The angles made by f with the x-axis, y-axis and z-axis are 38.32°, 107.19° and 51.39°.
Given vector is f = 80i - 40j + 60k.
We need to calculate the angles made by f with the x, y and z-axis.Let us calculate the magnitude of the vector f:
Magnitude of f = √(80²+(-40)²+60²)lb
Magnitude of f = √(6400+1600+3600)lb
Magnitude of f = √(11600)lb
Magnitude of f = 107.68 lb
We can use the direction cosines to find the angles made by f with the x, y and z-axis.
Let l, m and n be the direction cosines of f.
cos²θ + cos²φ + cos²γ = 1
Where θ, φ and γ are the angles made by f with the x, y and z-axis.
We know that,
f = 80i - 40j + 60k
∴ l = 80/107.68,
m = -40/107.68 and
n = 60/107.68
cos²θ + cos²φ + cos²γ = 1
(80/107.68)² + (-40/107.68)² + (60/107.68)² = 1
cos²θ = (80/107.68)²
cosθ = ±(80/107.68)
cos²φ = (-40/107.68)²
cosφ = ±(-40/107.68)
cos²γ = (60/107.68)²
cosγ = ±(60/107.68)
Therefore, the angles made by f with the x-axis, y-axis and z-axis are
cosθ = ±(80/107.68)
cosφ = ±(-40/107.68)
cosγ = ±(60/107.68)
Since we have two possible solutions, let us calculate the angles with both the positive and negative values.
θ = cos⁻¹(80/107.68)
θ = 38.32° and
θ = 141.68°
φ = cos⁻¹(-40/107.68)
φ = 107.19° and
φ = 252.81°
γ = cos⁻¹(60/107.68)
γ = 51.39° and γ = 128.61°
Therefore, the angles made by f with the x-axis, y-axis and z-axis are 38.32°, 107.19° and 51.39°.
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How can i explain to children what the length of 2000 feet looks like to children
2000 feet can be represented as 20 blue whales in a row of each 100 feet.
What is a blue whale?
Blue whales are the largest mammals to have ever existed on Earth. With lengths of up to 100 feet and weights of up to 200 tons, these magnificent marine creatures command the waters.I feet = 30.48 centimeters
To explain what the length of 2000 feet looks like:
Take the example of a blue whale of 100 feet.
Now, imagine placing up 20 blue whales one behind the other.
20 × 100 feet = 2000 feet
Therefore, 2000 feet can be represented as 20 blue whales in a row of each 100 feet.
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Find the area of the shape. ( image of 3/4 circle with the radius of 6 ) Answer in exact terms of pi or use 3.14 as pi and enter as a decimal.
The area of the shape is 84.78 square units
Area of a circleThe formula for calculating the area of a circle is expressed as:
A = πr²
where
r is the radius of a circle
Given the following
r = 6 units
pi = 3.14
Substitute
A = 3.14 * 6²
A = 113.04 square units
If the figure is 3/4 of a circle, the area of the shape will be:
A = 3/4 * 113.04
A = 84.78 square units
Hence the area of the shape is 84.78 square units
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It cost jose $8.70 to send 87 text messages. How much would it cost to send 10 text messages?
Answer:
One dollar
Step-by-step explanation:
So you divide 8.70 by 87 to get 0.10. So then you multiply 0.10 by 10 to get a dollar
Sending 87 text messages cost $8.70. Sending 10 text messages would cost $1.00.
Use the concept of fraction defined as:
A fraction is a part of the whole number and a way to split up a number into equal parts. Or, A number which is expressed as a quotient is called a fraction. It can be written in the form of p: q, which is equivalent to p / q.
Given that,
Cost to send 87 text messages: $8.70
Number of text messages: 87
To find out how much it would cost to send 10 text messages,
Use a proportion.
If it cost Jose $8.70 to send 87 text messages,
Set up the proportion:
$8.70 / 87 messages = x / 10 messages
To solve for x,
Cross-multiply and divide:
($8.70 x 10) / 87 = $1.00
Therefore, it would cost $1.00 to send 10 text messages.
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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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The three steps for constructing a simple frequency distribution are Group of answer choices find the observed range, find the interval width, and construct the frequency distribution find the real range, count the scores, and construct the frequency distribution find the real range, find the interval width, and construct the frequency distribution all of the above
The correct option is C. find the real range, find the interval width, and construct the frequency distribution.
What is frequency distribution?To make the information easier to understand, frequency distributions are textured look which organise and present frequency counts. Absolute frequencies as well as frequency distributions, such as percentages or ratios, can be displayed in frequency distributions.
The real range is find by the following method-
Range is referred to as the difference between both the lower limit of the minimal interval and the upper limit of the maximal interval of the grouped data in the case of a continuous frequency distribution. That is true for the following ranges for X: 0–10, 10–20, 20–30, and 40–50; 4–0=40.The interval width is find by the following method-
The distance between both the lowest endpoint of one interval as well as the lowest endpoint of the following interval is known as the class interval width. Therefore, if the continuous intervals in our study range 0 to 4, 5 through 9, etc., the very first five intervals' widths are 5 and the final interval is open because no larger endpoints is allocated to it.To know more about the frequency distribution table, here
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The correct question is-
The three steps for constructing a simple frequency distribution are -
Group of answer choices
A. find the observed range, find the interval width, and construct the frequency distribution
B. find the real range, count the scores, and construct the frequency distribution
C. find the real range, find the interval width, and construct the frequency distribution
D. all of the above
Which of these statements is true for
f(x)=(1/2)^x
A. the domain of f(x) is x>0
B. it is always increasing
C. the range of f(x) is y>1/2
D. the y-intercept is (0,1)
Answer: D. the y-intercept is (0,1)
Step-by-step explanation:
For all exponential functions without a vertical shift, the y-intercept is (0,1).
Which of the following is a key property of the quadratic parent function?
OA. It is in quadrants III and IV.
OB. Its vertex is at the origin.
C. It is not a parabola.
OD. It is not a function.
The correct property of the quadratic parent function is given by:
B. Its vertex is at the origin.
What is the quadratic parent function?The quadratic parent function is modeled by:
y = x².
It has the vertex at the origin and increases to the left and to the right, into quadrants I and II, forming a function that is graphed by a parabola.
Hence the correct option regarding a property of the quadratic parent function is given by:
B. Its vertex is at the origin.
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A particular company's net sales, in billions, from 2008 to 2018 can be modeled by the expression t2 10t 68, where t is the number of years since the end of 2008. what does the constant term of the expression represent in terms of the context?
The company earned 68 billion dollars from 2008 to 2018.
According to the statement
we have given that the company's particular sales in expression are represented by (t² + 10t + 68)
And we have to find the constant terms in this expression which represent the sale of the company.
So, The given expression is (t² + 10t + 68)
From above expression it is clear that the it is in the form of Quadratic Equations.
So, The Quadratic equation, is an equation containing a single variable of degree 2. Its general form is ax^2 + bx + c = 0, where x is the variable and a, b,c are the known numbers and C is the Constant.
The general representation of Quadratic equation is
ax^2 + bx + c = 0 -(1)
and the given expression are
(t² + 10t + 68) -(2)
After comparing both equations
Here C =68 = Constant.
So, 68 is the Constant term of the expression represent in terms of the context.
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Disclaimer: The question was incomplete. Please find the full content below.
Question:
A particular company's net sales, in billions, from 2008 to 2018 can be modeled by the expression t2 + 10t + 68, where t is the number of years since the end of 2008. What does the constant term of the expression represent in terms of the context?
The company earned 68 billion dollars from 2008 to 2018.
The company earned 68 billion dollars in 2008.
The company earned 10 billion dollars from 2008 to 2018.
The company earned 10 billion dollars in 2008.
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Function f is a rational function with a y-intercept at (0,2). if g(x) = 4f(x), which point represents the y-intercept of function g?
Answer:
(0,8)
Step-by-step explanation:
[tex]g(0)=4f(0)=4(2)=8[/tex]
please help me it is a fairly easy problem
Answer:
a=5
b=6
c=3
d=4
surface area: 105.24
Answer:
Area of the right triangular prism = 84 cm²
Step-by-step explanation:
A is the width of the right side rectangle = 4 cm
B is the length of the rectangle = 6 cm
C is the base of the triangle = 3 cm
D is width of the middle rectangle = 5 cm
b) Area of rectangle = length *width
[tex]\sf Area \ of \ triangle = \dfrac{1}{2}*base *height\\\\Area of 2 triangles = 2 *\dfrac{1}{2}*base * height = base * height[/tex]
Surface area of the right triangular prism = area of 2 triangle + area of left rectangle + area of middle rectangle + area of right rectangle
= 3 * 4 + 3*6 + 5 * 6 + 4 *6
= 12 + 18 + 30 + 24
= 84 cm²
At a community fund raiser event, bracelets cost $3 and necklaces cost $6. Complete the patterns in
the first two columns for the cost of 0, 1, 2, 3, and 4 pieces of jewelry. Write the ordered pair for the
corresponding terms in the third column. Then graph the ordered pairs.
I need to make a sequence for the cost of braclets (in dollars) and I also need to make an sequence for the cost of necklaces (in dollars)
please help :)
The sequence for the cost of bracelet in dollars = $0, $3, $6, $9 and $12.
Also the sequence for the cost of necklace is = $0, $6, $12, $18, $24.
Calculation of the various cost sequenceThe column given concerning the products contains the following amounts of products such as 0, 1, 2, 3, and 4 pieces.
Therefore to calculate the amount to be used to buy each product of bracelet and necklace in each column multiplication is carried out.
That is, 0×3, 1 ×3, 2×3, 3×3, and 4×3
which is = $0, $3, $6, $9 and $12 respectively for the bracelet.
0×6, 1 ×6, 2×6, 3×6, and 4×6
Which is = $0, $6, $12, $18, $24. respectively for the necklace.
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find the sum of all 2 digit multiples of 6
Answer:
The sum of all 2 digit multiples of 6 is 810
Step-by-step explanation:
12+18+24+30+36+42+48+54+60+66+72+78+84+90+96 = 810
Answer:
810
Step-by-step explanation:
2 digit multiples of 6 are 12,18,24...90,96
You can just add them on your calculator.
is 3(2x + 1) equal to 5x + 4
Answer:no but maybe
Step-by-step explanation:if you distribute the 3 and multiply the 3 with box the 2 and 1 you would get 6x+3 which is not equal to 5x+4 but if you take one from the 6 and add it to the 3 it would be even so i dont know but maybe
Select the correct statement.
A. seventeen thousand four hundred ninety two multiplied by seven over five is less than seventeen thousand four hundred ninety two
B. seventeen thousand four hundred ninety two multiplied by seven over five is equal to seventeen thousand four hundred ninety two
C. seventeen thousand four hundred ninety two is greater than seventeen thousand four hundred ninety two multiplied by seven over five
D. seventeen thousand four hundred ninety two multiplied by seven over five is greater than seventeen thousand four hundred ninety two
HELPPPPP
Answer:
I think it's (B) and (D)
Step-by-step explanation:
As to what I have studied
please help im in a rush
Answer:
a)
Surface Area: 172 cm^2
Volume: 112 cm^3
b) Surface Area
c) Volume
Step-by-step explanation:
The dollar-mart grocery stores sales six bars of soap for $1. how many soaps can a customer buy with $9.
Answer:
54
Step-by-step explanation:
6 bars of soap = $1
x bars of soap = $9
We need to multiply both sides of the equation by the same amount so that $9 is on the right.
$1 x 9 = $9
6 x 9 = 54
So, you can buy 54 soaps with $9.
Hope this makes sense. Feel free to ask further if need be.
In which situation could the quotient of -24 divided by 3 be used to answer the question?
A. The temperature of a substance decreased by 24∘C per minute for 3 minutes. What was the overall change of the temperature of the substance?
B. A football team lost 24 yards on one play, then gained 3 yards on the next play. How many total yards did the team gain on the two plays?
C. Julia withdrew a total of $24 from her bank account over 3 days. She withdrew the same amount each day. By how much did the amount in her bank account change each day?
D. A cookie jar contains 24 cookies. Each child receives 3 cookies. How many children are there?
The situation in which the quotient of -24 divided by 3 be used to answer the question is;
Julia withdrew a total of $24 from her bank account over 3 days. She withdrew the same amount each day. By how much did the amount in her bank account change each day? AlgebraCheck all options:
A. The temperature of a substance decreased by 24∘C per minute for 3 minutes. What was the overall change of the temperature of the substance?
Temperature decrease = 24°C per minutes × 3 minutes
= 72°C decrease in temperature
B. A football team lost 24 yards on one play, then gained 3 yards on the next play. How many total yards did the team gain on the two plays?
Loss = -24 yards
Gain = 3 yards
Total gain = -24 + 3
= -21 yards
C. Julia withdrew a total of $24 from her bank account over 3 days. She withdrew the same amount each day. By how much did the amount in her bank account change each day?
Amount withdrawn per day = -$24 / 3
= -$8 per day
Change in her account per day = $8
D. A cookie jar contains 24 cookies. Each child receives 3 cookies. How many children are there?
Number of children = 24 cookies / 3 cookies
= 8 cookies per children
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Given: JKLM is an isosceles trapezoid, KL ∥ JM
Prove: KM ≅ JL
Trapezoid J K L M with diagonals is shown. Sides K L and J M are parallel.
What is the missing reason in step 4?
Statements
Reasons
1. JKLM is an isosceles trapezoid,
KL ∥ JM 1. given
2. JK ≅ LM 2. definition of isosceles trapezoid
3. KL ≅ KL 3. reflexive property
4. ∠JKL ≅ ∠MLK 4. ?
5. △JKL ≅ △MLK 5. SAS ≅ theorem
6. KM ≅ JL 6. CPCTC
definition of linear pair
definition of congruence
base angles theorem
sufficient base angles theorem
Third option Base angles theorem is the correct answer as it contains the information that statement 4 is missing.
About The Base Angle Theorem
The two base angles in an isosceles triangle that are opposite the congruent sides must be identical in measure or congruent, according to the base angle theorem.
As a result, JKL and MLK are the base angles of the recognized isosceles triangle. Thus, the two angles, ∠JKL and ∠MLK are congruent or equal to each other by the means of the base angles theorem, and third option - base angles theorem - the missing justification for statement 4.
About the other options
When two lines cross at one point, a linear pair of angles is created. If the angles follow the intersection of the two lines in a straight line, they are said to be linear. A linear pair's total angles are always equal to 180°.In geometry, two figures or objects are said to be congruent if their shapes and sizes match, or if one is the mirror image of the other.According to the sufficient base angles theorem, if a triangle's sides (such as those of an isosceles triangle) are congruent, then its opposing angles must also be congruent.Learn more abut the base angles theorem here:
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Answer: option c
Step-by-step explanation:
Dad has 15 oak and maple boards from the lumber yard. each oak board is 6 centimetres thick. each maple board is 9 centimetres thick. when my dad stacks the boards, the stack is 96 centimetres thick. how many of each type of board does he have?
Answer:
He has 2 maple boards and 13 oak boards.
Step-by-step explanation:
Assuming all boards are oak : 6*15=90
Maple minus oak = 3
96-90=6 6/3=2 maple boards
Verification : 2*9+6*13 = 96
Answer:
13 oak, 2 maple
Step-by-step explanation:
o = oak
m = maple
o + m = 15
6o + 9m = 96
o + m = 15
o = 15 - m
6o + 9m = 96
6(15 - m) + 9m = 96
90 - 6m + 9m = 96
90 + 3m = 96
3m = 6
m = 2
o + 2 = 15
o = 13
6(13) + 9(2) = 96 (Yes, correct!)
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You spend 0.88 of your allowance this week. what percent of your allowance did you spend?
Answer: 88%
Step-by-step explanation: So, to convert the decimal into a percentage, multiply .88 by 100. And BOOM! you have 88% as your answer.
We spent 88% of your allowance this week.
Since we know that,
A figure or ratio that may be stated as a fraction of 100 is a percentage. If we need to calculate a percentage of a number, we should divide it by its entirety and then multiply it by 100. The proportion therefore refers to a component per hundred. Per 100 is what the word percent means. The letter "%" stands for it.
To find out what percentage of your allowance you spent,
Divide the amount you spent by your total allowance,
then multiply by 100 to get the percentage.
Here are the steps:
1. Identify the amount you spent: 0.88
2. Identify your total allowance: 1.00 (assuming your allowance is $1.00) 3. Divide the amount spent by the total allowance,
⇒ 0.88 ÷ 1.00 = 0.88 4.
Multiply the result by 100 to get the percentage,
⇒ 0.88 x 100 = 88%
Therefore, we spent 88% of your allowance this week.
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Question 1 (1 point)
m2IJK-570 and mZIJL-20°. Find mZLJK.
3
→K
O a MZLJK-77°
Ob mZLJK-37°
Oc mZLJK-40°
Od mZLJK=-37°
Answer:
37°
Step-by-step explanation:
Subtract angle IJL from angle IJK.
A car dealership needs a program to store information about the cars for sale. For each car, they want to keep track of the following information: number of doors (2 or 4), whether the car has air conditioning, and its average number of miles per gallon. Which of the following is the best design
The best object-oriented program design for this car dealership's software program is to use a single class Car, with three instance variables (fields):
numDoorshasAirmilesPerGallon.What is a class?A class can be defined as a user-defined blueprint (prototype) or template that is typically used by software programmers to create objects and define the data types, categories, and methods that should be associated with these objects.
In object-oriented programming (OOP) language, a class that is written or created to implement an interface in a software would most likely implement all of that interface's method declarations without any coding error.
In this scenario, the best object-oriented program design for this car dealership's software program is to use a single class Car, with three instance variables (fields):
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Complete Question:
A car dealership needs a program to store information about the cars for sale. For each car, they want to keep track of the following information: number of doors (2 or 4), whether the car has air conditioning, and its average number of miles per gallon. Which of the following is the best design?
(A) Use four unrelated classes: Car, Doors, AirConditioning, and MilesPerGallon.
(B) Use a class Car with three subclasses: Doors, AirConditioning, and MilesPerGallon.
(C) Use a class Car, with fields: numDoors, hasAir, and milesPerGallon.
(D) Use a class Car, with subclasses of Doors, AirConditioning, and MilesPerGallon.
(E) Use classes: Doors, AirConditioning, and MilesPerGallon, each with a subclass Car.
The 90% confidence interval for the mean one week amusing time in new york city is 5.22 less than mean less than 5.98 minutes construct a 95% confidence interval based on the same data which interval provides more information
The 95% confidence interval based on the same data is 5.15 < μ < 6.05
Because it has a longer interval, the 95 percent confidence interval offers more details.
What is a confidence interval?The likelihood that a parameter will fall between two values near the mean is shown by a confidence interval.
Confidence intervals quantify how certain or uncertain a sampling technique is.
Confidence intervals are used by statisticians to gauge the degree of uncertainty in a sample variable.
The 90% confidence interval is given by:
Mean - 1.645(S.D./[tex]\sqrt{n}[/tex]) < μ < Mean + 1.645(S.D./[tex]\sqrt{n}[/tex])
2μ = 5.22 + 5.98 = 11.2
μ = 11.2 / 2 = 5.6
Thus,
5.6 - 5.22 = 1.645(S.D./[tex]\sqrt{n}[/tex]) = 0.38
S.D./[tex]\sqrt{n}[/tex] = 0.38 / 1.645 = 0.231
The 95% confidence interval is given by:
Mean - 1.96(S.D./[tex]\sqrt{n}[/tex]) < μ < Mean + 1.96(S.D./[tex]\sqrt{n}[/tex])
=5.6 - 1.96(0.231) < μ < 5.6 + 1.96(0.231)
= 5.6 - 0.4528 < μ < 5.6 + 0.4528
= 5.15 < μ < 6.05
The 95% confidence interval provides more information because it has a larger interval.
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Cones and prisms both have only one base.
A.true
B.False
Answer:
False
Step-by-step explanation:
Prisms can have more than one base.
The expression $\frac{4k 8}{4}$ simplifies to an expression of the form $ak b$ where $a$ and $b$ are integers. Find $\frac{a}{b}$ .
The value of integers a and b is 1 and 2 and the value of a/b is 1/2.
According to the statement
we have a given a expression 4k+8 / 4 and we have to simplify this expression in the form of ak+b and find the value of a/b where a and b are the integers.
An integer is a whole number (not a fractional number) that can be positive, negative, or zero.
Here 4k+8/4 becomes
4k/4 +8/4 -(1)
and general form is
ak+b -(2)
Now, compare these equations then we get
a = 1 and b = 2.
Now we find the value of a/b
then
a/b = 1/2.
So, The value of integers a and b is 1 and 2 and the value of a/b is 1/2.
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Determine where each piece below belongs to create a rational expression equivalent to the one shown above.
An example of a rational expression is 2x+3/x-2.
What is a rational expression?The complete question wasn't found. Therefore, a overview will be given. It should be noted that a rational expression simply means quotient of two polynomials.
A rational expression is a fraction whose numerator and denominator are polynomials.
In this case, an example of a rational expression is 2x+3/x-2
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five more than the product of two and a number,
decreased by seven;
evaluate when a = 8
Key Words
five
more than
product
two
a number
decreased by
seven
Intro
Replace With
5
+
X
2
a
7
Write and evaluate the expression. Then, complete the
statements.
First, write the expression
Second,
Third,
The answer is
V
8 in for the variable, a
by using
Done
P
Answer:
It was difficult reading the question, as written. Check my assumptions.
Step-by-step explanation:
"five more than the product of two and a number,
decreased by seven"
---
Let the "number" be a.
We are told "five more than . ." means +5,
"the product of two and a number" means 2a,
"decreased by seven" mean -7
Put the statements together to obtain:
+5+2a-7
Simplify to 2a-2
Evaluate when a = 8:
2a-2
2(8)-2
16-2
= 14
Use the factor theorem to determine 2x^3+3x^2-5x-6 of the polynomials is divisible by(x+1)
The quotient of the polynomials [tex]2x^{3} +3x^{2} -5x-6[/tex] divided by (x+1) is [tex]2x^{2} -x-6[/tex].
Given two polynomials [tex]2x^{3}+3x^{2} -5x-6[/tex] and (x+1).
We have to determine whether [tex]2x^{3}+3x^{2} -5x-6[/tex] is divisible by (x+1).
Factors are the numbers which are multiplied to get the number whose factors are given. The factor theorem is a theorem which links factors and zeroes of a polynomial.
Polynomial is an expression of more than two algebraic terms may be in addition or subtraction.
We have to first find the factors of polynomial [tex]2x^{3}+3x^{2} -5x-6[/tex].
[tex]2x^{3}+3x^{2} -5x-6[/tex]=[tex]2x^{3}+3x^{2} -4x-x-6[/tex]
=[tex]x^{2}[/tex](2x+3)-2(2x+3)-x
=[tex](x^{2} -x-2)[/tex](2x+3)
=([tex]x^{2}[/tex]-x-2)(2x+3)
=[[tex]x^{2}[/tex]-2x+x-2](2x+3)
=[x(x-2)+1(x-2)](2x+3)
=(x+1)(x-2)(2x+3)
Because (x+1) is a factor of [tex]2x^{3}+3x^{2} -4x-x-6[/tex],so it is divisible by (x+1).
Hence using factor theorem we can say that [tex]2x^{3}+3x^{2} -4x-x-6[/tex]is divisible by (x+1).
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Richard went to a zipline park for his birthday. richard travels on the zipline at a constant rate.
the points on the following coordinate plane show how many feet he travelled in 2, 3, and 4 seconds while riding one of the ziplines.
how far does richard travel on the zip line in 1 second?
Richard Travel 40 feet on the zip line in 1 sec
The diagram is missing and hence the diagram.is shown below
Distance is the total movement of an object without any regard to direction.
A coordinate plane is a two-dimensional plane formed by the intersection of two number lines. One of these number lines is a horizontal number line called the x-axis and the other number line is a vertical number line called the y-axis
According to given Diagram
120 - 80 = 40 feet
On the Zip line
80-40=40
So the richard travels 40 feet on the zip line in 1 sec
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Pentagon ABCDE and pentagon A″B″C″D″E″ are shown on the coordinate plane below:
Which two transformations are applied to pentagon ABCDE to create A″B″C″D″E″?
The transformations that are applied to pentagon ABCDE to create A"B"C"D"E" are:
1) Translation (x, y) → (x + 8, y + 2)
2) Reflection across the x-axis (x, y) → (x, -y)
So, the overall transformation given in the graph is (x, y) → {(x + 8), -(y + 2)}.
What are the transformation rules?The transformation rules are:
Reflection across x-axis: (x, y) → (x, -y)Reflection across y-axis: (x, y) → (-x, y)Translation: (x, y) → (x + a, y + b)Dilation: (x, y) → (kx, ky)Calculation:The pentagons in the graph have vertices as
For the pentagon ABCDE: A(-4, 5), B(-6, 4), C(-5, 1), D(-2, 2), and (-2, 4)
For the pentagon A"B"C"D"E": A"(4, -7), B"(2, -6), C"(3, -3), D"(6, -4), and E"(6, -6)
Consider the vertices A(-4, 5) from the pentagon ABCDE and A"(4, -7) from the pentagon A"B"C"D"E".
Applying the Translation rule for the pentagon ABCDE:
The rule is (x, y) → (x + a, y + b)
So, the variation is
-4 + a = 4
⇒ a = 4 + 4 = 8
5 + b = 7
⇒ b = 7 - 5 = 2
So, the pentagon ABCDE is translated by (x + 8, y + 2).
Applying the Reflection rule for the translated pentagon:
The translated pentagon has vertices (x + 8, y + 2).
When applying the reflection across the x-axis,
(x + 8, y + 2) → {(x + 8), -(y + 2)}
Therefore, the complete transformation of the pentagon ABCDE to the pentagon A"B"C"D"E" is (x, y) → {(x + 8), -(y + 2)}
Verification:
A(-4, 5) → ((-4 + 8), -(5 + 2)) = (4, -7)A"
B(-6, 4) → ((-6 + 8), -(4 + 2)) = (2, -6)B"
C(-5, 1) → ((-5 + 8), -(1 + 2)) = (3, -3)C"
D(-2, 2) → ((-2 + 8), -(2 + 2)) = (6, -4)D"
E(-2, 4) → ((-2 + 8), -(4 + 2)) = (6, -6)E"
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