Find the equation for the ellipse

Note that the graphing form and vertex of each quadratic function are given as follows:


a) for f(x)=x² +6x+15, the graphing form of the quadratic function is f(x) = (x + 3)² + 6. The vertex is (-3, 6).

b) for y= x²-4x+9 the graphing form of the quadratic function is y = (x - 2)² + 5. The vertex is (2, 5).

c) for  f(x)= x² +8x, the graphing form of the quadratic function is f(x) = (x + 4)² - 16. The vertex is (-4, -16).

d) for . y=x² - 4x+9, the graphing form of the quadratic function is y = (x - 2)² + 5. The vertex is (2, 5).

a. To complete the square for the quadratic function f(x)=x² +6x+15, we add and subtract the square of half of the coefficient of x, which is (6/2)² = 9:

f(x) = x² + 6x + 15

= (x² + 6x + 9) + 6

= (x + 3)² + 6

Therefore, the graphing form of the quadratic function is f(x) = (x + 3)² + 6. The vertex is (-3, 6).

b. To complete the square for the quadratic function y= x^2-4x+9, we add and subtract the square of half of the coefficient of x, which is (-4/2)² = 4:

y = x^2 - 4x + 9

= (x^2 - 4x + 4) + 5

= (x - 2)² + 5

Therefore, the graphing form of the quadratic function is y = (x - 2)² + 5. The vertex is (2, 5).

c. To complete the square for the quadratic function f(x)= x² +8x, we add and subtract the square of half of the coefficient of x, which is (8/2)² = 16:

f(x) = x² + 8x

= (x² + 8x + 16) - 16

= (x + 4)² - 16

Therefore, the graphing form of the quadratic function is f(x) = (x + 4)² - 16. The vertex is (-4, -16).

d. To complete the square for the quadratic function y=x² - 4x+9, we add and subtract the square of half of the coefficient of x, which is (-4/2)² = 4:

y = x² - 4x + 9

= (x² - 4x + 4) + 5

= (x - 2)² + 5

Therefore, the graphing form of the quadratic function is y = (x - 2)² + 5. The vertex is (2, 5).

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The features of the given function are;

y-intercept at (0,10)

vertical asymptote of x = -4

A function is a relation in which each possible input value leads to exactly one output value. We say “the output is a function of the input.” The input values make up the domain, and the output values make up the range.

The function is given as;

g(x) = f(x + 4) + 8

The domain of a function is a set of all possible input values while the range of a function is defined as a set of all possible output values.

A vertical asymptote of a graph is a vertical line x = a where the graph tends toward positive or negative infinity as the inputs approach a.

Thus, vertical asymptote is x = -4.

The he feat will be at (0, 10)

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Answer:

YOU HAVE TO USE THE VECTORS AS WELL MATRIX TO GET THE SOLUTION OF THIS QUESTION

Answer:

To find the price of 3 tons of concrete, we need to first figure out how much Trey paid per ton of concrete.

Since Trey paid 42 dollars for 2/3 ton of concrete, we can set up a proportion:

2/3 ton = 42 dollars

1 ton = x dollars

To solve for x, we can cross-multiply:

(2/3) * x = 42 * 1

2x = 126

x = 63

So, Trey paid 63 dollars per ton of concrete.

To find the price of 3 tons of concrete, we can multiply the cost per ton by the number of tons:

3 tons * $63/ton = $189

Therefore, the price of 3 tons of concrete is $189.

The general form of the equation of this quadratic function is f(x) = x² - 2x + 1.

In Mathematics, the standard form of a quadratic equation is represented by the following mathematical equation;

ax² + bx + c = 0

Based on the information provided about this quadratic equation, it comprises the following ordered pairs;

Center (h, k) = (1, 0),

(x, y) = (0, 1).

By substituting the ordered pairs into the standard form of a quadratic equation, we have:

f(x) = a(x - h)² + k

1 = a(0 - 1)² + 0

1 = a

Therefore, the required quadratic equation is given by:

f(x) = 1(x - 1)² + 0

f(x) = (x - 1)(x - 1)

f(x) = x² - x - x + 1

f(x) = x² - 2x + 1

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The formula d=b2−4ac when solved for b = ± [tex]\sqrt{d +4ac}[/tex]. Option B

Subject of formula is defined as a variable that is being solved for in an equation.

It is the variable that is made to stand out in an equation. It is the variable that is made to stand alone on one end of the equality sign.

From the information given, we have that;

d=b2−4ac

Now, collect the like terms

d + 4ac = b²

Find the square root of both sides, we have that;

b = [tex]\sqrt{d + 4ac}[/tex]

Insert the plus-minus sign to show the interval, we have;

b = ± [tex]\sqrt{d +4ac}[/tex]

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According to the solving Triangle The Value of X in the given figure is 20°

Triangles are three-sided polygons with three vertices. The angles of the triangle are formed by connecting the three sides end to end at a point. The total of the triangle's three angles equals 180 degrees which is called interior angles.

The three angles are given as ∠1=84°, ∠2=41°, ∠3=2x+15°.

Adding three angles

84° + 41° + 2x + 15° = 180° (sum of interior angle is 180°)

140° + 2x = 180°

2x = 180°-140°

2x = 40°

x = 20°

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In triangles ABC and PQR, the ratio of the measure of side AB and side PQ is 2 : 3 and the ratio of the measure of sides BC and QR is 2 : 3. Thus, AB/PQ = BC/QR. Also, the m∠B = m∠Q. The ratio of two pairs of corresponding sides is congruent and the included angle is equal. Hence, ΔABC  ~ ΔPQR by SAS similarity theorem.

SAS similarity theorem is a theorem in geometry that stands for "Side-Angle-Side Similarity Theorem".

This theorem states that if two triangles have two pairs of corresponding sides that are proportional in length, and the included angles between these sides are congruent (have the same measure), then the two triangles are similar.

We can see that examining the two triangles, it is clear that it is in line with SAS similarity theorem.

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Answer:

It is not clear what is meant by "a log can Kate use to fill remaining 1/3 meters." However, if we assume that Kate has a gap of 1/3 meters that she needs to fill, she could use a log that is 1/3 meters long to fill the gap. Alternatively, she could use any other object that is 1/3 meters long to fill the gap.

Answer:

Step-by-step explanation:

[tex]i=80[/tex]° (opposite angles in a cyclic quadrilateral are supplementary)

[tex]e=84[/tex]° (An exterior angle of a cyclic quadrilateral is equal to the interior

             angle at the opposite vertex.)      

The domain and range of given graph is Domain: (-∞, ∞) Range: (0, ∞)

What is Function ?

Function can be defined in which it relates an input to output.

The horizontal axis (x-axis) represents the input values, while the vertical axis (y-axis) represents the output values.

To identify the domain and range of this function from the graph, we can look at the values that the function takes on as well as the values that are excluded.

From the graph, we can see that the function appears to be increasing without bound as the input values increase. This means that the domain of the function is all real numbers or (-∞, ∞).

We can also see that the function appears to be taking on only positive values, and never touches or crosses the x-axis. This means that the range of the function is all positive numbers or (0, ∞).

Therefore, The domain and range of given graph is Domain: (-∞, ∞) Range: (0, ∞)

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Answer:

C.

Step-by-step:

The measure of angle A is congruent to the measure of angle P.

Answer:

The answer is (f.g)(x) = 12x^2 - 9.

Step-by-step explanation:

To find (f.g)(x), we need to apply the composition of functions formula, which is:

(f.g)(x) = f(g(x))

First, we need to find g(x):

g(x) = 3x^2

Now, we need to plug g(x) into f(x) in place of x:

f(g(x)) = 4(g(x)) - 9

= 4(3x^2) - 9

= 12x^2 - 9

Therefore, (f.g)(x) = 12x^2 - 9.

Hope this helps, I'm sorry if it doesn't! If you need more help, ask me! :]

The inverse functions and the domains and ranges of the inverse functions can be presented as follows;

A. The inverse function of f(x) is; [tex]f^{-1}(x) = \frac{3-x}{7\cdot x}[/tex]

B. f(g(x)) = [tex]\frac{3}{7\times \frac{(3-x)}{7\cdot x} + 1}[/tex] = x

C. The inverse function reverses the effect of the function. The inverse function is therefore;

x [tex]{}[/tex]      y

10[tex]{}[/tex]     5

6[tex]{}[/tex]      8

5[tex]{}[/tex]      9

4[tex]{}[/tex]      13

2) Domain; 10, 6, 5, 4

Range; 5, 8, 9, 13

An inverse function maps the output, or y-value of a function to the input value or x-value of the function.

2. A. f(x) = 3/(7·x + 1)

The inverse function can be found as follows;

Let f(x) = y

y = 3/(7·x + 1)

(7·x + 1) = 3/y

7·x = (3/y) - 1 = (3 - y)/y

x = (3 - y)/(7·y)

Plugging in x = f⁻¹(x), and y = x, to get;

B. g(x) = (3 - x)/(7·x)

f(x) = 3/(7·x + 1)

f(g(x)) = 3/(7 × ((3 - x)/(7·x)) + 1) = 3/(((3 - x)/(x)) + 1) = 3/(3/x - 1 + 1)

3/((3/x - 1 + 1) = 3/(3/x) = 3 × x/3 = x

Therefore; f(g(x)) = 3/(7 × ((3 - x)/(7·x)) + 1) = x

C. The inverse of the function is the function that produces the input of the function from the function's output, therefore, the inverse of the function can be presented as follows;

x [tex]{}[/tex]     y

10[tex]{}[/tex]    5

6[tex]{}[/tex]     8

5[tex]{}[/tex]     9

4[tex]{}[/tex]     13

2) The domain is the set of the possible inputs of the function, and the range is the set of the possible outputs of the function.

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The equation would be y = 0.50x + 10.00, and we have drawn the graph for the same.

What is the graph of the equation?

The graph of an equation is a visual representation of the set of all solutions (points) that satisfy the equation.

To find the equation of the line, we use the slope-intercept form of a linear equation, which is:

y = mx + b

where m is the slope of the line and b is the y-intercept (the point where the line crosses the y-axis).

In this case, we are given the slope of the line (0.50) and the y-intercept (10.00), so we can simply substitute these values into the equation:

y = 0.50x + 10.00

This gives us the equation of the line in slope-intercept form. We can use this equation to find the cost of a cab ride for any distance x (in miles) by substituting the value of x into the equation and solving for y (the fare).

Now let's draw the graph of the equation y = 0.50x + 10.00, we get

Hence, the equation would be y = 0.50x + 10.00, and we have drawn the graph for the same.

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The value of the expression -x³ + 3x² - x + 20 when x = -1 is 23.

Given the expression in the question;

-x³ + 3x² - x + 20

To determine the value of the expression,

First, we substitute -1 for x in the expression:

-( -1 )³ + 3( -1 )² − ( -1 )) + 20

Next, we simplify each term using the order of operations (also known as PEMDAS):

-( -1 )³ = -( -1 ) = 1

(recall that the exponent is evaluated first, then the negative sign is applied)

3( -1 )² = 3(1) = 3

(recall that the exponent is evaluated first)

-( -1 ) = 1

(recall that subtracting a negative is the same as adding a positive)

Now, putting these simplified terms back into the original expression, we get:

1 + 3 + 1 + 20

Finally, we add these terms together to get the final answer:

23

Therefore, the value of the expression is 23.

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Answer:

See below.

Step-by-step explanation:

We're asked to prove that ΔSTV ≅ ΔTUW.

Let's begin by reviewing what was given to us.

Because we already have one side and an angle congruent, we need to prove that another angle or side is congruent.

Our goal is to find a Congruency Postulate to prove that ΔSTV ≅ ΔTUW.

As we can see in the diagram, ∠STV ≅ ∠UTV as they're vertical angles. This is possible because Line Segment SV ≅ & ║ Line Segment TW.

We are now able to prove that ΔSTV ≅ ΔTUW with the Angle-Side-Angle Triangle Congruency Postulate (ASA). We proved, and were given 2 angles, and 1 side.

Summary:

∠STV ≅ ∠UTV | Vertical Angles.

ΔSTV ≅ ΔTUW | ASA.

Answer:

James needs 5 gallons of water.

Step-by-step explanation:

1 gallon = 8 pints

40 ÷ 8 = 5

5 gallons

[tex]\dfrac{x^{2} + 10x + 21}{x + 7} = \dfrac{(x+3)(x+7)}{x + 7} = \bf x + 3[/tex]

___

[tex]x^{2} + 10x + 21 = x^{2} + 3x + 7x + 21 = x(x+3)+7(x+3) = (x+3)(x+7)\\[/tex]

Answer:

A

Step-by-step explanation:

Range referes to the smallest and largest values, so the answer you have circles is correct.

Good Job!

Answer: A) Range

Step-by-step explanation:

The range is the difference between the biggest and smallest numbers (or, in this case, folders). If Jack wants to find a notebook that will fit a small and large folder, he will need to find the range.

The ball strikes the grοund after apprοximately 1.07 secοnds.

Time: a measured οr measurable periοd during which an actiοn, prοcess, οr cοnditiοn exists οr cοntinues.

Speed: the rate at which sοmething mοves, οperates, οr happens.

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

We can sοlve this prοblem using the kinematic equatiοns οf mοtiοn. We knοw the initial velοcity (u = 7.30 m/s), the initial height (h = 2.91 m), and we want tο find the time taken fοr the ball tο hit the grοund (t).

Let's use the kinematic equatiοn that relates the final pοsitiοn (s), initial pοsitiοn (h), initial velοcity (u), acceleratiοn (a), and time (t):

[tex]s = h + ut + (1/2)at^2[/tex]

Since the ball is thrοwn vertically dοwnwards, the acceleratiοn is [tex]-9.81 m/s^2[/tex] (negative because it is in the οppοsite directiοn tο the initial velοcity).

At the mοment the ball hits the grοund, its final pοsitiοn s is zerο. Therefοre, we can rewrite the equatiοn as:

[tex]0 = h + ut + (1/2)at^2[/tex]

Solving for t, we get:

[tex]t = [-u\±\sqrt{(u^2 - 2ah)} ] / a[/tex]

We take the positive solution, since time cannot be negative. Substituting the values we get:

[tex]t = [ -7.30 \± \sqrt{((7.30)^2 - 2(-9.81)(2.91))} ] / (-9.81)[/tex]

[tex]t = [ -7.30 \± \sqrt{ (53.25)} ] / (-9.81)[/tex]

t ≈ 1.07 s οr t ≈ 0.31 s

The negative sοlutiοn dοesn't make physical sense in this cοntext, sο we discard it. Therefοre, the ball strikes the grοund after apprοximately 1.07 secοnds.

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Therefore, the likelihood of a vehicle accident just on road is 3/25. The ultimate solution is 3/25 because there is no other way to simplify this fraction.

The probability is typically defined as the proportion of favourable findings to all results inside the poll area. The formula for probability of an event is P(E) = (Number of positive occurrences) (Sample space).

If the odds against a car crash on the road are stated as "22:3", it means that for every 22 unfavorable outcomes (no car crash), there are 3 favorable outcomes (car crash).

The total number of outcomes is 22 + 3 = 25.

Therefore, the probability of a car crash on the road is:

3 favorable outcomes / 25 total outcomes = 3/25

So, the probability of a car crash on the road is 3/25. This fraction cannot be reduced further, so the final answer is 3/25.

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By the squeeze theοrem, lim f(x) = 0 as x apprοaches -5.

The squeeze theοrem, alsο knοwn as the sandwich theοrem, is a theοrem in calculus that allοws us tο find the limit οf a functiοn using the limits οf twο οther functiοns that "squeeze" the οriginal functiοn between them.

We have the inequality:

-2x² - 20x - 51 ≤ f(x) ≤ 2x² + 20x + 49

We can rewrite this inequality as:

-2(x+5)(x+4.5) ≤ f(x) ≤ 2(x+5)(x+4.5) + 1

Nοte that we have factοred -2x² - 20x - 51 as -2(x+5)(x+4.5) and 2x² + 20x + 49 as 2(x+5)(x+4.5) + 1.

As x apprοaches -5, we can see that (x+5) and (x+4.5) apprοach 0, sο we have:

-2(0)(0) ≤ lim f(x) ≤ 2(0)(0) + 1

0 ≤ lim f(x) ≤ 1

Therefοre, by the squeeze theοrem, we have:

lim f(x) = 0 as x apprοaches -5.

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]Therefore , the solution of the given problem of deviation comes out to be  the confidence interval contains both positive and negative values.

Statistics uses variance as a gauge of difference. The image of that figure is used to compute the average variance between the collection and the mean. It includes those data points itself into the calculations, unlike many other real measures of variability, by comparing also every figure to the mean.

Here,

Since the population standard deviation is unknown and the sample number is less than 30, we can use a t-distribution to build the confidence interval. The confidence interval's calculation is:

=>CI =  x t /2 × (s/n).

Since we need a 99% confidence range, = 0.01 and /2 = 0.005, and (n-1) = (49-1) = 48 is the number of degrees of freedom.

We discover that t/2 = 2.678 using the t-distribution table with 48 degrees of freedom and 0.005 significance level.

When we enter the numbers, we obtain:

=> CI = 3.1 ± 2.678 * (16.5/√49) = 3.1 ± 7.34

The estimated mean net change in LDL cholesterol following treatment, according to the 99% confidence range, is (4.24, 10.44) mg/dL.

We cannot say with 99% certainty that garlic is effective at lowering cholesterol because the confidence interval contains both positive and negative values.

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Answer:

38°

Step-by-step explanation:

this equal to 38° because angle EGD and angle IGJ are corresponding angles

The distributive property expressions 3(20 + 16) and 4(13 + 14) are equal to 108

An equation is an expression that shows how two or more numbers and variables are related using mathematical operations of addition, subtraction, multiplication, division, exponents and so on.

The distributive property is given as:

a(b + c) = ab + ac

3(20 + 16) = 3(20) + 3(16) = 60 + 48 = 108

Also:

4(13 + 14) = 4(13) + 4(14) = 52 + 56 = 108

The distributive property expressions 3(20 + 16) and 4(13 + 14) are equal to 108

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Answer:

We can use the distance formula to find the distance between the center of the circle (2, -3) and the point (-1, -6), which should be equal to the radius of the circle.

Step-by-step explanation:

Distance = √[(x2 - x1)² + (y2 - y1)²]

Distance = √[(-1 - 2)² + (-6 - (-3))²]

Distance = √[(-3)² + (-3)²]

Distance = √(18)

So the radius of the circle is equal to √(18) or approximately 4.24.

Based on the above, the Perimeter of the triangle DEF is 43.3 units

A mathematical statement known as an equation demonstrates the relationship between two or more numbers and variables by utilizing mathematical operations such as addition, reduction, multiplication, division, exponents, and so forth.

Given the triangle DEF. Using trigonometric ratio:

sin(38) = EF / 18EF = 11.1

Also: cos(38) = DF / 18DF = 14.2

The perimeter of triangle

DEF = DE + EF + DF

= 18 + 11.1 + 14.2 = 43.3 units

Hence, the Perimeter is 43.3 units

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