Part 5: Identify key features and graph a hyperbola from general form.
Answer the following questions to determine the key features of the hyperbola based on the equation
shown, and then graph it.
4x^2 + 8x – 9y^2 – 36y – 68 = 0
a) Write the standard equation of the hyperbola. (3 points)
b) What is the center of the hyperbola? (1 point)
c) What are the foci of the hyperbola? (2 points)
d) What are the asymptotes of the hyperbola? (2 points)
e) Sketch a graph of the hyperbola and label the center, foci, guiding box, and asymptotes. (4 points)
Answers
a) [tex]\frac{(x+1)^{2}}{3^{2}}-\frac{(y+2)^{2}}{2^{2}} = 1[/tex].
b) (h, k) = (-1, -2).
c) F₁ (x, y) = (-1 - √13, -2), F₂ (x, y) = (-1 + √13, -2).
d) y = ±2 · (x + 1) / 3 - 2.
e) See the image attached below.
How to analyze a hiperbolaIn this question we must analyze the general equation of a hyperbola and derive all the required information by algebraic and graphic means. All this information is derived from the standard equation of the hyperbola, which is presented below:
[tex]\frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}} = 1[/tex] (1)
Where:
a - Horizontal axis distanceb - Vertical axis distance h, k - Coordinates of the center of the hyperbola.The foci are located at the following two points: [tex]F_{1} (x, y) = (h - c, k)[/tex], [tex]F_{2} (x,y) = (h + c, k)[/tex].
Where [tex]c = \sqrt{a^{2}+b^{2}}[/tex].
And the asymptotes of the hyperbola are described by the following formulas:
y₁ = b · (x - h) / a + k (2)
y₂ = - b · (x - h) / a + k (3)
a) We transform the general equation into its standard form:
4 · x² + 8 · x - 9 · y² - 36 · y - 68 = 0 Given(2 · x)² + 4 · (2 · x) - (3 · y)² - 12 · (3 · y) - 68 = 0 Definition of power/Associative and commutative properties/Definition of multiplication[(2 · x)²+ 2 · 2 · (2 · x) + 4] - [(3 · y)² + 2 · 6 · (3 · y) + 36] + 36 - 68 - 4 = 0 Compatiblity with addition/Definition of multiplication/Commutative and associative properties/(-1) · a = - a(2 · x + 2)² - (3 · y + 6)² - 36 = 0 Perfect square trinomial/Definition of addition4 · (x + 1)² - 9 · (y + 2)² = 36 Compatibility with addition/Existence of additive inverse/Modulative property[tex]\frac{(x+1)^{2}}{3^{2}}-\frac{(y+2)^{2}}{2^{2}} = 1[/tex] Definition of power/Compatibility with multiplication/Existence of multiplicative inverse/ResultThe standard equation of the hiperbola is [tex]\frac{(x+1)^{2}}{3^{2}}-\frac{(y+2)^{2}}{2^{2}} = 1[/tex]. [tex]\blacksquare[/tex]
b) From the result found in a) we conclude that the center of the hyperbola is (h, k) = (-1, -2).
c) The foci of the hyperbola are located at the following two points:
[tex]c = \sqrt{3^{2}+2^{2}}[/tex]
[tex]c = \sqrt{13}[/tex]
F₁ (x, y) = (-1 - √13, -2), F₂ (x, y) = (-1 + √13, -2)
The foci of the hyperbola are F₁ (x, y) = (-1 - √13, -2) and F₂ (x, y) = (-1 + √13, -2). [tex]\blacksquare[/tex]
d) And the asymptotes of the hyperbola are presented below:
y₁ = 2 · x / 3 (2)
y₂ = - 2 · x / 3 (3)
The asymptotes of the hyperbola are y₁ = 2 · (x + 1) / 3 - 2 and y₂ = - 2 · (x + 1) / 3 - 2. [tex]\blacksquare[/tex]
e) The graph of the hyperbola and all its characteristics are presented by means of graphing tool. The result is presented in the image attached below.
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Answer:
The equation of the hyperbola = (x + 1)^2/3^2 - (y + 2)^2/2^2 = 1
The center of the hyperbola = (-1, -2) (h, k)
The vertices of the hyperbola (my assingment asked me for vertices) = (2, -2) and (-4, 2)
The asymptotes = one at y = (2x+2/3) - 2 and the other is at
y = -((2x+2/3) -2)).
Step-by-step explanation:
For the graphing part plug the equation into desm0s and the asymptote equation. For the guiding box part, you use the B variable. Since b = 2 in this problem, you count up two spaces from the center (-1,-2) and then down two spaces. These will be the top and bottom of your guiding box. Then use your vertices as the sides and make a box that connects.
Good luck!
Related Questions
Can anyone tell me why this is wrong
Answers
Answer:
i, n, e, q, u, a, l, t, y
Step-by-step explanation:
You repeated some letters.
pls give me brainliest and have a nice day! Hope it helps!
What is the range for this set of data? 38, 17, 55, 40 02 38 39 O 72
Answers
Answer:
38
Step-by-step explanation:
The range is the difference between the highest value and the lowest value. Here the highest value is 55 and the lowest 17. Therefore, the range is take away 17 from 55 which equals to 38
The range for this set of data is 38.
Option B is the correct answer.
What is range in a set of data?The range is a measure of dispersion or spread in a set of data.
It is defined as the difference between the largest and the smallest values in the data set.
Range provides an idea of how much the data values are spread out or dispersed.
If the range is small, it indicates that the data values are close together, while a large range indicates that the data values are spread over a wider range of values.
We have,
To find the range of a set of data, we need to subtract the smallest value from the largest value.
Here, the smallest value is 17 and the largest value is 55.
So,
The range of the data set is:
55 - 17 = 38
Thus,
The range for this set of data is 38.
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It takes 3 eggs to make a cake .how many cakes can you make with 37 eggs
Answers
Answer:
12 cakes
Step-by-step explanation:
37 / 3 = 12.33333
Refer to the diagram shown.
I need help with this please help me
Answers
Answer: Angle-Side-Angle Theorem
Step-by-step explanation:
The two triangles are shown to have congruent angles. By the Reflexive Property of Congruence, Side PM is shared and congruent between the two.
Because the angles are on either side (left and right) of the shared side, the triangles are congruent by the Angle-Side-Angle Theorem.
Two car dealerships pay their employees a monthly salary plus a commission on cars that they sell.
Answers
The person who is right on the dealership that pays a higher commission, is Tyson because Dealership A pays the higher commission.
How to find the commission paid ?The commission paid on Dealership A cars can be found by the formula :
= ( Monthly income for 4 cars - Monthly income for 1 car ) / Difference in number of cars
= ( 3, 850 - 1, 900 ) / ( 4 - 1 )
= $ 650
The commission paid on Dealership B cars would be:
= ( Monthly income for 4 cars - Monthly income for 1 car ) / Difference in number of cars
= ( 3, 500 - 2, 000 ) / ( 4 - 1 )
= 1, 500 / 3
= $ 500
Dealership A pays the higher commission.
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There are 5 yellow golf balls, 3 red golf balls, and 10 white golf balls in a bag. What are the odds of randomly choosing a yellow golf ball?
Answers
Answer:
5/18
Step-by-step explanation:
Answer:
2
Step-by-step explanation:
1+1=2
The number of views for a video on a social media website doubles every day. You begin observing the
number of views 8 days after the video was introduced. The number y of views x days after the video is introduced is
represented by the function = 22(2^x-8). When will there be one million views?
Answers
The total number of times social media browsers have been showed your content.
How is social media value calculated?A general, but simplified earned media value formula is: EMV = impressions x cost per 1000 impressions x adjustable variable. The adjustable variable can be anything you are looking to track like engagement or impressions.
Simplifying
22 = 2(x + 8)
Reorder the terms:
22 = 2(8 + x)
22 = (8 * 2 + x * 2)
22 = (16 + 2x)
Solving
22 = 16 + 2x
Solving for variable ‘x’.
Move all terms containing x to the left, all other terms to the right.
Add ‘-2x’ to each side of the equation.
22 + -2x = 16 + 2x + -2x
Combine like terms: 2x + -2x = 0
22 + -2x = 16 + 0
22 + -2x = 16
Add ‘-22’ to each side of the equation.
22 + -22 + -2x = 16 + -22
Combine like terms: 22 + -22 = 0
0 + -2x = 16 + -22
-2x = 16 + -22
Combine like terms: 16 + -22 = -6
-2x = -6
Divide each side by ‘-2’.
X = 3
Simplifying
X = 3
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salve for y √(7-x)²+(9-y)²
Answers
The solution for y in the equation √(7 - x)² + (9 - y)² is infinite many solutions
How to determine the solution for yFrom the question, we have the following parameters that can be used in our computation:
√(7-x)²+(9-y)²
Express tthe equation properly
So, we have the following representation
√(7 - x)² + (9 - y)²
The above is an expression, not an equation
This means that the value of the variable y can take any value and the equation will be true
In other words, it means that the variable y has infinite many solutions because it is not an equation but it is an expression
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Review the diagram of a locket in the shape of an ellipse.
The major and minor axes are labeled in the diagram.
If the center of the locket is at the origin of a coordinate
plane, which equation represents the locket?
x2
361
+
72
= 1
169
OK
V2
361
+
X2
= 1
169
38 mm
x2
1.444
1
676
26 mm
0 ' '
x2
676
= 1
1,444
Answers
Answer:
D
Step-by-step explanation:
If a = 38 and b = 26 then the equation would be y^2/1444 + x^2/676 = 1
also just graph it or smth
The equation of the ellipse in the figure is x^2/676 + y^2/1296 = 1
How to determine the equation?The figure is an ellipse, and it has the following parameters:
Center, (h,k) = (0,0)Major radius = 36Minor radius = 26The equation of the ellipse is then calculated using:
x^2/r^2 + y^2/R^2 = 1
So, we have:
x^2/26^2 + y^2/36^2 = 1
Evaluate the exponents
x^2/676 + y^2/1296 = 1
Hence, the equation of the figure is x^2/676 + y^2/1296 = 1
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A movie theater sells popcorn in bags that open to form similar prisms, as shown. The large bag has a volume of 576 cubic inches. The small bag is 12 in tall and the large bag is 16 in tall. What is the volume of the small bag?
Answers
Which of the following statements must be true based on the diagram below?
Select all that apply. (Diagram is not to scale.)
Answers
As a result, after studying the diagram below of triangle and considering the reasons that follow, the appropriate answers are 1, 2, 3, and 4.
Define triangle.Any three points in a plane can be connected to create triangles, which are polygonal (shapes) with three sides and three angles. They are among the first shapes in geometry that are explored. Since each polygon (with 4, 5, 6, or n sides) may be divided into triangles, triangles are particularly significant.
Here,
A segment bisector in this case is 1)OP.
Since segment LKMN is not divided into two equal sections, NO OP is not a segment bisector.
A perpendicular bisector is
2)OP.
NO,Because a perpendicular bisector creates an angle to the normal and divides a segment into two equal halves, OP is not a perpendicular bisector.
The angle bisector NO is OP.
Due to the fact that OP they do not divide any angles into two equal pieces.
4) The diagram's vertex of a pair of congruent angles is O.
YES Since O lines connect two angles with the same degree ()
5)O is the right angle's vertex.
NO
As a result of things not being at a straight angle.
6) A segment's midway in the diagram
By drawing diagonals across the four vertices LMNK, it is possible to determine where this segment's midpoint lies.
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AB=25 , BD=4, Solve for the radius?
Please help!!!!!
Answers
9514 1404 393
Answer:
r = 14.5
Step-by-step explanation:
AD is the perpendicular bisector of chord EC, so is the diameter of the circle. The radius is half the diameter.
r = AD/2 = (AB +BD)/2 = (25 +4)/2 = 29/2
r = 14.5
__
Segments BE and BC are 10 = √(25·4)
speeeeeeeedddddddddddddddddddddddddddddddd
Answers
Answer:
I would be ...
75/8
multiple 9 by 8 and then add 3
Hope this helps!
Step-by-step explanation:
;)
Answer:
75/8!
Step-by-step explanation:
If f(x) = 2x4 - 3x + 3, then what is the remainder when f(x) is divided by x - 1?
Answers
The remainder on dividing 2x⁴-3x+3 by x-1 is 2
What are functions?Function, in mathematics, is an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).
Given here: 2x⁴-3x+3 by x-1
The remainder theorem states that when a polynomial p(x) is divided by (x - a), then the remainder = f(a)
∴ The remainder would is f(1) = 2×1-3+3
=2
Hence, The remainder on dividing 2x⁴-3x+3 by x-1 is 2
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Which of the following fractions is equivalent to 10/12
Answers
Answer:
I don't know what the options are so I'm just gonna list a few and if they aren't on there lmk the options you have to choose from
Step-by-step explanation:
5/6, 15/18, 20/24, 25/30, 30/36, 35/42, 40/48, 45/54, 50/60, 55/66, 60/72, 65/78, 70/84, 75/90, 80/96, 85/102, 90/108, 95/114, 100/120
Answer:
5/6
Step-by-step explanation:
the equivalent fraction of 10/12 is 5/6 because both the numerator and the denominator are divisible by 2.
You find a mutual fund that offers approximately 6% APR compounded
monthly. How much will you need to invest each month for the next year in
order to have $1000?
Answers
Answer:
$81.06
A P E X :)
- Joy and John MacAllister have agreed to purchase a house for $79,900. First
National Savings & Loan is willing to lend the money at 7.5 percent for
25 years, provided the MacAllisters can make a $19,900 down payment. The
total closing cost is 3.5 percent of the amount of the mortgage. What is their
total closing cost?
Answers
Answer:
Step-by-step explanation:
8. Laurie graphed two lines, as shown on the right.
One line has a slope of -2 and the other has a slope of
1/2. Laurie noticed that although the slopes are
negative reciprocals of each other, the lines do not
appear to be perpendicular. Explain why this result
occurs, even though she graphed the lines correctly.
Answers
It's possible that the lines do not appear to be perpendicular, even though the slopes are negative reciprocals, because the lines may not be intersecting at 90 degrees angle,
What are the slopes of the two lines?
The slope of a straight line between two points says (x1,y1) and (x2,y2) can be easily determined by finding the difference between the coordinates of the points. The slope is usually represented by the letter m.
The slopes of the two lines that Laurie graphed, -2 and 1/2, are negative reciprocals of each other, which means that their product is -1.
In a Cartesian coordinate system, lines that are perpendicular to each other will have slopes that are negative reciprocals of each other, and the product of their slopes will be -1.
However, it's possible that the lines do not appear to be perpendicular, even though the slopes are negative reciprocals, because the lines may not be intersecting at 90 degrees angle.
The two lines could be skew lines that never intersect and thus cannot be perpendicular.
Another reason could be that the lines may not be in the same coordinate system or not graphed on the same set of axes.
When working with slope, we always assume that the lines are defined on a 2D Cartesian coordinate system, if that is not the case, it could cause the lines to not appear as perpendicular.
Hence, it's possible that the lines do not appear to be perpendicular, even though the slopes are negative reciprocals, because the lines may not be intersecting at 90 degrees angle, another reason could be that the lines may not be in the same coordinate system or not graphed on the same set of axes.
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Each other 11 letters of the word mathematics with written on a separate piece of paper and placed in a bowl what is the probability of drawing
Answers
Therefore, the probability of drawing "OBJ" from the bowl is (1/11) * (1/11) * (1/11) = 1/1331
Define Probability.The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1, where, broadly speaking, 0 denotes the event's impossibility and 1 denotes certainty.
Define Statistics.The study of statistics focuses on gathering, organizing, organizing, analyzing, interpreting, and presenting data. It is customary to start with a statistical population or a statistical model to be researched when applying statistics to a scientific, industrial, or social problem.
The word "mathematics" has 11 letters, so there are 11 letters in the bowl. The probability of drawing the letter "O" is 1/11, since there is only 1 "O" in the 11 letters. Similarly, the probability of drawing the letter "B" is 1/11, and the probability of drawing the letter "J" is 1/11 as well.
Since the events of drawing the letter "O", "B", and "J" are independent of each other, the probability of drawing all three letters in order is the product of their individual probabilities.
Therefore, the probability of drawing "OBJ" from the bowl is (1/11) * (1/11) * (1/11) = 1/1331
It is a very low probability, the chances of drawing that sequence are very low.
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5 in. 3 8 in. What is the volume, in cubic inches, of the prism? cubic inches
Answers
Answer:
70in^3
Step-by-step explanation:
Ok so the the formula for volume for a cube is V= L x W x H
so if the dimensions are 5in by 3in by 8in, the volume of the cube is 5x8x3 = 70 inches^3
HELPPP MEE THANK U, 20 POINTS!~
Answers
Answer:
B
Step-by-step explanation:
The second box plot starts later then the first one on the number line.
HELLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLP
Answers
Write a letter to your friend describing your hobbies. (80 words)A2/32, Chengal Nagar
Tambaram
Chennai – 600023
May 5, 2021
Dearest Malu,
I hope this letter finds you in the best of health. I received your letter the other day asking how I am spending my leisure time these days.
I have developed many hobbies over the years but collecting stamps and embroidery have been my favourite so far. Stamp collection is something I started as a kid. I had just taken a few pages off my notebook and made it my stamp book. I get so excited every time I see the postman. I wait to see what kind of stamp I will get. Knowing that I have a liking for collection of stamps, my cousins have been sending it across to me every time they find a different stamp. I can’t wait to show you my collection.
Embroidery is another hobby I picked up recently. I always liked embroidered clothes. I just started trying out different stitching patterns. Slowly, I learned almost everything, and now I am selling embroidered frames and dresses. It is so fulfilling to see a piece of work come out so well. I am sending you a gift along with this letter, something I made, as your birthday is fast approaching.
Hope to meet you soon.
Your dearest friend,
Renitaالسلام عليكماخوي عارض شريطPandemic
if someone takes your food is it still sharing
Answers
Answer:
not unless you want them to have it :)
Step-by-step explanation:
Debby deposits 25% if the money that she earns each week in a savibgs account if debby earns $72 each week, how much money will she deposit next week
Answers
($72 x 25%)
If f(x)=-4x-5 and g(x)= 3-x, what is g(-4) +f(1)?
A.-10
B.7
C.-9
D.-2
Answers
Answer:
-2
Step-by-step explanation:
sustitue -4 into x for 3-x which would equal 7 because a negative times a negative equals positive. Then substitue 1 in for x in -4x-5 which would get you -9
give brainliest pls
help please! include how you did it and i will give you brainilist.
Answers
Answer: A' (3, 6)
Step-by-step explanation: It is given that the point at is at the coordinate of (1, 2). In this coordinate, 1 represents the x-coordinate while 2 represents the y-coordinate. All you have to do is plug the values of x and y into (x+2)(y+4); so, it will look like this, (1+2)(2+4). Therefore, the new x-coordinate is 3 and the new y-coordinate is 6, so A'(3, 6).
A house 20 meters long and 15 meters wide is surrounded by a verandah of uniform width of 3 meters. Find the cost of the flooring the verandah at RS 200 per square meter
Answers
Answer:
49200 rs. so the cost of flooring =246*200=49200 rs
Step-by-step explanation:
we measure the area of veranda by taking width in addition of length and breadth- house area
it means
area of veranda =outer rectangle-inner rectangle
⇒area= (20+3+3)*(15+3+3)-20*15
⇒area=26*21-300
⇒area= 546-300
⇒area=246 m²
Hence , the cost of flooring =246*200=49200 rs.
Given the number 254, 476, 951
What number is in the TEN THOUSAND's SPOT?
Answers
Step-by-step explanation: 1 is in the ten thousand places and has a place value of 10,000. 3 is in the thousand's place and has a place value of 3,000. 5 is in the hundreds and has a place value of 500.
i dont understand this help
Answers
laucs ori nisiquiera estoy en universidad estoy en eseyo pe
35×75(46×13-55)-76=?
Answers
Answer:
1,425,299
Step-by-step explanation:
Answer:
1425299
Step-by-step explanation:
35×75(46×13-55)-76
multiply stuff in parenthesis
35×75(598-55)-76
subtract stuff in parenthesis
35×75(543)-76
multiply
1425375-76
subtract
1425299