Contrapositive is an example for indirect proof.
Indirect proof;
When the conclusion from a hypothesis is taken to be false (or the opposite of what it states), indirect proof or proof by contradiction is achieved from the given or deduced statements.
Contrapositive method of proof;
Proof by contrapositive, also known as proof by contraposition, is an inference rule used in proofs in mathematics where a conditional statement is inferred from its contrapositive. To put it another way, the assertion "if B, then not A" is proven rather than the conclusion "if A, then B."
Proof by Contrapositive method is indirect.
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Proof by contrapositive is an indirect method of proof where we assume the negation of the conclusion and show that it leads to the negation of the hypothesis. It is a powerful tool in mathematical reasoning and can be used to prove various statements and theorems.
In mathematics, a direct proof is a method of proving a statement by providing a logical sequence of steps that directly leads from the given information to the desired conclusion. On the other hand, an indirect proof is a method of proving a statement by assuming the opposite of the desired conclusion and deriving a contradiction or an absurdity.
To prove a statement using the contrapositive, we start by assuming the negation of the conclusion. Then, we proceed to show that if the negation of the conclusion is true, then the negation of the hypothesis must also be true. In other words, we prove the contrapositive of the statement.
Here is an example to illustrate proof by contrapositive:
Statement: If a number is even, then its square is also even.
To prove this statement by contrapositive, we assume the negation of the conclusion: If a number's square is odd, then the number itself is odd.
We then proceed to show that if the negation of the conclusion is true, then the negation of the hypothesis is also true. In this case, if the square of a number is odd, then the number itself must be odd.
We can prove this by considering that any even number squared will always result in an even number. Therefore, the contrapositive of the original statement is true, which implies that the original statement is true as well.
In conclusion, proof by contrapositive is an indirect method of proof where we assume the negation of the conclusion and show that it leads to the negation of the hypothesis. It is a powerful tool in mathematical reasoning and can be used to prove various statements and theorems.
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When constructing a perpendicular bisector of a segment, why should you make arcs with the same radius from each endpoint? select the true statement.
Arcs with the same radius from each endpoint will help you locate the midpoint of the segment more accurately.
When constructing a perpendicular bisector of a segment, it is important to make arcs with the same radius from each endpoint. This will help you to locate the midpoint of the segment more accurately. By using arcs of the same radius, you can ensure that the bisector is perfectly perpendicular to the segment. When the two arcs intersect, the midpoint of the segment will be the point of intersection. This will also ensure that the bisector is exactly in the middle of the segment, as the two arcs with the same radius will be of equal length. By constructing arcs with the same radius, you can achieve a precise and accurate perpendicular bisector of the segment.
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What is the undefined terms of ray?.
The undefined terms of ray are-
PointLinePlaneThe best way to think of four geometric concepts is as "described and not defined." Since they are used to define other, more complex pieces, some fundamental elements in all branches of mathematics cannot be defined. The three undefined terms that serve as the foundation of Euclidean geometry in geometry can still be used to describe the other three terms. Mathematicians build worlds of concepts like angles, polygons, Platonic solids, Cartesian graphs, and more using descriptions of these four terms as a starting point.These terms are not any less useful or valid than other terms that emerge from them just because they are formally undefined. Theorems, proofs, and the definition of other words frequently make use of these four undefined terms.To learn more about rays here
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How do you find the inverse of a linear graph?.
The inverse of the linear graph can be found by interchanging the values of x and y
When a function is stated in the form y = mx + b, where m and b are real integers and m is not equal to 0, the function is said to be linear. A linear graph is the graph of a linear function. The inverse of a linear graph can be found in two steps -
The first step is to replace the value of y with x and x with y in the original equation of the linear function.The second step is to solve the equation for x in terms of y, which will give the inverse function.For example, if the linear function equation is y = 3x + s, the inverse function can be found by swapping x and y, and solving the equation for x: y = 3x + 2 and x = 3y + 2. Alternatively, the inverse function can also be determined by interchanging x and y values and changing the sign of the slope of the original function. Therefore, the inverse function of y = 3x + 2 is x = (2/3)y - 2/3
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How large does the strength of an earthquake differ for a 1. 0 increment in the Richter magnitude scale?.
The strength of an earthquake differ for a 1. 0 increment in the Richter magnitude scale by 31 units.
American seismologists Charles F. Richter and Beno Gutenberg developed the Richter scale (ML), a numerical indicator of an earthquake's size, in 1935. The height of the greatest seismic wave, as measured by the logarithm of its amplitude (height), is used to calculate the earthquake's magnitude.
Although the original Richter scale has been supplanted by other, more precise scales in modern science, the Richter scale is still frequently used incorrectly in news reporting of earthquake severity as a catch-all name for the logarithmic scale that earthquakes are measured on.
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Help is very much appreciated need this by tmr.
A triangle has vertices A (2,-3), B (-2,1) and C (1,2). Find the midpoint of the side AB.
The midpoint of side AB is (0,-1)
How to find the midpoint of the Triangle?
To find the midpoint of a line segment, you need to find the average of the x-coordinates and the average of the y-coordinates of the two endpoints of the segment.
To find the midpoint of a line segment, you need to find the average of the x-coordinates and the average of the y-coordinates of the two endpoints of the segment.
The midpoint of side AB can be found using the coordinates of points A and B:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Midpoint = ((2 -2)/2, (-3+1)/2)
Midpoint = (0,-1)
Hence, the midpoint of side AB is (0,-1)
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please help this is due today!!!!
The unit rate for the speed of the snail is 4/3 cm/s
What is unit rate?A unit rate means a rate for one of something. We write this as a ratio with a denominator of one. For example, if you ran 70 yards in 10 seconds, you ran on average 7 yards in 1 second. Both of the ratios, 70 yards in 10 seconds and 7 yards in 1 second, are rates, but the 7 yards in 1 second is a unit rate.
Given is a graph of distance vs time for the speed of a snail,
We need to find, unit rate for the speed of the snail by the given graph,
Since, we are given a coordinate (12, 16) therefore, the snail covers a distance of 16 cm in 12 sec,
Therefore, in 1 sec, it will cover = 16 / 12 = 4/3 cm/s
Hence, the unit rate for the speed of the snail is 4/3 cm/s
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Jake is buying baseball cards for his brother in college. The price of a card is based on its condition. The table shows the total cost of x number of cards for each condition.
Condition Total Cost
Poor x
Fair 1.75x
Good 9.5x
Excellent 20.5x
Like New 45.65x
He buys 6 that are in fair condition, 5 that are in good condition, and 2 that are in excellent condition. The shipping cost for the baseball cards is $4.00. What expression represents the total cost of buying and shipping the baseball cards?
The total cost of buying and shipping the baseball cards is given as follows:
$396.
How to obtain the total cost?The total cost is obtained applying the proportions in the context of this problem, considering the state of each card.
The amounts, along with their cost, are given as follows:
Fair: 6 x 1.75 x 4 = $42.Good: 5 x 9.5 x 4 = $190.Excellent: 2 x 20.5 x 4 = $164.Hence the total cost of buying and shipping the baseball cards is obtained as follows:
42 + 190 + 164 = $396.
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help me with this topic pls i need to finish 15 topics and i olny have 6 done :')
The amount of water in the tank at 0 minutes is equal to 150 liters.
As time increases, the amount of water in the tank increases by 100 liters per minute.
How to determine the amount of water in the tank at 0 minutes?Mathematically, the point-slope form of a straight line can be calculated by using this mathematical expression:
y - y₁ = m(x - x₁) or y - y₁ = (y₂ - y₁)/(x₂ - x₁)(x - x₁)
Where:
m represents the slope.x and y are the data points.Next, we would determine the linear equation representing the line which passes through the points (0, 150) and (2, 350) by using the point-slope form as follows:
y - y₁ = (y₂ - y₁)/(x₂ - x₁)(x - x₁)
y - 150 = (350 - 150)/(2 - 0)(x - 0)
y - 150 = 100(x - 0)
y = 100x + 150
When x = 0 minutes, the amount of water in the tank is given by;
y = 100(0) + 150
y = 150 liters.
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A gymnast joined a yoga studio to improve his flexibility and balance. He pays a monthly fee and a fee per class he attends. The equation y = 30 + 10x represents the amount the gymnast pays for his membership to the yoga studio per month for a certain number of classes.
Which graph represents this situation?
For the above case, the graph of the linear equation y = 20 + 10x
Which graph represents this situation?Each solution inside the scenario involving two variables may be thought of as the Cartesian coordinates of a specific point on the Euclidean plane.The answers to a linear equation create a line inside the Euclidean plane, and each line may be considered as the sum of all solutions to a linear model with two variables.
This is how this type of equation became known as “linear.” The solutions to an equation system with integer variables form a hyperplane in Euclidean space with size n (a subdomain of dimension n1).
Because linear equations typically properly reflect nonlinear systems, they are widely employed in all fields.
Mathematics as well as in physics and engineering.
Here the equation is given as y = 20 + 10x
Therefore at x = 0 , y = 20 and at y = 0, x = -2
This represents that the fixed monthly fee is 20 and the daily fee is 10.
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[tex]y=kx[/tex]
If [tex]y=42[/tex] when [tex]x=70[/tex], what is the value of K?
Give any decimal answers to 1 d.p
Answer:
k=0.6
Step-by-step explanation:
y=kx
42=k(70)
42=70k
k=0.6
What are some of the reasons why our bags would differ from the expected distribution?
The reasons why our bags would differ from the expected distribution are sampling error, measurement error, selection bias, quality control issues, human error, environmental factors and external factors
Reasons why our bags would differ from the expected distribution?There are several reasons why the distribution of bags might differ from the expected distribution. Some possible causes include:
1. Sampling error: If the sample size of bags is small, the distribution of bags may not accurately reflect the overall population.
2. Measurement error: If the bags were not measured or counted correctly, the distribution may be affected.
3. Selection bias: If the bags were not selected randomly or if certain bags were excluded from the sample, the distribution may not reflect the true population.
4. Quality control issues: If there were issues with the production of the bags, such as variations in size or weight, the distribution may be affected.
5. Human error: If mistakes were made in the handling, counting, or measuring of the bags, the distribution may not be accurate.
6. Environmental factors: External factors, such as temperature, humidity, or transportation, could cause changes in the weight of bags.
7. External factors: The distribution could also differ because of external factors such as competition, marketing, seasonality, etc.
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How do you find the domain and range of an absolute value graph?.
First of all, we will need to find the set of values of x for which we will get the possible value of y. This is comprehended as the domain of a function. Secondly, to find the range, we need to find all the possible values of y, which we will get after putting the values of x.
Now, to find the domain:-
Case 1: If the equation doesn’t have a denominator, then the domain will be (−∞,∞).
Case 2: If the equation has a denominator, then the domain will be all values other than those where the denominator will become 0.
Case 2: If the equation has a root in the denominator, the domain will be all values except the ones where the value is inside root<0.
To find the range:-
Case 1: If the equation doesn’t have a denominator, the range will be [0,∞). This is because whenever the value of y tends to be negative, the modulus sign will make it positive.
Case 2: If the equation has a denominator, then the range will be all values above 0 except the points where the denominator is equal to 0.
Case 3: If the equation has a root in the denominator, the domain will be all values more than 0 except the ones where the value is inside root<0.
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Charlotte has some number cards that each
have a whole number on them.
If she chooses a card at random, the
probability that the number on it is even is 4/9
Work out the ratio of even cards to odd cards
that Charlotte has.
Give your answer in its simplest form.
When a die is thrown, getting an even number, The probability of getting an even number when a die is thrown is 1/2.
How do you find the probability of even?Some number cards with a whole number on each are in Charlotte's possession. There is a 4/9 chance that the card she chooses at random will have an even number on it.
Charlotte's balance of even to odd cards. Your response should be as brief as possible. Three out of six, or three-sixths, of a die will provide an even number.
As both the numerator and denominator of this fraction can be divided by three, it can be simplified. A definition of an even number An even integer is one that can be divided by two while leaving a 0 as the final digit. Two, four, six, eight, ten, and other even numbers are examples. The first and present definitions are equivalent.
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In the triangle below, m/A = x°, m/B = 3x°, m/C= 24°. Find the value of mA =?
29°
39°
36.5°
O 41.5°
Answer:
39°
Step-by-step explanation:
Basically, what you have to do is x+3x+24°=180° (since the measures of all sides inside all triangles is 180°). 4x+24°=180°. 4x=156°. x=39°.
Is inverse function concave?.
The concept is the Concave function. The answer is yes if the inverse function is convex in nature.
In mathematics, a concave function is the opposite of a convex function. A concave feature is also referred to as downward concave, downward concave, upward convex, vertex convex, or top convex.
The slope of the inverse decreases as the slope of f increases. Thus the inverse of a convex differentiable function with a non-zero derivative is strictly concave.
In general, it can be shown that the inverse of a convex function is concave. Geometrically, a function is concave if its graph lies across its tangents. A function is concave if its graph lies below its tangent.
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write the word sentence as an inequality
A number b multiplied by is at most -3 2/3
Please help,
thank you,
An inequality is b > -3 The symbol > represents "greater than OR equal to".
What is inequality ?The occurrence of an unequal and/or unequal distribution of opportunities and resources among the people that make up a society is referred to as inequality. To different individuals and in various settings, the word "inequality" may indicate different things.B cannot be less than -3, but it can be more than, as it must be at least -3. The greater than sign derives from this. The inequality in word form would be that B is bigger than or equal to -3 since the lowest number you may have is -3 (the number itself still counts).An inequality is b > -3 The symbol > represents "greater than OR equal to".To learn more about inequality refer to:
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It takes 1200 workers 8 years to build 3 ziggurats. How many years would it take 1600 workers to build 25 ziggurats? (Assume all workers work at the same constant rate.) You can solve this problem without knowing what a ziggurat is!
The number for years that it would take 1600 workers to build 25 ziggurats is 66.67 years.
How to calculate the number of years?To find out how many years it would take 1600 workers to build 25 ziggurats, you can use the formula:
(time taken by 1200 workers to build 3 ziggurats) x (1600 workers / 1200 workers) x (25 ziggurats / 3 ziggurats) = time taken by 1600 workers to build 25 ziggurats
Plugging in the given values:
(8 years) x (1600 / 1200) x (25 / 3) = 8 years x (4/3) x (25/3)
= 8*25/3
= 200/3
= 66.67 years
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The rental fee for a Quad to ride off road and on the beach is $80, plus $10.00 for each hour that you ride. Write an algebraic expression for the total cost of renting the Quad for h hours?
Expression: ____________
Use the expression to determine the cost of renting the Quad for three and a half hours.
Total Cost is $__________
Algebraic expression for the total cost of renting the Quad for h hours is $80 + $10h. Cost of renting the Quad for three and a half hours is $115.
Short note on algebraic expressions?A mathematical sentence that incorporates coefficients, unknowable variables, algebraic operations, and constants is called an algebraic expression. The equals symbol, however, is not allowed in an expression. A mathematical phrase known as an algebraic expression combines variables and constants using the operational (+, -, & &) symbols. The equal (=) symbol is absent from algebraic symbols. Algebraic expressions include, for example, 10x + 63 and 5x - 3.
The rental fee for a Quad is $80 and it costs an additional $10.00 for each hour that you ride. So the total cost of renting the Quad for h hours can be represented by the algebraic expression:
$80 + $10h
where h is the number of hours you rent.
To find the cost of renting the Quad for three and a half hours, we can substitute h = 3.5 into the expression:
$80 + $10(3.5) = $80 + $35 = $115
So the total cost of renting the Quad for three and a half hours is $115.
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Is 4 is a perfect number?.
4 is not a perfect number because the sum of its factors (besides 4 itself), 1+2, is less than 4. Numbers like 4 are known as deficient numbers.
In number theory, a natural number is called deficient if it is greater than the sum of its divisors, excluding itself. For example, the number 8 is deficient because its divisors are 1, 2, and 4, and 1 + 2 + 4 = 7, which is less than 8. A number that is not deficient is called abundant.
The concept of deficient numbers was first studied by the ancient Greek mathematician Nicomachus of Gerasa in his work "Introduction to Arithmetic". He classified numbers into three categories: abundant, perfect, and deficient numbers. He defined a perfect number as a number whose divisors (excluding the number itself) sum up to the number and abundant if the sum of divisors (excluding the number itself) is greater than the number.
The study of deficient numbers is closely related to the study of divisors of numbers, which is a fundamental area of number theory. The study of deficient numbers also has connections to other areas of mathematics such as algebraic number theory and the theory of quadratic forms. In addition, the study of deficient numbers has applications in computer science and cryptography.
It is important to note that only a small percentage of numbers are deficient, and the study of deficient numbers is not as widely studied as the study of prime numbers or perfect numbers, but it is still an interesting area of research in number theory.
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In a right angled triangle, the lenght of the right angle is 11 and the lenght of the other two sides are positive integers. what is the perimeter of the triangle?
Answer:
66
Step-by-step explanation:
2m = 11
m = 11/2
m = 5.5
m² - 1 = 29.25
m² + 1 = 31.25
The perimeter = 29.25+31.25+5.5 = 66
How to add two rational expressions with different denominators?.
Find the lowest common denominator before adding rational expressions. It is obvious that this is the common denominator because the denominator of the first fraction factors to 2(x+2).
Least Common Denominator (LCD): The smallest expression that is divisible by the original denominators is known as the LCD of two or more rational expressions. We need to locate all the denominators of our rational expressions in order to find the LCD.The addition and subtraction of rational expressions resembles those of fractions. Remember that if the denominators are the same, we can add or subtract the numerators and then write the result over the shared denominator. When dealing with rational expressions, a polynomial will serve as the common denominator.To learn more about Rational expression here
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What is product of division?.
The number being divided is known as the dividend, the number dividing it is known as the divisor, and the result of the division is known as the quotient.
Factors are the numbers that are being multiplied, and the result of the multiplication is known as the product. Dividend, divisor, and quotient are the terms used to describe the three numbers involved in division: the dividend, the divisor, and the result of the division, the quotient.
The arithmetic procedures of multiplication and division are related and developed from commonplace situations. Multiplication is the same as repeated addition for whole numbers, hence repeated addition exercises are frequently used to teach multiplication.
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In two or more complete sentences, explain how to solve the cube root equation, x + 1 3 − 2 = 0 .
The value of x is 7 in the equation ∛x+1 - 2 =0
What is Equation?Two or more expressions with an Equal sign is called as Equation.
The given equation is ∛x+1 - 2 =0
The first step is to figure out what the equation is. When unconventional math symbols are used, and when there are no grouping symbols identifying operands, that can be the most difficult step
∛x+1 - 2 =0
Add 2 on both the sides
∛x+1 = 2
Take cube on both the sides
x+1=2³
x+1=8
Subtract 1 from both sides
x=7
Hence, the value of x is 7 in the equation ∛x+1 - 2 =0
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Find a bae and exponent form for the following number without uing the exponent 1
The exponent form of 1,000,000 is [tex]10^{6}[/tex] and the base of 1,000,000 is 10.
The term 'power' in mathematics refers to increasing a base number to its exponent. It signifies the "base number" and "exponent" as the two fundamental elements of powers.
The exponent represents the number of times the base number is multiplied, whereas the base number is defined as a number multiplied by itself. We can see exponent terms in many scientific notations to denote the large numbers as the powers of 10.
In a nutshell, power is a number expressed using exponents. It means that the same factor is multiplied multiple times.
Thus, The exponent form of 1,000,000 is [tex]10^{6}[/tex] and the base of 1,000,000 is 10.
Complete question: Find a base and exponent form for the following number without using the exponent 1,000,000.
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The base number is described as a number multiplied by itself, whereas the exponent shows how many times the base number has been multiplied. Many scientific notations use exponent terms to identify large values as powers of 10.
In a nutshell, a power is an exponentiated integer. It denotes numerous times of multiplying the same factor.
write an equivalent expression for 4x + 9 + 3y + 3x + 4
The equivalent expression for 4x + 9 + 3y + 3x + 4 is 7x + 3y + 13.
What is a good example of an equivalent expression?
Example Expressions Expressions That Are Equivalent Equivalent
Because both expressions have the same value for any value of x, 3(x + 2) and 3x + 6 are identical expressions. 3x + 6 = 3 × 4 + 6 = 18.
To write an equivalent expression, we can simplify the expression by combining like terms:
4x + 3x = 7x
9 + 4 = 13
So, the equivalent expression for 4x + 9 + 3y + 3x + 4 is 7x + 3y + 13.
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Zahara Smith i wondering how he can make her check tretch enough to afford her monthly cable, water, grocery, and cell phone bill. Zahara i concerned with
need. Limit. Need and want. Want and limit
Answer: Sam here it
Step-by-step explanation:
If you were to graph the inequality would the arrow point to the left or right? x<17
The arrow point to the left for the inequality x < 17.
What is inequality?When two expressions are connected by a sign like "not equal to," "greater than," or "less than," it is said to be inequitable. The inequality shows the greater than and less than relationship between variables and the numbers.
Given that inequality is x < 17. When the inequality is plotted on the graph it will highlight the values that are less than the number 17. The graph of the inequality is attached with the answer below.
Hence, the arrow for the given inequality will point to the left side.
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(5c³) - (2c⁵) = ?
please help
Answer:
[tex](5-2c^{2}) c^{3}[/tex]
Step-by-step explanation:
Factor out [tex]c^{3}[/tex]
[tex](5-2c^{2})[/tex] is not factored since it does not have any rational roots.
I really need help like tonight
Answer:
Angle 1- 36
Angle 2- 90
Angle 3- 54
Angle 4- 36
Angle 5- 90
Angle 6- 54
Angle 7- 128
Angle 8- 54
Angle 9- 128
Angle 10- 54
Angle 11- 36
Angle 12- 144
Angle 13- 36
Angle 14- 144
(Let R, denotes the reflection on y-axis and R2 denotes a rotation of 90° about the center O(0.0), then for any point A(3. 4). find RR, (A).)
The solution of RR(A) is (-3, -4).
What is the reflection and rotation on a point? Reflection and rotation are two types of transformations that can be applied to a point. Reflection is when a point is mirrored across a line of symmetry. The line of symmetry acts as a mirror and the point is flipped across it. This produces a reflection in which the point is now at the exact opposite side of the line of symmetry. Rotation is when a point is spun around a fixed axis. The point can be rotated in either a clockwise or counter-clockwise direction. When the point is rotated, it is still in the same location, only rotated around the axis. Both of these transformations can be used to move a point in a certain direction or to change the shape of an object. Reflections and rotations can be used to create symmetry in a shape or to move a point to a different location. They are both useful transformations that can be used to manipulate the position or shape of an object.To learn more about reflection on y-axis refer to:
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