[tex]$\mathbb{Q}(\sqrt{2})$[/tex] is isomorphic to [tex]$\mathbb{Q}[x] /(x^2-2)$[/tex], as desired.
What is the equivalent expression?
Equivalent expressions are expressions that perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.
Show that [tex]$\mathbb{Q}(\sqrt{2})$[/tex] is isomorphic to [tex]$\mathbb{Q}[x] /(x^2-2)$[/tex] :
We define a function [tex]$\phi[/tex] : [tex]\mathbb{Q}[x] \to \mathbb{Q}(\sqrt{2})$[/tex] by [tex]$\phi(f(x)) = f(\sqrt{2})$[/tex].
This function is clearly a homomorphism since it preserves addition and multiplication.
Furthermore, we see that [tex]$\phi(x^2-2) = (\sqrt{2})^2-2 = 0$[/tex],
so the kernel of [tex]$\phi[/tex] contains the ideal generated by [tex]$x^2-2$[/tex].
By the first isomorphism theorem, there exists an isomorphism [tex]$\operator{deg}(r) < \operator{deg}(x^2-2) = 2$[/tex][tex]$\tilde{\phi} : \mathbb{Q}[x] /(x^2-2) \to[/tex][tex]\operator{im}(\phi)$.[/tex]
It remains to show that [tex]$\tilde{\phi}$[/tex] is surjective. Let [tex]$a+b\sqrt{2} \in \mathbb{Q}(\sqrt{2})$[/tex] be an arbitrary element. Since [tex]$\mathbb{Q}[x]$[/tex] is a polynomial ring, we can apply the division algorithm to find [tex]$q(x),r(x) \in \mathbb{Q}[x]$[/tex] such that [tex]$a+b\sqrt{2} = q(\sqrt{2}) + r(\sqrt{2})$[/tex] where [tex]$\operator{deg}(r) < \operator{deg}(x^2-2) = 2[/tex].
But then [tex]$r(\sqrt{2}) = a+b\sqrt{2} - q(\sqrt{2}) \in \operator{im}(\phi)[/tex], so [tex]$\tilde{\phi}$[/tex] is surjective.
Therefore, [tex]$\mathbb{Q}(\sqrt{2})$[/tex] is isomorphic to [tex]$\mathbb{Q}[x] /(x^2-2)$[/tex], as desired.
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Four glasses of milk and 2 snack bars have a total of 86 carbohydrates (carbs), and 2 glasses of milk and 3 snack bars have a total of 69 carbs. Determine how many carbs are in one glass of milk and in one snack bar.
Answer:
One glass of milk: 15 carbs
One snack bar: 13 carbs
Step-by-step explanation:
This problem is an example of systems of linear equations. We can set this problem up by defining variable. I used [tex]m[/tex] for the glass of milk and [tex]b[/tex] for the snack bar.
First, we have: 4 glasses of milk and 2 snacks bar have a total of 86 carbs. We can write this as
[tex]4m+2b=86[/tex]
Next: 2 glasses of milk and 3 snack bars have a total of 69 carbs.
We can write this as
[tex]2m+3b=69[/tex]
Now we set it up.
[tex]4m+2b=86[/tex]
[tex]2m+3b=69[/tex]
By dividing both sides of the first equation by 2, we can use the subtraction method to get rid of the variable [tex]m[/tex]. Now we have
[tex]2m+b=43[/tex]
[tex]2m+3b=69[/tex]
Subtract like terms and solve for [tex]b[/tex]
[tex]-2b=-26[/tex]
[tex]b=13[/tex]
Now we substitute [tex]b[/tex] back into the original equation to find [tex]m[/tex].
[tex]2m+(13)=43[/tex]
[tex]2m=30[/tex]
[tex]m=15[/tex]
There you have it. I hope this helps!
(Linear Relationships MC) An indoor soccer field can be rented for personal use. The total cost for renting the field can be found by using the equation y=225x + 200 The x-variable is the number of hours the field is being rented, the constant is the flat rate for renting the field, and the y-variable is the total cost, in dollars Which statement is true based on the given equation? The equation shows a linear relationship, but not a proportional relationship The equation shows a linear relationship and a proportional relationship The equation does not show a linear relationship or a proportional relationship The equation shows a proportional relationship but not a linear relationship
first statement is true, the equation shows a linear but not proportional.
what is linear equation?The equation which has two variables with first order term and only a constant available in the equation is termed as linear equation. Example of a linear equation, y =mx + b
Which statement is true for this question?
Given, an equation, y = 225x + 200
where x and y are two variables
x denotes the number of hours the field is being rented
y denotes the total cost
constant 200 implies the flat rate for renting the field.
If we compare this equation to the standard form of linear equation, y = mx + b, where m = slope and b is the y intercept.
we will find that the above two equation is similar and therefore the equation given in question is a linear relationship.
the coefficient 225 is the slope of equation and the constant 200 is the y intercept.
this given equation is not a proportional relationship.
The first statement is true.
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Solve the inequality and write the solution in set-builder notation. r - 8 < 7
r<15 because I am smart dont doubt me
John wants to measure the height of a tree. He walks exactly 100 feet from the base of the tree and looks up. The angle from the ground to the top of the tree is 33 degrees. How tall is the tree?
The height of the tree, given the angle from the ground and the distance from the base of the tree, is 64. 94 feet.
How to find the height of the tree ?Assuming that this situation was modelled as a right angled triangle, then the angle from the ground to the top of the tree would be such that the distance from the base of the tree would be the adjacent measure and the height of the tree would be the opposite measure.
This means that the relevant operation to use would be Tan.
Using the Tan operation, and assuming the height of the tree was x, the formula would be:
Tan (33 degrees ) = x / Distance from base
Tan (33 degrees ) x / 100
x = 64. 94 ft
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find x and mut pleaase
The formula for a circle's sector's area is /360o (r2), where r is the circle's radius and is the sector's angle.
Formula for a Sector's Circumference
The following is the formula for a circle's sector's perimeter:
Sector perimeter equals radius plus radius plus arc length.
Sector perimeter equals 2 radius plus arc length.
The relation is used to compute arc length:
Arc length is equal to l = (/360) 2r.
Therefore,
Sector perimeter: 2 Radius + ((/360) 2r)
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Assuming that r is the circle's radius and that is the sector's angle, the formula for the area of a circle's sector is /360o (r2).
What are sectors in a circle?A circular sector (also known as a circle sector or a disc sector; symbol: ) is the area of a disc (a closed region bordered by a circle) enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger as the major sector.
The radius of the circle, r, and the sector angle (in degrees) that the arcs at its centre subtend are also given. A = 1/2 r2 gives the area if the subtended angle is provided in radians.
The area, A, of just the segment may be calculated using the following formula if you know the circle's radius, r, and the sector's central angle,, in degrees.
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What is the coefficient of x² in the polynomial x³ 4x² 7x 2?
So on solving the provided question we can say that coefficient of x^2 in x³ +4x² -7x -2 is 1.
What is the coefficient?A coefficient in mathematics is a polynomial, a series, or the multiplicative coefficient of a particular term in an expression. Typically numeric, however any expression is permitted. The term "parameter" can also refer to the coefficients themselves if they are variables. A number times a variable equals a coefficient. Coefficient examples include: The coefficient is 14 in phrase 14c 14c 14c. The coefficient is 1 for word g. multiplies the variable by this amount. example Given that "z" is a variable and that 6z is the definition of the term, the coefficient is 6. One is the coefficient of the square of x.
here,
coefficient of x^2 in x³ +4x² -7x -2 is 1
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graph 2y-4x<8 please
We have drawn the graph of the given linear inequality.
What is linear inequality?
A linear inequality is a mathematical statement involving a linear function and one of the symbols <, >, ≤, or ≥. The general form of a linear inequality in one variable is:
ax + b <, >, ≤, or ≥ c
where a, b, and c are constants, and x is the variable.
To graph the inequality 2y - 4x < 8, we first need to put it in slope-intercept form, y = mx + b, so we can see the slope and y-intercept of the line. We can do this by adding 4x to both sides:
2y - 4x < 8
2y < 4x + 8
Then we divide both sides by 2 to get y by itself:
y < 2x + 4
Now we have y = 2x + 4 in slope-intercept form, where the slope is 2 and the y-intercept is 4. To graph the line, we can use the y-intercept as a starting point and use the slope to find additional points on the line.
To graph the line, we can follow these steps:
Plot the y-intercept, which is (0,4) on the graph.
Using the slope of 2, find a second point by going up 2 units and over 1 unit. This point is (1,6)
Draw a line through these two points.
Hence, we have drawn the graph of the given linear inequality.
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The volume of a sphere is given by the equation $V=\frac{1}{6\sqrt{\pi}}S^{3/2}$V=
1
6√πS3/2 , where cap s$S$S is the surface area of the sphere. Find the volume of a sphere, to the nearest cubic meter, that has a surface area of 60 square meters. Use 3.14 for pi$\pi$π .
The volume of a sphere, to the nearest cubic meter is; 44 cubic meter
How to find the Volume of a sphere?
We are given that the formula for the volume of a sphere is;
V = [1/(6√π)]S^(3/2)
where;
V is volume
S is the surface area of the sphere
π = 3.14
We are given that surface area is 60 square meters. Thus, we will now have;
V = [1/(6√3.14)]60^(3/2)
V = 43.713 m³
Approximating to the nearest cubic meter gives;
V = 44 cubic meter
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unit 8 right triangles and trigonometry homework 7 law of cosines
The value of x will be cos^-1 549/900.
What is Law of Cosine ?
The relationship between a triangle's side lengths and the cosine of its angle is known as the law of cosines. Additionally known as the cosine rule.
We know that ,
a²= b²+c² - 2bc cosα, where α is the angle between b and c.
Here, b = 18 , c= 25, a = 20 and cos x
So, a²= b²+c² - 2bc cosα
(20)² = (18)² + (25)² - 2×18×25 cos x
400 = 324 + 625 - 900 cos x
900cos x = 549
cos x = 549/900
hence, x = cos^-1 549/900
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What is the additive inverse of 8 /- 13?
The additive inverse of 8 /- 13 is -8 /+ 13. This means that when you add 8 /- 13 to its additive inverse, the result is 0.
The additive inverse of 8 /- 13 is its opposite number. This means that the sign of the number must be flipped. The inverse of 8 /- 13 is -8 /+ 13. To find the inverse of a number, you must switch the sign of the number from positive to negative or from negative to positive.
When two numbers are added together, the result will be zero if the numbers are opposites. This means that 8 /- 13 and its inverse, -8 /+ 13, when added together, will equal 0. This is because the two numbers have the same value but opposite signs. The positive 8 is cancelled out by the negative 8, and the negative 13 is cancelled out by the positive 13, leaving 0.
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can somebody help me I been doing this all day please dont guess its for a test
decide if thee absolute value inequality is an "and" compound inequality, an "or" compound inequality or nether.
1. |5+m|<13
a. and b. or c. neither
2. 4|y+6|≥3
a. and b. or c. neither
3. |g - 3| = -2
a. g=5 or -5 b. g = 1 or -1 c. null set
Answer: its number 2
Step-by-step explanation:
You swam 4 sets of 200 yards and your friend swam 5 sets of 150 yards. How many more yards did you swim?
You swam 50 yards more than your friend.
What is the unitary method?The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
Given that, you swam 4 sets of 200 yards and your friend swam 5 sets of 150 yards.
Total number of yards you swam 4×200
= 800 yards
Total number of yards your friend swam 5×150
= 750 yards
Number of yards more you swam =800-750
= 50 yards
Therefore, you swam 50 yards more than your friend.
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A newsletter publisher believes that more than 31% of their readers own a personal computer. Is there sufficient evidence at the 0.02 level to substantiate the publisher's claim?
State the null and alternative hypotheses for the above scenario
The null hypothesis is: H0: p ≤ 0.31
The alternative hypothesis is : H1: p > 0.31
What is a null hypothesis?A common statistical theory called the null hypothesis asserts that there is no statistical link or significance between two sets of observed data and measurable events for a given single observed variable.
The null states that the probability of the readers have the personal computers have to be lower than or equal to 31 percent.
What is the alternative hypothesis?
An alternate hypothesis is a claim that the two variables have some kind of statistical relationship.
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All three members of the Harrison family want to take a cruise with Luxury Starline. Estimate how much the family would spend if a cruise costs $1,825 per person by rounding the cost per person to the nearest thousand. $
In the word problem ,the total cost for cruise for total family is $5475.
What is word problem?
Word problems are often described verbally as instances where a problem exists and one or more questions are posed, the solutions to which can be found by applying mathematical operations to the numerical information provided in the problem statement. Determining whether two provided statements are equal with respect to a collection of rewritings is known as a word problem in computational mathematics.
Here the Total members of the Harrison family = 3
Cost of cruise per person = $1825
Now, Total cost for cruise of Harrison family is,
=> Total members × Cost per person
=> 3 × 1825
=> $5475
Hence the total cost for cruise for total family is $5475.
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x^2 = 361 Which represents the solution of the given equation?
Answer: X=19
Step-by-step explanation:
x^2=361
Square root both sides to get rid of the exponent above x.
[tex]\sqrt{x^2}[/tex]=[tex]\sqrt{361}[/tex]
x=19
I am atempting my math homework but I do not understand this question It says "Vera spent 0.04 of of the earnings on a new phone apps What percent of monthly earnings did she spend on phone apps" Can someone help?
The percent of monthly earnings that Vera spends on phone apps is 4%.
How to express the percentage?.The percentage simply has to do with the a value or ratio which can be stated as a fraction of 100. It should be noted that when we want to we calculate a percentage of a number, we simply divide it and then multiply the value that is gotten by 100.
Since Vera spent 0.04 of of the earnings on a new phone apps, the percent of monthly earnings did she spend on phone apps will be:
= 0.04.
= 4/100
= 4%
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What is the total area of the three triangle?
Answer: triangle 1's 1/2 length* triangle 1's width+triangle 2's 1/2 length* triangle 2's width+triangle 3's 1/2 length* triangle 3's width
Step-by-step explanation: triangle 1/2 length* triangle width=triangle area.
A company manufactures two types of cabinets, type 1 and type 2. It produces 110 total cabinets each week. Last week, the number of type 2 cabinets produced exceeded twice the number of type 1 cabinets produced by 20. If x is the number of type 1 cabinets produced and y is the number of type 2 cabinets produced, the system of equations that represents this situation is x + y = 110 and y = 2x + 20.
X for type 1 cabinets = 30
Y for type 2 cabinets = 80
And that proven that type 2 cabinets (Y) produced exceeded more than twice the number of type 1 cabinets (X)
As per the equations stated
X + Y = 110 ……….. (1)
Y = 2x + 20 …...... (2)
We substitute Y at equation (1) with equation (2), to find X
X + Y = 110
X + (2X + 20) = 110
3X + 20 = 110
3X = 110 – 20
3X = 90
X = 90 / 3
X = 30
Then, We substitute value X at equation (1 ) with the value that we have discovered before
X + Y = 110
30 + Y = 110
Y = 110 – 30
Y = 80
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Translate the following arguments into symbolic form. Then determine whether each is valid or invalid by constructing a truth table for each.
If national elections deteriorate into TV popularity contests, then smooth-talking morons will get elected. Therefore, if national elections do not deteriorate into TV popularity contests, then smooth-talking morons will not get elected.
Symbolic Form:
P: National elections deteriorate into TV popularity contests
Q: Smooth-talking morons will get elected
The argument can be expressed as:
If P then Q
Therefore, if not P then not Q
This is a valid argument as it follows the logical form of Modus Ponens. We can verify this using a truth table:
P | Q | P → Q | ¬P | ¬P → ¬Q
T | T | T | F | T
T | F | F | F | T
F | T | T | T | T
F | F | T | T | T
The truth table shows that the argument is valid because the final column (¬P → ¬Q) is true for all four cases. This indicates that if the symbolic form of premise (P) is false, then the conclusion (Q) must also be false.
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A manufacturer records the number of errors each work station makes during the week. The data are as follows 6 3 2 3 5 20 254201 Which of these choices display the correct dotplot?
O A. Number of Errors for the Week for Workstations
O B. Number of Errors for the Week for Workstations 0 10 10
O C. Number of Errors for the Week
O D. Number of Errors for the Week for Workstations for Workstations 10 10
The correct dot plot for the given data would be option A. Number of Errors for the Week for Workstations.
A dot plot is a graph that displays the frequency of a data set by plotting a dot above the number for each occurrence of that number. The x-axis represents the data values and the y-axis represents the frequency of each data value.
In this case, the data represents the number of errors each workstation makes during the week, so the correct way to display the data in a dot plot would be to have the x-axis represent the number of errors and the y-axis represents the frequency of each error. Each dot on the dot plot would be above the number of errors for each workstation.
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Are these answers correct?? If not, please help
Answer:
Step-by-step explanation:
Question 4 I got something different x= 5, y= -4 (double check my work to make sure i used correct equation and all)
Also for slopes does your teacher want you to include the "x". Cuz when I think of slope its only the "m" part which means it should only be -1, -4 but its your call.
How many types of images are there in class 8?
There are 2 types of images namely virtual and real image
The point where light rays from an object meet or appear to contact after reflection or refraction is sometimes referred to as the image. In this concept, a "object" might be anything that light rays are emanating from.
Real images and virtual images are the two types of images. The method by which they are created distinguishes a genuine image from a virtual image. Light beams must converge to create an actual image, whereas they must diverge to create a virtual image.
Real images are those that are created when light rays collide at a specific location after being reflected or refracted by a mirror or lens.
Virtual pictures are those that only seem to form when viewed from behind a mirror. Behind the mirror, the picture is not actually there. When light beams are refracted or reflected, a virtual picture is created.
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Jose swam 8 kilometers against the current in the same amount of time it took him to swim 16 kilometers with the current. The rate of the current was 1 kilometer per hour. How fast would Jose swim if there were no current?
The rate at which Jose will swim if there was no current would be = 2 km / hour
What is speed ?Speed is defined as the scalar quantity that is used the express the relationship of the distance of a moving object with respect to its time.
Jose swims with the current of the water and against the current of the water within the same time.
When swimming with the current he covers = 8 km
The rate he used with the current = 1 km / hour
When he is swimming against the current he covers = 16 km
Therefore the rate at which he swims against the current = 16/8 = 2km/hour.
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A line pae through the point (–6,5) and (8,–2). What i it equation in point-lope form?
The equation of the line passing through the point (-6, 5) and (8, -2) in point-slope form is y - 5 = -0.5(x + 6).
The point-slope form of a line is
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope of the line.
To find the equation of a line given two points, we can use either point and find the slope of the line using the formula:
m = (y2 - y1) / (x2 - x1)
So, we can use the point (-6, 5) and (8, -2)
m = (y2 - y1) / (x2 - x1) = (-2 - 5) / (8 - (-6))
m = -7/14
m = -1/2
Now we can use this slope and the point (–6,5) to find the equation of the line in point-slope form:
y - 5 = -1/2(x - (-6))
y - 5 = -0.5(x + 6)
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What does range () function returns Mcq?
The range is found by finding the vertex and noting that it is the minimum or maximum (depending on direction) of the function. You can find the vertex by using -b/2a for the x value and then plugging in to get y value.
What is a function?
A mathematical phrase, rule, or rule that establishes the link between explanatory variables or a dependent variable In mathematics, curves exist everywhere, and they are crucial for constructing physical links in the sciences.
A Method can be defined by a module, an object, or another function. An internal feature of a class is called a method. Methods are listed inside a base class to make the connections between it class and thus the method clear. A method can be called or invoked using different syntax. We can convert a type into an object before being able to access its methods.
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the perimeter of a rectangle is 70 cm. if its length is decreased by 5 cm and its width is increased by 5 cm,its area will increase by 50 cm^2.find the length and the width of the original rectangle
The length of the rectangle is 25 cm and the width is 10 cm.
How to evaluate for the length and width of the original rectangleWe shall represent the length and width of the original rectangle with x and y respectively so that the perimeter is;
2(x + y) = 70 {divide through by 2}
x + y = 35...(1)
length of new rectangle = x - 5
width of new rectangle = y + 5
area of new rectangle:
(x - 5)(y + 5) = xy + 50
xy + 5x - 5y - 25 = xy + 50
xy - xy + 5x - 5y = 50 + 25
5x - 5y = 75 {divide through by 5}
x - y = 15...(2)
add equations (1) and (2) we have;
x + y + x - y = 35 + 15
2x = 50 {divide through by 2}
x = 25
substitute 25 for x in equation (1), we have;
25 + y = 35
y = 35 - 25 {subtract 25 from both sides}
y = 10
Therefore, have the length and width of the original rectangle to be x = 25 cm and y = 10 cm respectively.
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Maisie knows that she needs 3 kg of grass seed to make a rectangular lawn 5 m by 9m.
Grass seed is sold in 2 kg boxes
Maisie wants to make a rectangular lawn 10 m by 14m.
She has 5 boxes of grass seed.
(a) Has Maisie got enough grass seed to make a lawn 10 m by 14m!
You must show all
your working
(4)
Total Seeds required 9.33 kg and she has 10 kg.so, she has enough grass seed to make a lawn 10m by 14 m.
If you're fortunate enough to have a rectangular lawn, determining the size is straightforward. Multiply the length and breadth measurements together to get the total. You now have your region/area.
So, Maisie knows that she needs 3 kg of grass seed to make a rectangular lawn 5 m by 9m.
Grass seed is sold in 2 kg boxes,
Maisie wants to make a rectangular lawn 10 m by 14 m,
She has 5 boxes of grass seed
To Find, Has Maisie got enough grass seed to make a lawn 10m by 14 m
Solution:
3 kg of grass seed to make a rectangular lawn 5 m by 9m.
=> 3 kg of grass seed needed for 5 x 9 = 45 m²
=> 1 m² required = 3/45 kg seed
a rectangular lawn 10 m by 14 m,
area = 10 x 14 = 140 m²
Hence 140 m² required = 140 x 3/45 = 9.33 kg
Grass seed is sold in 2 kg boxes,
She has 5 boxes of grass seed
Hence total seeds = 5 x 2 = 10 kg
Therefore, total seeds required 9.33 kg and she has 10 kg.so, she has enough grass seed to make a lawn 10m by 14 m.
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help asap please math hw
Answer:
First question:
y intercept = 3
slope = -1/3
y = -1/3x+3
Second Question:
y intercept = -3
slope = -3/2
y = -3/2x=3
Third Question:
y intercept = -1
slope = 3
y = 3x-1
youre welcome buddy :)
Peter stands at the vertex A of a square ABCD. He can make one step at a time either clockwise or counter-clockwise (the length of the step equals the side-length of the square). In how many ways he can make 8 steps and land at the same vertex A?
Therefore 128 ways that Peter can make 8 steps and land back at vertex A.
How many ways can Peter make 8 steps to Vertex A?There are two possible directions for each step that Peter can take, either clockwise or counter-clockwise.
Therefore, for 8 steps, there are 2^8 = 256 possible ways that Peter can make the steps. However, only half of these ways will end with Peter back at vertex A, since the other half will have him end at the opposite vertex.
Therefore, there are 256/2 = 128 ways that Peter can make 8 steps and land back at vertex A.
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What vertical translation is applied to y = x2 if the transformed graph passes through the point (5, 18)?
Step-by-step explanation:
well,
y = x²
would give us
y = 5² = 25
(5, 25)
but we have
(5, 18) so, what is the difference ?
we see that y is reduced by 7 units.
so,
y = x² - 7
the whole curve is moved down by 7 units.