothesis is that the sample was not drawn from a population with a mean of 98

To perform the t-test, we can first calculate the t-statistic

t = (X - μ) / (s / sqrt(n))

where X is the sample mean, μ is the population mean (which is given as 98), s is the sample standard deviation, and n is the sample size.

Plugging in the values, we get

t = (95 - 98) / (12 / sqrt(400)) = -5

The degrees of freedom for this test are (n - 1) = 399. Using a t-table with 399 degrees of freedom and a significance level of 0.05, we find that the critical t-value is -1.965 (assuming a two-tailed test).

Since our calculated t-value of -5 is less than the critical t-value of -1.965, we can reject the null hypothesis and conclude that the sample was not drawn from a population with a mean of 98 at a significance level of 0.05. Therefore, we can conclude that the mean of the population from which the sample was drawn is unlikely to be 98.

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a) $250

b) C(x) =$200 + x*$50

c) The graph is in the image at the end.

d) The equation is not proportional.

There is a fixed charge of $200 and $50 per month, so the cost after 1 month will be:

C(1) = $200 + $50 = $250

b) Now we want to get the cost after x months, it will be $200 plus x times the $50 charged per month, then we will get:

C(x) =$200 + x*$50

That linear equation models the cost.

c) To graph this, evaluate the line in two different values of x, then graph the obtained points, and connect them with a line.

c(0) = $200 + 0*$50 = $200

So we have the points (0, 200) and (1, 250), with these two we can get the graph of the line you can see at the end.

d) No, the relation is not proportional because there is a fixed value of $200.

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The probability that fewer than two of the seven households are one-person households is:P(x ≤ 1) = P(0) + P(1) = 0.002 + 0.0322 = 0.0342.

The statistical abstract of the United States reports that 38 percent of the country's households are composed of one person. If 7 randomly selected homes are to participate in a Nielsen survey to determine television ratings, the probability that fewer than two of these homes are one-person households is 0.0813.Explanation:Given that the statistical abstract of the United States reports that 38% of the country's households are composed of one person.Now, we have to find the probability that fewer than two of these homes are one-person households when 7 randomly selected homes participate in a Nielsen survey to determine television ratings.

We are given: The proportion of one-person households = 38% = 0.38. The proportion of households with more than one person = 100% - 38% = 62% = 0.62. We are supposed to find the probability that fewer than two of the seven households are one-person households.The required probability is the sum of the probabilities of selecting 0 one-person households and 1 one-person household.

P(x ≤ 1) = P(0) + P(1)P(0) = probability of selecting 0 one-person households P(1) = probability of selecting 1 one-person householdn = 7, r = 0 P(0) = (7C0)(0.38)^0(0.62)^(7-0) = 0.002 P(1) = (7C1)(0.38)^1(0.62)^(7-1) = 0.0322

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Clara is earning money at a faster rate than Tiffany, with a difference of $4 per hour.

A graph in x-y axis plot is a visual representation of mathematical functions or data points on a Cartesian coordinate system.

To determine who is earning money at a faster rate, we need to calculate the hourly earnings for each person.

For Tiffany, the hourly earnings can be calculated as follows:

Hourly earnings = Earnings / Time

Hourly earnings = 40 / 5 = 8

Hourly earnings = 80 / 10 = 8

Hourly earnings = 120 / 15 = 8

So, Tiffany's hourly earnings are a constant $8 per hour.

For Clara, we can estimate her hourly earnings from the graph. The graph shows that Clara earns $60 for 5 hours of work, $120 for 10 hours of work, and $180 for 15 hours of work. Therefore, her hourly earnings can be calculated as follows:

Hourly earnings = Earnings / Time

Hourly earnings = 60 / 5 = 12

Hourly earnings = 120 / 10 = 12

Hourly earnings = 180 / 15 = 12

So, Clara's hourly earnings are a constant $12 per hour.

Thus, Clara is earning money at a faster rate than Tiffany, with a difference of $4 per hour.

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The statements that are true about the quadratic function and its graph include the following:

In Mathematics, the graph of any quadratic function or equation always forms a parabola because it is a u-shaped curve. For the given quadratic function, the graph is a upward parabola because the coefficient of x² is positive i.e when the value of "a" is greater than zero.

Next, we would determine the statements about the quadratic function and its graph that are true:

At point (-10, 82), we have:

[tex]f(x) = \dfrac{1}{5} \ x^2 - 5x + 12[/tex]

[tex]f(x) = x^2/5 - 5x + 12[/tex]

[tex]f(-10) = -10^2/5 - 5(-10) + 12[/tex]

[tex]f(-10) = 82[/tex]

At point (20, -8), we have:

[tex]f(x) = \dfrac{1}{5} \ x^2 - 5x + 12[/tex]

[tex]f(x) = x^2/5 - 5x + 12[/tex]

[tex]f(20) = 20^2/5 - 5(20) + 12[/tex]

[tex]f(20) = -8[/tex]

In conclusion, the graph of this quadratic function does not contain the point (0, 0) as shown in the image attached below.

Answer wil be (4/14) .

.......

Answer:

Answer 4/14

Step-by-step explanation: