The p-value(0.0538) is greater than the given significance level of 0.01. So, there is no sufficient evidence to show that the manufacturer needs to recalibrate the machines.
What is the test statistics formula for finding the p-value approach of hypothesis testing?The test statistics formula for finding the z-score is,
z = (X - μ)/(σ/√n)
Where,
X = sample mean
μ = population mean
σ = population standard deviation
n = sample size
With this z-score, the p-value is calculated. If p > α, then the null hypothesis is not rejected, and if p < α, then the null hypothesis is rejected.
Calculation:It is given that,
The population mean μ = 4.00
The sample size n = 81
The sample mean X = 3.97
The standard deviation σ = 0.14
and the significance level α = 0.01
So, the hypothesis is:
The null hypothesis: H0: μ = 4
The alternative hypothesis: Ha: μ ≠ 4
Thus, the test statistic value is,
z = (X - μ)/(σ/√n)
= (3.97 - 4)/(0.14/√81)
= - 0.03/0.015
= -1.928
So, for z = -1.928, the p-value is 0.053855
Since p > 0.01, there is no sufficient evidence to show that the manufacturer needs to recalibrate the machines.
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Cora is wrapping a ribbon around a cylinder-shaped gift box. The box has a diameter of 15 inches and the ribbon is 60 inches long. Cora is able to wrap the ribbon all the way around the box once, and then continue so that the second end of the ribbon passes the first end. What is the central angle formed between the ends of the ribbon? Round your answer to the nearest tenth of a degree.
The central angle formed between the ends of the ribbon is 98.6°.
How to illustrate the angle?The circumference will be:
= πd
= 3.14 × 15 = 47.1
Ribbon = 60 - 47.1 = 12.9
Length of arc = x/360 × πd
12.9 = x/360 × 3.14 × 15
x = 98.6°
The angle is 98.6°
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For how many positive integers $n$ less than or equal to $24$ is $n!$ evenly divisible by $1 2 \dots n$
The value of positive integers in the set are 16.
According to the statement
we have given that the there is a set of numbers from 1 to n and we have to find that the how many integers in this set. and there is one condition that the numbers in the set are less than or equal to 24.
So, For this purpose,
Since [tex]$1 + 2 + \cdots + n = \frac{n(n+1)}{2}$[/tex]
the condition is equivalent to having an integer value for [tex]$\frac{n!} {\frac{n(n+1)}{2}}$.[/tex]
This reduces, when [tex]$n\ge 1$[/tex], to having an integer value for [tex]$\frac{2(n-1)!}{n+1}$[/tex]
This fraction is an integer unless n+1 is an odd prime. There are 8 odd primes less than or equal to 24,
so there are 24-8 = 16.
So, The value of positive integers in the set are 16.
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How many distinct triangles can be formed for which m∠E = 64°, g = 9, and e = 10?
triangle(s)
Using the law of sines, it is found that 1 triangle can be formed with the given conditions.
What is the law of sines?Suppose we have a triangle in which:
The length of the side opposite to angle A is a.The length of the side opposite to angle B is b.The length of the side opposite to angle C is c.The lengths and the sine of the angles are related as follows:
[tex]\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}[/tex]
Hence, for this problem, we have that the relation is:
[tex]\frac{\sin{64^\circ}}{10} = \frac{\sin{G}}{9}[/tex]
[tex]\sin{G} = 0.9\sin{64^\circ}[/tex]
[tex]\sin{G} = 0.8089[/tex]
[tex]G = \arcsin{0.8089}[/tex]
G = 54º
A triangle can have one angle of 64º and another of 54º, hence 1 triangle can be formed with the given conditions.
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determine which two of the three given triangles are similar. Find the scale factor for the pair.
22 A circle passes through the points
P(3, 0) and Q(0, 5). Its centre lies on
the line y = x + 2.
(i) Find the equation of the perpendicular bisector of PQ.
(ii) Hence show that the coordinates of the centre of the circle are (-1, 1).
(iii) Find the equation of the circle.
A second circle with equation
2x² + y² + ax + by - 14 = 0 has the
same centre as the first circle.
(iv) Write down the value of a and of b.
(v) Show that the second circle lies
inside the first circle.
The equation of the first circle is (x + 1)^2 + (y - 1)^2 = r^2 and the equation of the second circle is (x + 1)² + (y - 1)² = 16
The equation of the perpendicular bisectorThe points are given as:
P(3, 0) and Q(0, 5)
The midpoint of PQ is
Midpoint = 0.5(3 + 0, 0 + 5)
Midpoint = (1.5, 2.5)
Calculate the slope of PQ
m = (y2 - y1)/(x2 - x1)
m = (5 - 0)/(0 - 3)
m = -5/3
A line perpendicular to PQ would have a slope (n) of
n = -1/m
This gives
n = -1/(-5/3)
n = 0.6
The equation is then calculated as:
y = n(x - x1) + y1
Where
(x1, y1) = (1.5, 2.5)
So, we have:
y = 0.6(x - 1.5) + 2.5
y = 0.6x - 0.9 + 2.5
Evaluate the sum
y = 0.6x + 1.6
Hence, the equation of the perpendicular bisector of PQ is y = 0.6x + 1.6
The center of the circleWe have:
y = x + 2
Substitute y = x + 2 in y = 0.6x + 1.6
x + 2 = 0.6x + 1.6
Evaluate the like terms
0.4x = -0.4
Divide
x = -1
Substitute x = -1 in y = x + 2
y = -1 + 2
y = 1
Hence, the center of the circle is (-1, 1)
The circle equationWe have:
Center, (a, b) = (-1, 1)
Point, (x, y) = (0, 5) and (3, 0)
A circle equation is represented as:
(x - a)^2 + (y - b)^2 = r^2
Where r represents the radius.
Substitute (a, b) = (-1, 1) in (x - a)^2 + (y - b)^2 = r^2
(x + 1)^2 + (y - 1)^2 = r^2
Substitute (x, y) = (0, 5) in (x + 1)^2 + (y - 1)^2 = r^2
(0 + 1)^2 + (5 - 1)^2 = r^2
This gives
r^2 = 17
Substitute r^2 = 17 in (x + 1)^2 + (y - 1)^2 = r^2
(x + 1)^2 + (y - 1)^2 = r^2
Hence, the circle equation is (x + 1)^2 + (y - 1)^2 = r^2
The value of a and bThe equation of the second circle is
2x² + y² + ax + by - 14 = 0
Rewrite as:
2x² + ax + y² + by = 14
For x and y, we use the following assumptions
2x² + ax = 0 and y² + by = 0
Divide through by 2
x² + 0.5ax = 0 and y² + by = 0
Take the coefficients of x and y
k = 0.5a k = b
Divide by 2
k/2 = 0.25a k/2 = 0.5b
Square both sides
(k/2)² = 0.0625a² (k/2)² = 0.25b²
Add the above to both sides of the equations
x² + 0.5ax +0.0625a² = 0.0625a² and y² + by + 0.25b² = 0.25b²
Express as perfect squares
(x + 0.25a)² = 0.0625a² and (y + 0.5b)² = 0.25b²
Add both equations
(x + 0.25a)² + (y + 0.5b)² = 0.0625a² + 0.25b²
So, we have:
2x² + ax + y² + by = 14 becomes
(x + 0.25a)² + (y + 0.5b)² = 0.0625a² + 0.25b²+ 14
Comparing the above equation and (x + 1)^2 + (y - 1)^2 = r^2, we have:
0.25a = 1 and 0.5b = -1
Solve for a
a = 4 and b = -2
This means that the value of a is 4 and b is -2
Show that the second circle is in the firstWe have:
a = 4 and b = -2
Substitute these values in (x + 0.25a)² + (y + 0.5b)² = 0.0625a² + 0.25b²+ 14
This gives
(x + 0.25*4)² + (y - 0.5*2)² = 0.0625*4² + 0.25*(-2)²+ 14
(x + 1)² + (y - 1)² = 16
The equation of the first circle is
(x + 1)² + (y - 1)² = 17
The radii of the first and the second circles are
R = √17
r = √16
√17 is greater than √16
Since they have the same center, and the radius of the first circle exceeds the radius of the second circle, then the second circle lies inside the first circle.
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Select all the correct systems of equations. Which systems of equations have infinite solutions? 2x + 5y = 31 6x - y = 13 y = 14 - 2x 6x + 3y = 42 2x + y = 14 x = 13-2y Reset 2x + y = 10 -6x = 3y + 7 y = 13-2x 4x - 3y = -19 2x + y = 17 Next -6x = 3y - 51
Answer:
The system of equations y=14-2x and 6x+3y=42.
Step-by-step explanation:
Please see attachment to see the answer as a graph.
Hope this helps!
Consider the quadratic function f(x) = x2 – 5x + 12. Which statements are true about the function and its graph? Select three options. The value of f(–10) = 82 The graph of the function is a parabola. The graph of the function opens down. The graph contains the point (20, –8). The graph contains the point (0, 0).
The graph of the quadratic function f(x)=x² - 5x + 12 is a parabola which opens up.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
The standard form of a quadratic equation is:
y = ax² + bx + c
The graph of a quadratic equation is a parabola.
The graph of the quadratic function f(x)=x² - 5x + 12 is a parabola which opens up.
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Which expression is equivalent to -7(y - 2)
The expression equivalent to -7(y - 2) is -7y + 14
Which expression is equivalent to -7(y - 2)?
Given the expression; -7(y - 2)
We expand the expression by multiplying each element inside the parentheses the element out i.e -7
-7(y - 2)
[ -7 × y and -7 × -2 ]
-7y + 14
Therefore, the expression equivalent to -7(y - 2) is -7y + 14.
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Determine whether the point (1,5) is a solution to the system of equations g(x)=3x+2 f(x)=|x-1|+1
Point [tex](1,5)[/tex] is a solution of the equation [tex]g(x)=3x+2[/tex] and point [tex](1,5)[/tex] is not a solution of equation [tex]f(x)=|x-1|+1[/tex]
Why point [tex](1,5)[/tex] is solution of the equation [tex]g(x)=3x+2[/tex] ?
If point [tex](1,5)[/tex] is solution of the equation then they must satisfy the equation.
So we need to check one by one
[tex](x,y)=(1,5)[/tex]
equation [tex]g(x)=3x+2[/tex]
We can write as
[tex]y=3x+2\\5=3*1+2\\5=5[/tex]
So point [tex](1,5)[/tex] is solution of this equation.
Equation [tex]f(x)=|x-1|+1[/tex]
We can write as
[tex]y=|x-1|+1\\5=|1-1|+1\\5=0+1\\5\neq 1[/tex]
So point [tex](1,5)[/tex] is not solution of equation [tex]f(x)=|x-1|+1[/tex]
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Circle O is shown. Line segments B O and O C are radii. Point A is on the circle and is not within angle B O C.
The measure of Arc B A C is 240. What is the ratio of the measure of the major arc to the measure of the minor arc?
3:1
1:3
2:1
1:2
The ratio of major to minor arc = 2 : 1
What is a Circle?
A circle is the locus of a point such that its distance from a fixed point (center) is always constant.
Calculation
The length of an arc (L) with center angle θ is given by:
L = (θ/360) * 2πr
where r is the radius.
Putting the values in the above formula.
For the major arc:
L = (240/360) * 2πr
For the major arc:
l = ((360-240)/360) * 2πr
l = (120/360) * 2πr
The ratio of major to minor arc = [(240/360) * 2πr] / [(120/360) *2πr] = 2 : 1
The ratio of major to minor arc = 2 : 1.
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Answer:c
Step-by-step explanation:
I'm having a hard time with this question so someone help please
Answer:
7.7 km
Step-by-step explanation:
This question is asking you to find the third side of a triangle in which two sides and an angle not between them are given. The law of sines provides the necessary relationships.
Law of SinesThe Law of Sines tells you that triangle side lengths are proportional to the sine of the opposite angle. For this triangle TPJ, the relation is ...
t/sin(T) = p/sin(P) = j/sin(J)
We are given T=68°, p=5.2 km, t=7.5 km, and we are asked to find the length of j, the TP distance.
SetupWe know the sum of angles in a triangle is 180°, and the sine of an angle is equal to the sine of its supplement. This lets us fill in the Law of Sines equation like this:
7.5/sin(68°) = 5.2/sin(P) = j/sin(68°+P) . . . . . solve for j
SolutionThis can be solved in two steps. First, we find angle P, then we use that to find length j.
sin(P) = 5.2/7.5 × sin(68°) ≈ 0.642847
P = arcsin(0.642847) ≈ 40.0045°
Now, we can find j:
j = 7.5 × sin(68° +40.0045°)/sin(68°) ≈ 7.693 . . . km
The distance from the tower to the plane is about 7.7 km.
__
Additional comment
We have used T, P, and J to refer to the vertices at the tower, plane, and jet.
The triangle can also be solved using the Law of Cosines. In that case, the missing side will be the solution to a quadratic equation with irrational coefficients.
Could someone help me with the first two parts of this geometry question? I can do the third and fourth myself.
Answer:
see image
Step-by-step explanation:
see image. We can use the given info and definition of isosceles (congruent sides) and reflexive property (something is congruent to itself) to prove (i) by SSS.
The using "Corresponding Parts of Congruent Triangles are Conguent" (CPCTC) to get more info to prove (ii) by SAS
Find the difference. write your answer as a fraction in simplest form. 5/8-(-7/8)
Answer:
1 1/2
Step-by-step explanation:
When you are subtracting a negative number is is the same as adding a positive. So the problem is really 5/8 + 7/8, since the denominators are the same, just add the numerators to get 12/8 divide the top and bottom by 4 to get 3/2 or 1 1/2, if I divide 3 by 2.
On a coordinate plane, square P Q R S is shown. It has points (0, 4), (4, 4), (0, 0), and (4, 0).
Prove the diagonals of the square with vertices P(0, 4), Q(4, 4), R(0, 0), and S(4, 0) are perpendicular bisectors of each other.
Step 1: Calculate the slope of the diagonals.
The slope of diagonal PS is
.
The slope of diagonal QR is
.
Step 2: Calculate the midpoint of the diagonals.
The midpoint of PS is
.
The midpoint of QR is
.
The diagonals of the square are perpendicular bisectors because the diagonals are
.
The slope of diagonal PS is -1 and the slope of diagonal QR is 1 based on the information about the coordinate plane.
How to illustrate the information?Step 1: Calculate the slope of the diagonals.
The slope of diagonal PS is -1
The slope of diagonal QR is 1
Step 2: Calculate the midpoint of the diagonals.
The midpoint of PS is (2,2)
The midpoint of QR is (2,2)
The diagonals of the square are perpendicular bisectors because the diagonals are perpendicular and share the same midpoint
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Answer:
1. The slope of diagonal PS is -1.
2. The slope of diagonal QR is 1.
3. The midpoint of PS is (2,2).
4. The midpoint of QR is (2,2).
5. The diagonals of the square are perpendicular bisectors because the diagonals are perpendicular and share the same midpoint.
Step-by-step explanation: I just did it on edge 2023. Hope this helps!
What is the circumference of a circle (to the nearest whole number) whose radius is 5?
Answer:
31
Step-by-step explanation:
C= 2πr We will used 3.14 for pi and the radius is 5
C = 2(3.14)(5)
C =31.4 Then round to the nearest whole number which would be 31. 31.4 is closer to 31 than 32
Check all that apply
X^2+7x-8
Answer:
x = -8, 1.
Step-by-step explanation:
x^2 + 7x - 8 = 0
(x + 8)(x - 1) = 0
x = -8, 1.
Answer:
-8 and 1
Step-by-step explanation:
Quadratic expression.
A quadratic expression is an expression with 2 as the highest power of the unknown.
Solving the question:
[tex]x {}^{2} + 7x - 8 \\ x {}^{2} + 8x - x - 8 \\ x(x + 8) - 1(x + 8) \\ (x - 1)(x + 8) \\ x = 1 \: or \: x = - 8[/tex]
A music store owner noted that cd sales had dropped 23% from one quarter to the next. if the owner sold 715 units in the first quarter, how many did she sell in the next quarter?
round your answer to the nearest whole unit of cds. do not include units with your answer.
help please what's the correct answer
If the sales had dropped 23% then music store had sold 551 units in the second quarter.
Given that a music store had sold 715 units in first quarter and sales had dropped by 23%.
We are required to determine the sales in the second quarter.
Percentage is the number or ratio expressed in terms of 100. It is expressed as %.
Sales in first quarter=715 units
Decrease in sales=23%
In units ,decrease=715*23%
=715*23/100
=164.45
Sales of second quarter=715-164.45
=550.55
After rounding we will get 551.
Hence if the music store sold 715 units in first quarter then the units sold by music store in second quarter 551 units.
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A waste management survey collected data about the types of materials that were recycled by three different states in
2008
20082008. The table at left shows the approximate amount of each material recycled measured in thousands of tons. Based on the information in the table at left, as a percentage, approximately what is the relative frequency of plastics recycled in New Mexico in plastics recycled in all three states combined?
As a percentage, the relative frequency of plastics recycling in New Mexico in 2008 compared to plastics recycled in all three states combined is approximately 11%.
What is relative frequency?A relative frequency measures the proportion of events that occur within a class out of the total number of observations.
When using relative frequency, the results are usually displayed in percentages, proportions, and fractions.
Data and Calculations:Year State Recycled Plastics Relative
(thousands of tons) Frequency
2008 New Mexico 500 0.111 (500/4,500)
2008 New York 1,500 0.333 (1,500/4,500)
2008 Texas 2,500 0.566 (2,500/4,500)
Total 4,500 1
Thus, the relative frequency of plastics recycling in New Mexico in 2008 compared to plastics recycled in all three states combined is approximately 11%.
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Answer: 9%
Step-by-step explanation:
just did the question
Three less than a number x is more than 15. an inequality that represents this word sentence is
Answer:
x - 3 > 15
Step-by-step explanation:
please help! this is the last one i need and i forgot how to do it!
Based on the information, Christian would have $5525.5 of annuity.
How to calculate the annuity?According to given information, the number of coffees per week is 3 then, per month is 3x4 = 12
For each coffee is $4.5. Then monthly expenditure for coffees is 12 x 4.5 = $54
Rate of interest r = 1.6% = 1.6/100 = 0.016 and for monthly compounding r = 0.016/12 = 0.00133
n = number of payments = 8 x 12 = 96
We can use the formula for finding the future value as below
FV = C x [ ( 1 + r )n-1 ] / ( r )
FV = 54 x [ ( 1 + 0.00133 )96 – 1 ] / (0.00133)
= 54 x [ (1.13609 - 1)] / (0.00133)
= 54 x 0.13609 / (0.00133)
= 54 x 102.3233
= 5525.5
Therefore Christian would have $5525.5 of annuity.
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Determine the perimeter of the figure.
Answer:
[tex]\huge\boxed{\sf 7x + 10\ cm}[/tex]
Step-by-step explanation:
Perimeter:Sum of all sides of a figure is the perimeter of the figure.Perimeter of the figure:= 5 + 2x + 2x + 5 + 3x
Combine like terms
= 2x + 2x + 3x + 5 + 5
= 7x + 10 cm
[tex]\rule[225]{225}{2}[/tex]
Answer:
7x + 10
Step-by-step explanation:
Perimeter is the sum of all the sides together, the distance around a shape.
3x + 5 + 2x + 2x + 5
Organize so the variables are on one side
3x + 2x + 2x + 5 + 5
Combine like terms:
7x + 10
Your answer would be: 7x + 10
I hope it helps! Have a great day!
bren~
can you help me with this
2. Using the figure, find and y.
Answer: [tex]x=8, y=2\sqrt 2[/tex]
Step-by-step explanation:
By the Pythagorean theorem, x=8.
So, by the geometric mean theorem,
[tex]y^2 = (6)(2)=8\\\\y=2\sqrt{2}[/tex]
A bag contains 1 white (W), 3 blue (B1, B2, B3), and 2 red (R1, R2) marbles. (4 points)
Use a tree diagram to list all the possible outcomes for tossing a coin and then drawing a marble from the bag.
show all work
insert picture of the tree diagram pls
The possible outcomes are WH, BH, BH, BH, RH, RH, WL, BL, BL, BL, RL, RL
How to determine the possible outcomes?The given parameters are:
Coin = Head (H) and Tail (T)
Marble = W, B, B, B, R, R
See attachment for the tree diagram
From the tree diagram, we have the following possible outcomes
WH, BH, BH, BH, RH, RH, WL, BL, BL, BL, RL, RL
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Let a and b be positive integers such that the product of all positive divisors of a equals the product of all positive divisors of b. Prove that a
The property A*B = (A' + A") * ( B'+B") is equal to each other.
According to the statement
We have given that the a and b is the positive integers and we have to prove that the product of all positive divisors of a equals the product of all positive divisors of b.
And in this statement
Now we use Distributive property
Here A and B is the positive integer and
A' and A" are the divisors of A and B' and B" are the divisors of B.
Now,
A = A' + A" and B = B'+B"
Now, Multiply both with each other then
A*B = (A' + A") * ( B'+B")
then
A*B = (A' B'+ A'B") + ( A"B'+A'B")
by this way it is equal to each other
Hence proved.
So, The property A*B = (A' + A") * ( B'+B") is equal to each other.
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210÷[5+16÷2{6−18÷(7+2)}]−15
Answer:
[tex]-\frac{345}{37}[/tex]
Step-by-step explanation:
Just follow the order of operations
210÷[5+16÷2{6−18÷(7+2)}]−15
= 210÷[5+16÷2{6−18÷9}]−15 (calculate 7+2)
= 210÷[5+16÷2{6−2}]−15 (calculate 18÷9)
= 210÷[5+16÷2{4}]−15 (calculate 6-4)
= 210÷[5+16÷2×4]−15
= 210÷[5+8×4]−15 (calculate 16÷2)
= 210÷[5+32]−15 (calculate 8×4)
= 210÷[37]−15 (calculate 5+32)
= 210÷37−15
= 210÷37−(15×37)÷37 (put on the same denominator)
= 210÷37−555÷37 (calculate 15×37)
= (210−555)÷37
=-345÷37 (calculate 210-555)
I’ve tried everything and yet don’t understand this
Snow reaches at a 15 inches after 16 hours. and 8 inches melt in next 6 hours.
According to the statement
we have a given that there was 1 inch snow on the ground before the snowstorm starts.
then snow fell at constant rate of 3 inches every 2 hours until the snow level reaches at 12 inch had fallen
It means after that there was 13 inch snow on the ground.
and in 1 hour only 1.5 inches snow increases
and if snow takes 2 hours to increase at 3 inches then
it takes 10 hours to reach at 12 inches.
So, snow falls for 10 hours to increase at 12 inches.
And then the snow falls at constant rate of 1 inch per hour for 2 hours.
It means now snow reaches at a 15 inches after 16 hours.
Then sun melt the snow at a 4 inches per 3 hours and sun came for 6 hours then
it means now the level of snow reduced and came at the 7 inches.
because sun reduced 8 inches snow in 6 hours.
So, this is the data collected by a person.
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PLEASE ANSWERERRREEERRR
the operation:
[tex]2*10^1 - 4*10^0 = 20 - 4 = 16[/tex]
Gives a square number.
How to get a perfect square?
A perfect square is the product of a number and itself.
As you can see in the example, 16 is a square number. Then let's try to get 16.
To get 16 we can use the difference:
20 - 4
So the first therm needs to be equal to 20, and the second equal to 4.
To write 20 in scientific notation we have:
[tex]2*10^1[/tex]
To write 4 in in scientific notation:
[tex]4*10^0[/tex]
Then the operation:
[tex]2*10^1 - 4*10^0 = 20 - 4 = 16[/tex]
Gives a square number.
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Theo earns a base salary plus commission. At the end of September, he was paid $3,835.25 based on his monthly sales of $12,850. At the end of October, he earned $3,641 based on $9,875 in sales. What was Theo’s base salary?
Theo's base salary using a simultaneous equation approach is $2,996.22 .
In order to determine the base salary of Theo, we can express the two earnings and sales amounts using simultaneous equation, that we would solve to arrive at the base salary
Bear in mind that total earnings are determined as base salary plus commission
Let X be the base salary
Let R be the rate of commission
commission =sales*rate of commission
When earnings were $3,835.25 and sales is $12,850, we have the below equation:
X+(12,850*R)=3,835.25
X+12850R=3,835.25 .................... equation (1)
When earnings were $3,641 and sales is $9,875, we have the below equation:
X+(9,875*R)=3641
X+9875R=3641 ......................... equation (2)
subtract equation 2 from 1
2975R=194.25
R=194.25/2975
R=commission rate=6.52941176470588%
substitute for R in equation 1 to arrive at X(base salary)
X+(12850*6.52941176470588%)=3,835.25
X=3,835.25-(12850*6.52941176470588%)
X=base salary=$2,996.22
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Find an equation whose graph is a Parabola with vertex (1, 1) and focus (2, 2).
The matricial equation of the degenerate parabola is [tex]\left[\begin{array}{c}x'\\y'\end{array}\right] = \left[\begin{array}{c}\frac{ \sqrt{2} }{2}\cdot x + \frac{1}{8}\cdot x^{2}+1 \\\-\frac{\sqrt{2}}{2}\cdot x + \frac{1}{8}\cdot x^{2} + 1 \end{array}\right][/tex].
How to determine the equation of a degenerate parabolaA parabola is degenerated when its axis of symmetry is not parallel to any orthogonal axis (i.e. x, y). First, we have to find an "equivalent" parabola whose axis of symmetry is parallel to the y-axis and whose vertex is along that axis.
Direction of the degenerate parabola
[tex]\tan \theta = \frac{2 - 1}{2 - 1}[/tex]
tan θ = 1
θ = 45°
This means that the degenerate parabola must be rotated 45° clockwise (- 45°) around the origin with respect to the "equivalent" parabola.
Distance from the vertex to the focus
[tex]p = \sqrt{(2 - 1)^{2}+(2 - 1)^{2}}[/tex]
[tex]p = \sqrt{2}[/tex]
Vertex of the "equivalent" parabola
[tex](h', k') = (0, \sqrt{2})[/tex]
Equation of the "equivalent" parabola
[tex]y = \frac{1}{4\cdot p}\cdot x^{2} + k'[/tex]
[tex]y = \frac{1}{4\sqrt{2}}\cdot x^{2}+\sqrt{2}[/tex]
Now we apply following rotation formula:
[tex]\left[\begin{array}{c}x\\y\end{array}\right] = \left[\begin{array}{cc}\cos (-45^{\circ})&-\sin (-45^{\circ})\\\sin (-45^{\circ})&\cos (-45^{\circ})\end{array}\right] \cdot \left[\begin{array}{c}x\\\frac{1}{4\sqrt{2}}\cdot x^{2}+\sqrt{2}} \end{array}\right][/tex]
[tex]\left[\begin{array}{c}x'\\y'\end{array}\right] = \left[\begin{array}{c}\frac{ \sqrt{2} }{2}\cdot x + \frac{1}{8}\cdot x^{2}+1 \\\-\frac{\sqrt{2}}{2}\cdot x + \frac{1}{8}\cdot x^{2} + 1 \end{array}\right][/tex]
A representation of the degenerate parabola is shown in the figure attached below.
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