The true statements are:
The graph of the function is a parabola.
The graph contains the point (0, 0).
The graph does not contain the point (20, -8).
How to deal with quadratic equation?The quadratic function is f(x) = x^2 - 5x + 12. Here are the statements that are true:
The value of f(-10) = 82:
To find f(-10), we substitute -10 for x in the function:
[tex]$$f(-10) = (-10)^2 - 5(-10) + 12 = 100 + 50 + 12 = 162$$[/tex]
Therefore, the statement "The value of f(-10) = 82" is false.
The graph of the function is a parabola:
Since the highest power of x in the function is 2, the graph of the function will be a parabola. Therefore, the statement "The graph of the function is a parabola" is true.
The graph of the function opens down:
The coefficient of [tex]$x^2$[/tex] in the function is positive (+1), which means the parabola opens upwards. Therefore, the statement "The graph of the function opens down" is false.
The graph contains the point (20, –8):
To see whether the point (20, -8) is on the graph of the function, we substitute x=20 into the function:
[tex]$$f(20) = (20)^2 - 5(20) + 12 = 400 - 100 + 12 = 312$$[/tex]
Since the y-coordinate of the point (20, -8) is not equal to 312, the statement "The graph contains the point (20, –8)" is false.
The graph contains the point (0, 0):
To see whether the point (0, 0) is on the graph of the function, we substitute x=0 into the function:
[tex]$$f(0) = (0)^2 - 5(0) + 12 = 12$$[/tex]
Since the y-coordinate of the point (0, 0) is equal to 12, the statement "The graph contains the point (0, 0)" is true.
Therefore, the true statements are:
The graph of the function is a parabola.
The graph contains the point (0, 0).
The graph does not contain the point (20, -8).
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What is the slope of the line in the following graph?
Answer:
A. Proofs attached to answer
Step-by-step explanation:
Proofs attached to answer
algebraically determine the values of x that satisfy the system of equations below
The solutions to the system of equations are (x,y) = (0,1) and (5/2,3)
What is Algebraic Equations?
An algebraic equation is also known as a polynomial equation because both sides of the equal sign contain polynomials. An algebraic equation is built up of variables, coefficients, constants as well as algebraic operations such as addition, subtraction, multiplication, division, exponentiation, etc.
To solve the system of equations algebraically, we can set the two expressions for y equal to each other and solve for x:
[tex]-2x + 1 = -2x^2 + 3x + 1[/tex]
Simplifying, we get:
[tex]-2x^2 + 3x + 1 + 2x - 1 = 0[/tex]
[tex]-2x^2 + 5x = 0[/tex]
Factorizing x, we get:
x(-2x + 5) = 0
Therefore, the solutions for x are:
x = 0 or x = 5/2
To find the corresponding values of y, we can substitute these values of x into either of the original equations for y:
For x = 0:
y = -2(0) + 1 = 1
For x = 5/2:
[tex]y = -2(5/2)^2 + 3(5/2) + 1 = -13/2 + 15/2 + 1 = 3[/tex]
Therefore, the solutions to the system of equations are:
(x,y) = (0,1) and (5/2,3)
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A worker earning an hourly wage changes positions within a company. The new position comes with a 20% raise in hourly wage. The
worker also receives a $2 increase in hourly wage. Use composition of functions to write a function that represents the worker's new
hourly wage when the 20% raise is applied before the $2 raise.
f(x) =
This function gives the same result as the previous function we derived, but it expresses the calculation as a composition of the two separate functions that represent each raise separately.
What is function?A function is a set of instructions or statements that perform a specific task or computation and returns a value or result. In programming, a function is typically defined using a specific syntax and can be called or invoked by other parts of the program to carry out a particular operation.
Functions are used in programming to break down complex tasks into smaller, more manageable pieces of code, which can then be reused in multiple parts of the program. They can also help to improve the organization, readability, and maintainability of code.
by the question.
let the worker's original hourly wage be represented by the variable x. Then the function for the worker's new hourly wage when the 20% raise is applied before the $2 raise can be expressed as follows:
[tex]f(x) = (1.20x) + 2[/tex]
In this function, the expression (1.20x) represents the worker's hourly wage after the 20% raise is applied, and the additional $2 raise is added to this amount to determine the final hourly wage.
We can also express this function using function composition notation. Let g(x) represent the function that applies the 20% raise to the worker's hourly wage and let h(x) represent the function that adds the $2 raise to the resulting hourly wage. Then we have:
[tex]g(x) = 1.20x[/tex]
[tex]h(x) = x + 2[/tex]
Using function composition, we can write the function for the worker's new hourly wage as:
[tex]f(x) = h(g(x)) = h(1.20x) = 1.20x + 2[/tex]
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Can someone please help with these 4
Answer:
Step-by-step explanation:
1) b (acute is less than 90)
2) a (obtuse: more than 90, less than 180)
3) c
4) c
Answer:
1. NOM, JOK, KOL
2. MOL, NOK, MOJ
3. NOJ, JOL
4. NOL, MOK
find the value of the indicated ratio
Check the picture below.
[tex]\cot(\gamma )=\cfrac{\stackrel{adjacent}{24}}{\underset{opposite}{18}}\implies \cot(\gamma )= \cfrac{4}{3}[/tex]
12. Nora and Sara both worked to make some money and earned a total of $95. If Sara earned $15
more than Nora, how much did they both earn? Step by Step Please!!
Answers:
Nora = $40
Sara = $55
=========================================================
Work Shown:
n = amount Nora earned
s = amount Sara earned
n+s = 95 .... total amount earned
s = n+15 .... since Sara earned $15 more than Nora
--------------
I'll use the substitution method to solve.
n+s = 95
n+n+15 = 95 ....... replaced "s" with n+15
2n+15 = 95
2n = 95-15
2n = 80
n = 80/2
n = 40 dollars is the amount Nora earned
s = n+15
s = 40+15
s = 55 dollars is the amount Sara earned
--------------
Check:
Nora+Sara = 40+55 = 95 dollars total earned
This confirms the answer. Also, we can see that Sara earned $15 more than Nora to help further confirm the answer.
Write each expression without the absolute value symbol |14-(x-6)|
To start, we need to recognize that absolute value is a way of expressing the magnitude of a number, regardless of its sign. Thus, to get rid of the absolute value, we need to consider the two cases in which the expression can have either a positive or a negative sign.
For the case of a positive sign, we can simply write the expression as: 14-(x-6)
For the case of a negative sign, we can multiply both sides of the equation by -1 to get: -14 + (x-6)
Overall, we can express the expression without the absolute value symbol as:
14-(x-6) or -14 + (x-6)
Suppose some computations were done on a calculator.
The result displayed was 7.97E33 for one computation.
The result displayed was 7.6E-37 for another computation.
Write these numbers in scientific notation.
(a) 7.97E33 =
(b) 7.6E-37 =
Answer:
(a) 7.97E33 = 7.97 x 10^33
(b) 7.6E-37 = 7.6 x 10^-37
Step-by-step explanation:
In scientific notation, a number is written as the product of a decimal number between 1 and 10 and a power of 10. The decimal number is called the mantissa, and the power of 10 is called the exponent. The mantissa indicates the significant digits of the number, while the exponent tells us the size of the number relative to 1. In the given calculator computations, the numbers are already written in scientific notation, where "E" represents the exponent notation. We just need to convert them into the standard form of scientific notation by expanding the notation as a product of the mantissa and the power of 10.
100 POINTS + BRAINLIEST PLS BE FAST!!
i) Find the mean, median, and mode of the frequency table as follows:
Mean = 6.6Median = 8Mode = 3.ii) The average that justifies the teacher's statement congratulating the class that 'over three quarters were above average' is the average mark of 10, which is 5.
What are the mean, median, and mode?The mean refers to the average or the quotient of the total values divided by the number of items.
The median is the middle value in the data, which occurs with marks 8 for the 13th and 14th students.
The mode is the value that occurs most frequently, which is 3 which occurs 6 times.
Frequency Table:
Mark Frequency Cumulative Frequency
3 6 18 (0 + 3 x 6)
4 3 30 (18 + 4 x 3)
5 1 35 (30 + 5 x 1)
6 2 47 (35 + 6 x 2)
7 0 47 (47 + 7 x 0)
8 5 87 (47 + 8 x 5)
9 5 132 (87 + 9 x 5)
10 4 172 (132 + 10 x 4)
Mean = 6.6 (172/26)
Median = 8
Mode = 3
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Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. Find the probability of obtaining a reading greater than -1.12°C.
The probability of getting a measurement above -1.12°C is 0.8686, or almost 87%.
The probability formula is what?Typically, the probability is defined as the ratio of favorable events to all other outcomes in the sample space.Probability of an event P(E) = (Number of favorable outcomes) is how it is stated (Sample space).
We must determine the area under the normal distribution curve to the right of -1.12°C in order to determine the likelihood of getting a reading higher than that value.
We may convert the value of -1.12°C to a z-score, which represents the amount of standard deviations from the mean, using a calculator or a typical normal distribution table:
z = (x - μ) / σ
z = (-1.12 - 0) / 1.00
z = -1.12
The area under the normal distribution curve to the right of a particular z-score can also be found using a calculator. For instance, we can type the following into a calculator that has a normal distribution function:
P(Z > -1.12)
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I will mark you brainiest!
Alternate interior angles are congruent.
A) False
B) True
Answer:
True.
Step-by-step explanation:
Alternate interior angles are congruent, meaning they have equal measure. When we have two parallel lines that are intersected by a transversal, and again my parallel lines are identified by using the same number of arrows, then two special angles are congruent and that is alternate interior angles.
34 M
The point (-1, 3) is the turning point of the graph with equation y = x² + ax + b₁
where a and b are integers.
Find the values of a and b.
In the equatiοn , the value οf a and b is 2 and 4.
What is equatiοn?A mathematical assertiοn that shοws the equality οf twο mathematical expressiοns is what an equatiοn in algebra is defined as. Fοr example, the equatiοn 3x + 5 = 14 is cοmpοsed οf the twο equatiοns 3x + 5 and 14, which are separated by the equal sign.
The equatiοn οf a parabοla in standard fοrm is
y = a(x - h)² + k
where (h, k) are the cοοrdinates οf the vertex and a is the cοefficient οf the x² term.
Here (h, k ) = (- 1, 3 ) and cοefficient οf x² term = 1 , then
=> y = (x - (- 1))² + 3 ,
=> y = (x + 1)² + 3
=> y = x² + 2x + 1 + 3
=> y = x² + 2x + 4
Nοw cοmpare tο y = x² + ax + b
Consequently, a = 2 and b = 4.
Thus, a and b have values of 2 and 4, respectively.
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A baseball team plays in a stadium that holds 60000 spectators. With the ticket price at $9 the average attendance has been 23000. When the price dropped to $7, the average attendance rose to 30000. Assume that attendance is linearly related to ticket price. What ticket price would maximize revenue?
Answer:
Step-by-step explanation:
We can start by assuming that the relationship between the ticket price and attendance is linear, so we can write the equation for the line that connects the two data points we have:
Point 1: (9, 23000)
Point 2: (7, 30000)
The slope of the line can be calculated as:
slope = (y2 - y1) / (x2 - x1)
slope = (30000 - 23000) / (7 - 9)
slope = 3500
So the equation for the line is:
y - y1 = m(x - x1)
y - 23000 = 3500(x - 9)
y = 3500x - 28700
Now we can use this equation to find the attendance for any ticket price. To maximize revenue, we need to find the ticket price that generates the highest revenue. Revenue is simply the product of attendance and ticket price:
R = P*A
R = P(3500P - 28700)
R = 3500P^2 - 28700P
To find the ticket price that maximizes revenue, we need to take the derivative of the revenue equation and set it equal to zero:
dR/dP = 7000P - 28700 = 0
7000P = 28700
P = 4.10
So the ticket price that would maximize revenue is $4.10. However, we need to make sure that this price is within a reasonable range, so we should check that the attendance at this price is between 23,000 and 30,000:
A = 3500(4.10) - 28700
A = 5730
Since 23,000 < 5,730 < 30,000, we can conclude that the ticket price that would maximize revenue is $4.10.
Julian Nestor is ready to purchase a new stroller. It is regularly priced at $127.99. The sale price is $88.54. What is the markdown?
Answer:
$39.45, about 31%
Step-by-step explanation:
You want to know the markdown represented by a sale price of $88.54 on a regular price of $127.99.
MarkdownThe dollar amount of the markdown is ...
88.54 -127.99 = -39.45
The price was marked down $39.45.
The percentage markdown from the original price is ...
-39.45/127.99 × 100% ≈ -30.823% ≈ -31%
The original price was marked down about 31% to get the sale price.
__
Additional comment
The negative price change means the price was marked down. If the change were positive, it would signify a markup.
Pls solve #5 will give brainliest for right answer
Answer: 200 texts
Step-by-step explanation:
If they paid the same amount, we can set the equations equal to each other and solve for x using inverse operations.
Given:
0.1x + 10 = 0.15x
Subtract 10 and 0.15x from both sides of the equation:
-0.05x = -10
Divide both sides of the equation by -0.05:
x = 200
Correct answer gets brainliest
Answer:
48 square feet
Step-by-step explanation:
Area of triangle = (1/2) · b · h
(1/2) · 3 · 4 = 6 square feet
There are 2 triangles, so we time 2
6 x 2 = 12 square feet
Area of rectangle = L × W
9 x 4 = 36 square feet
Now we take
12 + 36 = 48 square feet
Jonathan and Amber went to the store together to buy school supplies.
Jonathan purchased 2 notebooks and 5 elastic book covers for $6.75. Amber
purchased 4 notebooks and 2 elastic book covers for $7.50. What is the price
of a single notebook?
P
The price of a single notebook is $
Answer:
The answer is 1.5$
Step-by-step explanation:
Let the price of 1 notebook be x$ and 1 elastic book cover be y$
In first case,
2x+5y=$6.75
2x = $6.75-5y
x=($6.75-5y)/2------------- eqn i
In second case,
4x+2y=$7.50
4×($6.75-5y)/2 +2y=$7.50 [From eqn i]
($27-20y)/2 +2y=$7.50
($27-20y+4y)/2=$7.50
($27-16y)/2=$7.50
$27-16y=$7.50×2
$27-16y=$15
$27-$15=16y
$12=16y
y=$12/16
y=$0.75
The price of single elastic book cover is $0.75
Substituting the value of y in eqn i we get
x=($6.75-5y)/2
x=($6.75-5×$0.75)/2
x=($6.75-$3.75)/2
x=$3/2
x=$1.5
Hence, the price of single notebook is $1.5 Ans
Hey mate, please mark me as brainliest if you got the answer.
HELPPPPPP MEEEEE!!!!!!!
Question 1
What is the proper breakdown of 12x^2? (Make a prime factor tree for 12)
A. 12*x*x
B. 2*6*x*x
C. 2*2*3*x*x
D. 3*4*x
Question 2
What is the proper breakdown of 8x^3 - 12x^2 + 16x is the following:
2*2*2*x*x*x
2*2*3*x*x
2*2*2*2*x
What is the greasy common factor?
A. 2x
B. 4x
C. 4
D. 6
Question 3.
Rewrite 8x^3 -12x^2 + 16x, pulling the greatest common factor (gcf) out. (What is left?)
A. 4x(2x^2 - 3x + 4)
B. 4x (2x^2 + 3x + 4)
C. 4x (2^3 - 3x^2 + 4x)
D. 2x (4x^2 - 6x + 8)
Question 4.
What is the greatest common factor of 6x^2 + 16x + 10?
A. X
B. 2x
C. 6
D. 2
Question 5.
Factor: x^2 - 4x - 32
Remember what two numbers multiply to get -32 and add to get -4?
A. (X+8)(x-4)
B. (X+4)(x-8)
C. (X+8)(x+4)
D. (X+16)(x+2)
1. To break down 12x² into its prime factors,Start dividing 12 by 2, which gives you 6. Then, break 6 by 2, which gives 3. To write 12x² in terms of its prime factors, you simply need to add the x² term, giving you 2 * 2 * 3 * x * x or option C.
What is common factor?A common factor is a number or expression that divides two or more other numbers or expressions without leaving a remainder.
Since 3 is a prime number So the prime factorization of 12 is 2 * 2 * 3.
2. To find the greatest common factor of 8x³ - 12x² + 16x, you can start by factoring out any common factors that the terms share. In this case, all of the terms have a factor of 4x, so you can factor that out to get 4x(2x² - 3x + 4). Therefore, the greatest common factor is 4x or option B.
3. To rewrite 8x³ - 12x² + 16x, pulling out the greatest common factor, you can use the same method as in question 2. The greatest common factor of the terms is 4x, so you can factor that out to get 4x(2x² - 3x + 4) or option A.
4. To find the greatest common factor of 6x²+ 16x + 10, you can factor the expression using the quadratic formula or by factoring the expression as (3x + 5)(2x + 2). Since there is no other common factor between the terms, the greatest common factor is 1 or option A.
5. To factor x² - 4x - 32, you need to find two numbers that multiply to -32 and add to -4. These numbers are -8 and 4. Therefore, x² - 4x - 32 can be factored as (x - 8)(x + 4) or option C.
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Help please! I have no idea!!!!
The values of the function from the graph are h(7) = 10, h(0) = 9, h(t) = 8, t = 5 and h(t) = 0, t = 4
How to determine the value of the functionGiven that the graph is the graph of height as a function of time
To calculate the values of h(t) from the graph of g(x), we need to follow these steps
Identify the value of t on the x-axis where you want to calculate the value of h(t)Locate that point on the graphWrite the values from the pointUsing the above, we have the following
h(7) = 10
h(0) = 9
h(t) = 8, t = 5
h(t) = 0, t = 4
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Use the slope and y-intercept to identify the equation of this line.
The equation of the line is y = -2x. The correct option is the last option y = -2x
Writing the equation of the line in the given graphFrom the question, we are to write the equation of the line in the given graph using the slope and y-intercept from the graph
First, we will determine the slope of the graph
The slope of the graph calculated from the formula
Slope = Change in y / Change in x
Slope = (y₂ - y₁) / (x₂ - x₁)
Picking the points (-1, 2) and (0, 0)
Slope = (0 - 2) / (0 - (-1))
Slope = -2/(0 + 1)
Slope = -2/1
Thus,
Slope = -2
From the graph, the y-intercept of the graph is 0
Then,
From the slope-intercept form of the equation of a line,
y = mx + c
Where m is the slope
and c is the y-intercepts
The equation of the line is
y = -2x + 0
y = -2x
Hence, the equation is y = -2x
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Calculate the frequency in hertz of electromagnetic radiation that has a wavelength of 585.0 nm. (c = 3.00 X 10⁸ m/s)
The frequency in hertz of electromagnetic radiation which has a wavelength of 585nm is 5.13×10¹⁴ Hz.
The frequency(f) of electromagnetic radiation is related to its wavelength (λ) by the equation : f = c/λ;
Where, c = the speed of light in a vacuum, which is 3.00×10⁸ m/s.
So, to calculate the frequency of radiation with a wavelength of 585.0 nm, we need to convert the wavelength to meters:
⇒ 585.0 nm = 585.0×10⁻⁹ m;
Substituting the value in the above formula,
We get,
⇒ f = c/λ = 3.00×10⁸/(585.0×10⁻⁹)
⇒ 5.13×10¹⁴ Hz.
Therefore, the frequency of electromagnetic radiation is approximately 5.13 × 10¹⁴ Hz.
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What is the amount of interest to be paid if u got a reducing balance loan of $2700 and you will repay it over 5 years by quarterly installments of $1899.75 at interest rate 14% p.a. (compounded quarterly)
The total interest to be paid over the 5-year period is $403.22.
What is reducing balance loan?In a loan with a declining balance, interest is added to the outstanding balance at the start of each period, such as each month or each quarter. The outstanding balance, which decreases as the borrower makes repayments, is used to determine the interest rate. As a result, the borrower gradually pays less interest and more of each payment is used to lowering the main balance due. The lowering balance approach is frequently applied to mortgages, personal loans, and auto loans.
Given that, the quarterly installments are $1899.75.
For the entire year for a total of 5 years the total amount is:
Total amount = 4 * 5 * $1899.75 = $37995
For reducing balance loan the interest is calculated as:
For the first quarter, the interest to be paid is:
interest = (total amount - 0) * (0.14 / 4)
interest = (37995 - 0) * (0.14 / 4) = $94.50
Similarly for the remaining quarters we have:
Second quarter = $87.95
Third quarter: $81.05
Fourth quarter: $73.76
Fifth quarter: $65.96
The total interest to be paid over the 5-year period is:
$94.50 + $87.95 + $81.05 + $73.76 + $65.96 = $403.22
Hence, the total interest to be paid over the 5-year period is $403.22.
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Find the value of Tan 15 without using a table or calcutator
Answer:
Step-by-step explanation:
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Prove the following using a direct proof:
The sum of the squares of 4 consecutive integers is an even integer
Answer: A positive whole number multiplied by any whole number will remain positive. In the case of the squares of 4, it will always end in a 6 which is a positive number.
Step-by-step explanation:
4^2= 16
16^2 = 256
256^2= 65,536
etc.
3 minutes 4 seconds − 1 minute 16 seconds = minutes seconds
the result of the subtraction is 2 minutes 48 seconds
Convert minutes into seconds
To convert minutes into seconds, you can multiply the number of minutes by 60, since there are 60 seconds in a minute. For example, if you want to convert 1 minutes into seconds, you would do:
1 minutes ×60 seconds/minute = 60 seconds
Therefore, 5 minutes is equal to 300 seconds.
Solving the problem:3 minutes 4 seconds - 1 minute 16 seconds
We can start by subtracting the seconds:
3 minutes - 1 minute = 2 minutes
And add the borrowed 60 seconds to the seconds place:
-12 seconds + 60 seconds = 48 seconds
Therefore, the result of the subtraction is 2 minutes 48 seconds
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My cookbook's pancake recipe states that it makes 12 standard sized pancakes. The nutritional information says 2 pancakes is a serving containing 150 calories. For breakfast, I prepared half a recipe, but made smaller sized pancakes, so ended up preparing 8 pancakes. I ate 4 of them. How many calories did I consume?
Answer:
225 calories
Step-by-step explanation:
1 recipe makes
12 standard sized pancakes
2 pancakes are 1/6 of the recipe
1/6 of the recipe has 150 calories
6 × 150 calories = 900 calories
The full recipe of 12 pancakes has 900 calories
1/2 recipe was made
1/2 recipe has 1/2 × 900 calories = 450 calories
1/2 recipe made 8 pancakes
4 pancakes are half of the half recipe or 1/4 recipe
1/4 × 900 calories = 225 calories
Determine whether the function is linear. If it is, identify the rate of change. X -7, -5, -3, -1, 0 Y 11, 14, 17, 20, 23
Answer: Yes;
Step-by-step explanation: The function is linear in that the X values are increasing by 2 and the Y values are increasing by 3.
Answer:
Not linear
Step-by-step explanation:
X | -7 | -5 | -3 | -1 | 0
Y | 11 | 14 | 17 | 20 | 23
-5 - (-7) = +2
14 - 11 = +3
-3 - (-5) = +2
17 - 14 = +3
-1 - (-3) = +2
20 - 17 = +3
0 - (-1) = +1
23 - 20 = +3
There is a linear function graphed that would pass through the first 4 points, but not the last one;
You can discern this as there is a common pattern we can identify, everytime the x-value increases by 2 the y-value increases by 3, except with the last coordinates;
The rate of change, also known as the gradient, is the change in x divided by the change in y (Δx/Δy);
In the case of the first 4 points, the rate of change would be 3/2 or 1.5;
This simply means when x increases by 1, y increases by 1.5, IF the function is linear;
However, as mentioned the last point doesn't fit the pattern;
There, we can see the y-value increases by 3 when the x-value only increases by 1;
This means the point (0, 23) isnt on the graph of the function, the points (0, 21.5) and (1, 23), on the other hand, would be.
Find two vectors v1 and v2 whose sum is (0, -1, -3), where v1 is parallel to (5, 0, -4) while v2 i perpendicular to (5, 0, -4).
v1 = ?
v2 = ?
Two vectors v₁ and v₂ whose sum are (0, -1, -3), where v₁ is parallel to
(5, 0, -4) while v₂ is perpendicular to (5, 0, -4).
v1 = [40/13, -8/13 ]
v2 = [-1/13, -5/13]
Parallel vectors will be multiples of each other.
Perpendicular vectors will have a dot product of 0.
Let v₁ = [a, b] and v₂ = [c, d]
v₁ || [-5,1] means v₁ = N [-5,1] = [a, b]
So a = -5N and b = N
v₂ • [-5,1] = 0 (because they are perpendicular)
So [c, d] • [-5,1] = -5c + d = 0 or d = 5c
⇒ v₁ + v₂ = [a, b] + [c, d] = [3, -1]
So a + c = 3 and b + d = -1
Now,
Substitute -5N for
a. Substitute N for b.
and Substitute 5c for d.
This gives us a system of equations with two equations and 2 unknowns:
-5N + c = 3
N + 5c = -1
Multiply everything in the bottom equation by 5
-5N + c = 3
5N + 25c = -5
Now add vertically, eliminating the variable N and finding
26c = -2
or, c = -1/13
Now we work our way backwards.
-5N + c = -5N + (-1/13) = 3
-5N = 40/13
(-1/5) (-5N) = (40/13)(-1/5)
and
N = -8/13 and b = N so b = -8/13
d = 5c so d = -5/13
a = -5N, so a = 40/13
So the two vectors are:
v₁ = [40/13, -8/13 ]
v₂ = [-1/13, -5/13]
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Give a reason for your answer to the following question. Assume that all variables represent integers.
If n = 4k + 3, does 8 divide n2 − 1?
Yes, 8 divides n^2 - 1 when n = 4k + 3.
We are given that n = 4k + 3, where k is an integer. We want to determine if 8 divides n^2 - 1, which means we want to know if there exists an integer m such that n^2 - 1 = 8m.
To start, we can rewrite n^2 - 1 as (n + 1)(n - 1). This is a common trick in number theory problems involving divisibility, and it can often help us factorize expressions in a useful way.
Substituting n = 4k + 3, we get:
(n + 1)(n - 1) = (4k + 3 + 1)(4k + 3 - 1) = (4k + 4)(4k + 2)
Now, we want to determine whether 8 divides this expression. In other words, we want to know if there exists an integer m such that:
(4k + 4)(4k + 2) = 8m
We can simplify the left-hand side of this equation by factoring out 4
4(k + 1)(2k + 1) = 8m
Now, we see that 8 divides the left-hand side of the equation, since it is equal to 4 times an even number. Moreover, (k + 1) and (2k + 1) are both integers, so we can conclude that 8 divides n^2 - 1 when n = 4k + 3.
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Which of the following is true regarding cross-sectional data sets? Check all that apply. The data consist of a sample of multiple individuals. It can be assumed that the data were obtained through a random sampling of the underlying population. These data require attention to the frequency at which they are collected (weekly, monthly, yearly, etc.). The data are collected at approximately the same point in time.
The correct options are:
The data contain a sample of multiple individuals.
The data are collected are approximately at the same point in time.
Cross-sectional data sets are a type of research design used in statistical analysis. They consist of a sample of multiple individuals or entities observed at a single point in time. One of the characteristics of cross-sectional data sets is that they do not involve any observation or measurement over time, making them different from longitudinal or time-series data sets.
One advantage of cross-sectional data sets is that they are relatively easy and inexpensive to collect. It can be assumed that the data were obtained through a random sampling of the underlying population, making them representative of the larger population. However, these data require attention to the frequency at which they are collected (weekly, monthly, yearly, etc.), as the timing of data collection can impact the results.
Overall, cross-sectional data sets can provide a snapshot of a population or phenomenon at a specific point in time, making them useful for a wide range of research questions in social, economic, and political sciences.
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