A parabola has a focus of F(2, -0.5) and a directrix of y=-1.5 P(x,y) represents any point on the parabola, while D(x, -1.5) represents any point on the directrix.
What are the steps required to find the equation of the parabola?
Answers
We have the focus [tex](a,b) = (2, -0.5)[/tex]
The point [tex]P(x,y)[/tex]
The directrix, c at [tex]y=-1.5[/tex]
The steps to find the equation of the parabola are as follows
Step 1
Find the distance between the focus and the point P using Pythagoras. We have two coordinates; [tex](2, -0.5)[/tex] and [tex](x,y)[/tex].
We need the vertical and horizontal distances to find the hypotenuse (the diagram is shown in the second diagram).
The distance between the focus and point P is given by
[tex] \sqrt{ (x-a)^{2}+ (y-b)^{2} } [/tex]
Step 2
Find the distance between the point P to the directrix [tex]c[/tex]. It is a vertical distance between y and c, expressed as [tex]y-c[/tex]
Step 3
The equation of parabola is then given as
[tex] \sqrt{ (x-a)^{2}+ (y-b)^{2} } [/tex]=[tex]y-c[/tex]
[tex] (x-a)^{2}+ (y-b)^{2}= (y-c)^{2} [/tex] ⇒ substituting a, b and c
[tex] (x-2)^{2}+ (y--0.5)^{2} = (y--1.5)^{2} [/tex]
[tex] (x-2)^{2}+ (y+0.5)^{2}= (y+1.5)^{2} [/tex]⇒Rearranging and making [tex]y[/tex] the subject gives
[tex]y= \frac{ x^{2} }{2} -2x+1[/tex]
Related Questions
Aunt Mary is 6 times as old as Jane. If Aunt Mary is 72 years old, how old is Jane?
Answers
6x =Aunt Mary's age
6x = 72
x = 72/6
x = 12 years [Jane]
Because if Mary is 6 times older than just divide her age by 6. Which is 12.
And you can check like this:
-Jane is 12
-Mary is 6 times that
-so Mary is 72
A map of square piece of property is drawn at a scale of 1:500. If a side of the property on the map is 16 cm, what is the property's actual area?
Answers
s=500x
A=250000x^2, and since x=16cm
A=250000(16^2)
A=64000000 cm^2 But a more appropriate unit would be m^2 so:
A=64000000cm^2(m^2/10000cm^2)=6400 m^2
Answer:
Property's actual area = 6400 m²
Step-by-step explanation:
Scale = 1 : 500
Side on map = 16 cm
Side of property = scale x side on map = 16 x 500 = 8000 cm
Area of property = 8000 x 8000 = 64000000 cm²
Area of property = 6.4 x 10⁷ cm² = 6.4 x 10³ m²
Property's actual area = 6400 m²
A hotel chain charges $75 each night for the first two nights and $50 for each additional night’s stay. the total cost t is a function of the number of nights x that a guest stays.
Answers
you would need a function to determine whether to use a formula for 1 or 2 nights or a formula for more than 2 nights
T(x)={a if x <_ 2, b if x > 2}
if it's 2 nights or less use option a, a = 75x
if its longer than 2 days use option b, b= 50x + 50
Answer:
t = 75x {x ≤ 2}
t = 150 + 50x {x > 2}
Step-by-step explanation:
A hotel chain charges $75 per night for first two nights.
If the number of nights of the stay are x then charges for x nights
= 75x where x ≤ 2
If the stay is for than 2 nights then charges for x nights
= Charges for two first 2 nights(@$75 per night) + charges for other than initial 2 nights (@$50 per night)
= (75×2) + 50x where x > 2
Therefore total cost of the hotel if represented by t then the function will be
t = 75x {x ≤ 2}
t = 150 + 50x {x > 2}
Find the dot product, a • b. a = 6i + 5j, b = -5i + 4j
Answers
If i is the constant i, then it is 20j^2 -i*j + 30
Answer:
the answer is -10
Helllllllllllkklllllllkkkkkkkp
Answers
In other words, you can put it like this:
1(ab) + 1c = 1d
Since our answer choices, let's try to get a by itself.
ab + c = d
Subtract both sides by c.
ab = d - c
Now, a is being multiplied by b, so we need to perform the opposite order of operations (divide).
Divide both sides by b.
You're left with:
a = (d - c)/b
Your answer is D.)
I hope this helps!
a basketball player has already scored 56 points in a game the team record for point a in a game is 72 which equation would allow you to find out how many more points the player needs to tie the record
Answers
Answer: The player needs 16 more points to tie the record.
Step-by-step explanation:
Let the number of points the player needs to tie the record be 'x'.
Number of points that a basketball player has already scored = 56
Number of points a team need for record in a game = 72
According to question, it becomes,
[tex]x+56=72\\\\x=72-56\\\\x=16[/tex]
Hence, the player needs 16 more points to tie the record.
A bicycle lock has a four-digit code. The possible digits, 0 through 9, cannot be repeated. What is the probability that the lock code will begin with the number 5? . What is the probability that the lock code will not contain the number 0? .
Answers
The first digit is any of {0, 1, 2...,9}, that is 10 possibilities
the second digit is any of the remaining 9, after having picked one.
and so on...
so in total there are 10*9*8*7 = 5040 codes.
a. What is the probability that the lock code will begin with 5?
Lets fix the first number as 5. Then there are 9 possibilities for the second digit, 8 for the third on and 7 for the last digit.
Thus, there are 1*9*8*7=504 codes which start with 5.
so
P(first digit is five)=[tex] \frac{n(first -digit- is- 5)}{n(all-codes)}= \frac{1*9*8*7 }{10*9*8*7 }= \frac{1}{10}=0.1 [/tex]
b. What is the probability that the lock code will not contain the number 0?
from the set {0, 1, 2...., } we exclude 0, and we are left with {1, 2, ...9}
from which we can form in total 9*8*7*6 codes which do not contain 0.
P(codes without 0)=n(codes without 0)/n(all codes)=(9*8*7*6)/(10*9*8*7)=6/10=0.6
Answer:
0.1 ; 0.6
The probability that the lock code will begin with the number 5 is:
0.1The probability that the lock code will not contain the number 0 is:
0.6What is Probability?This refers to the use of permutations to find the possible occurrence of events.
The total number of elements is 10
Hence, the total codes is 5040.
P(five)= n(first number)/n(all codes)= 1/10= 0.1
Next,
P(codes without zero)= n(codes without zero)/n(all codes)
(9*8*7*6)/(10*9*8*7)=6/10=0.6
=0.6
Read more about probability here:
https://brainly.com/question/24756209
Rex weighed 80 pounds a year ago. Now he weighs 92 pounds. What is the percent increase in his weight?
(A)12%
(B)13%
(C)15%
(D)16%
Answers
.15~15%
Answer is 15%
%change=100(92-80)/80
%change=1200/80
%change=15%
What is the solution to this system of equations 2x-7=y y=2x+3
Answers
Answer:
No solution
Step-by-step explanation:
[tex]2x-7=y[/tex]
[tex]y=2x+3[/tex]
To solve the system of equations, we use substitution method
Lets substitute the second equation y=2x+3 for y in the first equation and solve for x
[tex]2x-7=2x+3[/tex]
Lets solve for x
subtract 2x on both sides
[tex]2x-2x-7=2x-2x+3[/tex]
[tex]-7=+3[/tex] is false
So, no solution for this system of equations
If 3x² -6=y for all real numbers x, then x² - 2 =
Answers
If 3(x² - 2) = y, then
[tex] \frac{3(x^2-2)}{3}= \frac{y}{3} \\ \\ x^2-2 = \frac{y}{3} [/tex]
Answer:
x² - 2 = y/3
Find the surface area of the figure below. Leave answer in terms of π.
A. 1060π in2
B. 1160π in2
C. 1260π in2
D. 1000π in2
Answers
Use SA=2πrh+2πr^2.
Plug in the numbers A=2π*10*40+2*π*10^2.
and you get 3141.59.
divide by Pi and you get 1000
2Pi *r*h + Pi*r^2 = 2Pi*10*40 + 10^2 *Pi
= 800Pi + 100Pi = 900Pi
Cone:
Height = SQRT(26^2 - 10^2) = 24
=2Pi*r*h/3 = 160Pi
Total Surface Area is 1060Pi
What is the simplified form of the expression?
5(14 - 2)2 ÷ 2
Answers
[tex] \frac{5(14 - 2)^2}{2} = \frac{5(12)^2}{2}=\frac{5*144}{2}=5*72=360[/tex]
12 times 5 and get 60
2 divided by 2 is 1
so i broke it down for you, you should figure it out from here hope i helped
Using the given points, determine the slope.
(0, 32) and (100, 212)
Answers
To find the slope of two given points, subtract the y values, and subtract the x values, or..
[tex] \frac{(y2 - y1)}{(x2 - x1)} [/tex]
[tex] \frac{(212 - 32)}{(100 - 0)} [/tex]
[tex] \frac{180}{100} [/tex]
The answer is 9/5 or 1.8.
Hope this helps! :)
The slope is the change in y divided by the change in x.
or
The slope is the ratio of rise to run.
so:slope = (212-32) / (100-0) = 9/5 = 1.8
If 7 is added to twice a number and this sum is multiplied by 5 â, the result is the same as if the number is multiplied by 6 and 11 is added to the product. what is theâ number?
Answers
(2x+7)*5 = 6x+11
10x+35 = 6x+11
4x+35 =11
4x=-24
x=-6
the number is -6
The base edge of an oblique square pyramid is represented as x cm. If the height is 9 cm, what is the volume of the pyramid in terms of x?
3x2 cm3
9x2 cm3
3x cm3
x cm3
Answers
Hence, the volume of pyramid is:
3x^2 cm^3
Step-by-step explanation:We know that a volume(V) of a oblique square pyramid with height 'h' and base area A is given by the help of formula:
[tex]V=\dfrac{1}{3}\times (Ah)[/tex]
Now, according to the ques we have:
The base edge of an oblique square pyramid is represented as x cm.
The height is 9 cm.
Hence, the base area (A) is calculated as:
[tex]A=x^2[/tex]
( Since the base is in the shape of a square and we know that the area of square with side length s is given as:
[tex]Area=s^2[/tex] )
So, the volume of the pyramid is calculated as:
[tex]V=\dfrac{1}{3}\times (x^2\times 9)\\\\V=3x^2 \ cm^3[/tex]
Hence, the volume of pyramid is:
3x^2 cm^3
A pyramid is a solid shape with several triangles and a base. The volume of the pyramid in terms of x is 3x^2 cm3
How to calculate the volume of a pyramid?The volume(V) of an oblique square pyramid with height 'h' and base area A is expressed according to the formula:
V = Bh/3
B is the base area = x * x = x^2
h is the height = 9cm
Substitute
V = x^2*9/3
V = 3x^2cubic cm
Hence the volume of the pyramid in terms of x is 3x^2 cm3
Learn more on volume of pyramid here: https://brainly.com/question/218706
The following is an arithmetic sequence.
-3, 3, -3, 3
true false
Answers
A(n)=a1 + d*(n-1)
So it should increase all the time linearly or decrease...
Answer: False
(05.01 MC)
The table and the graph below each show a different relationship between the same two variables, x and y:
How much more would the value of y be on the graph than its value in the table when x = 12?
20
30
60
70
Answers
For the table, y = 25x.
When x = 12, y = 25*12 = 300
For the graph, y = 30x
When x = 12, y = 30*12 = 360
When x = 12, the value of y on the graph is 60 more than its value in the table.
Answer:
The correct option is C. The value of y be on the graph than is 60 more than the y value in the table when x = 12.
Step-by-step explanation:
If a line passing through two points then the equation of line is
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
From the given table it is clear that the line passing thought the points (4,100) and (5,125).
[tex]y-100=\frac{125-100}{5-4}(x-4)[/tex]
[tex]y-100=25(x-4)[/tex]
[tex]y-100=25x-100[/tex]
[tex]y=25x[/tex]
The value of this function at x=12 is
[tex]y=25(12)=300[/tex]
From the given table it is clear that the line passing thought the points (0,0) and (2,60).
[tex]y-0=\frac{60-0}{2-0}(x-0)[/tex]
[tex]y=30x[/tex]
The value of this function at x=12 is
[tex]y=30(12)=360[/tex]
The difference between y value in the graph and in the table at x=12 is
[tex]360-300=60[/tex]
The value of y be on the graph than is 60 more than the y value in the table when x = 12. Therefore the correct option is C.
I need help with this problem
Answers
Find the measures of all of the angles in the parallelogram.
Answers
FED = DGF = 59x - 1
EDG = GFE = 121x + 1
And now since we know angles in a quadrilateral always add up to 360°, we can create an equation and solve for x:
2(59x - 1) + 2(121x + 1) = 360
118x - 2 + 242x + 2 = 360
360x = 360
x = 1
Now sub into our angle equations:
FED = DGF = 59(1) - 1 = 58°
EDG = GFE = 121(1) + 1 = 122°
what is the slope of a line that is parellel to the line represented by the equation y=2/5x+4/5
Answers
So, answer would be 2/5.
Joey has a full 40 coins consisting of only quarters and nickels if the total value is $5.00 how many of each does joey have.
Answers
0.25q + 0.05n = 5
0.25(40 - n) + 0.05n = 5
10 - 0.25n + 0.05n = 5
-0.25n + 0.05n = 5 - 10
- 0.20n = -5
n = -5 / -0.20
n = 25 <=== 25 nickels
q + n = 40
q + 25 = 40
q = 40 - 25
q = 15 <== 15 quarters
q=number of quarters
40 coins total so
n+q=40
total value is 500 cents
5n+25q=500
divide both sides by 5
n+5q=100
so we gots
n+q=40
n+5q=100
elimination
multily first equation by -1 and add to 2nd equation
-n-q=-40
n+5q=100 +
0n+4q=60
4q=60
divide both sides by 4
q=15
sub back
n+q=40
n+15=40
minus 15 both sides
n=25
25 nickles and 15 quarters
Find the equation of the linear function which passes through the points (-1, 3) and (4, -2). give your answer in y = mx + b form.
Answers
Hello : let
A(-1,3) B(4,-2)
the slope is : (YB - YA)/(XB -XA)
(-2-3)/(4+1) = -1
an equation is : y=ax+b a is a slope
y = -x +b
the line through
point B(4,-2) : -2 = -1(4)+b
b = 2
the equation is : y =-x+2
In pentagon ABCDE, angle A= 87, angle B=125, angle C= 63, angle D=169. Find angle E
Answers
m<E = 540 - (87+125+63+169)
m<E = 540 - 444
m<E = 96
answer
m<E = 96
Find the 6th term of a geometric sequence t3=444 t7=7104
Answers
Answer:
3552
Step-by-step explanation:
8. Find the congruence transformation that maps ∆ABC to ∆A’B’C’. Explain your reasoning
Answers
In this case, to flip any of the points to map up accurately to the other points, we would have to do a transformation of reflection about the y-axis. The reason behind this is that out of other transformation process, we can only achieve the image of triangle ∆A’B’C’ through reflection.
Dilation will not work since it only changes the size of the object. The shape and position will not affected.
A translation will not work because although the shape can move, it still needs to flip to achieve triangle ∆A’B’C’.
A rotation will also not work because when you rotate it, the shape will still not map over the new one.
Therefore the only answer to this question would be to
reflect triangle ∆ABC about the y-axis.
Triangle DEF is congruent to GHJ by the SSS theorem. Which rigid transformation is required to map DEF onto GHJ?
dilation
reflection
rotation
translation
Answers
Answer: The correct option is 4, i.e., translation.
Explanation:
It is given that the triangle DEF is congruent to GHJ by the SSS theorem.
From the given figure it is noticed that the triangle DEF and GHI are isosceles triangle.
Reason for correct option:
In transition the side and angles of the shape are not effected and the figure transfers from one place to another place without making any other change.
Since the both triangle are congruent and the position of all sides and angles are same, therefore option 4 is correct.
Reason for incorrect option:
In dilation the size of the figure is either stretched or compressed according to the scale factor k. But in the given figure the size of both triangles are same it means option A is incorrect.
In reflection the distance of all the point in preimage and all the image are same from the central line. Any central line does not exist in given figure. SO, the option B is incorrect.
In rotation the side length and angles are not affected but the position are not same, therefore the option C is incorrect.
Answer: Translation
Step-by-step explanation:
Given: Triangle DEF is congruent to GHJ by the SSS theorem.
From the given picture two sides of both triangles are equal therefore, the triangle DEF and GHI are isosceles triangle.
We know that a transition is a kind of rigid motion used in geometry to trace a function that moves an object a particular distance that maps produces images.
Since the both triangle are congruent and just position of triangle GHJ is shifted to the right and down.
Therefore, the transition is rigid transformation which is required to map DEF onto GHJ
In two or more complete sentences describe how to determine the appropriate model for the set of data, (0,2), (2,3), (5,4), (10,5).
Answers
Once you plot the data points, make a trendline. You are given with options. If you want a linear function, then you will have a linear model with a function equation of y = 0.2907x + 2.2643. It has a correlation coefficient of 0.9595. That's a strong correlation already. The R² value tells how good your model fits the data points. If you want to increase the R², a better model would be a quadratic function with the equation, y = -0.0209x²+0.506x+2.0232. As you can see the R² increase even more to 0.9992.
Zeus Industries bought a computer for $2740. It is expected to depreciate at a rate of 24% per year. What will the value of the computer be in 3 years?
Round to the nearest penny. Do not type the "$" sign in your answer
Answers
A=p (1-r)^t
A ?
P 2740
R 0.24
T 3years
A=2,740×(1−0.24)^(3)
A=2,740×(0.76)^(3)
A=1,202.8 round your answer to get
A=1203
Solve the equation. 8d + 3d – 3d – 8 = 3d
Answers
D = - 1.6
this is the answer simplified
Answer:
1.6
Step-by-step explanation:
If a and b are positive numbers such that 2b=3a, what's the value of ((a+b)/b)? I know you can plug in numbers for this, but i'm interested in solving this algebraically. Thanks.