What is the standard form of the equation of a quadratic function with roots of 4 and −1 that passes through (1, −9)?
y = 1.5x2 − 4.5x − 6
y = 1.5x2 − 4.5x + 6
y = −1.5x2 − 4.5x − 6
y = −1.5x2 − 4.5x + 6
Answers
Answer:
(a) y = 1.5x² -4.5x -6
Step-by-step explanation:
You want the standard form of the quadratic with roots 4 and -1, and passing through the point (1, -9).
CoefficientsFor roots p and q, the standard form can be found from the factored form:
(x -p)(x -q) = x² -(p+q)x +pq
For the given roots, the product pq will be ...
(4)(-1) = -4
The sum p+q is 4+(-1) = 3. The x-coefficient is the opposite of this.
The parent quadratic (before vertical scaling) will be ...
(x -4)(x +1) = x² -3x -4
The coefficients of the last two terms have the same sign. (Eliminates the 2nd and 4th answer choices.)
Leading coefficientThe given point has an x-value (1) between the given roots (-1, 4):
-1 < 1 < 4
The y-value at that point is negative. This means the vertex of the parabola will be below the x-axis, so the parabola opens upward and the leading coefficient is positive. (Eliminates the last two answer choices.)
The answer choices tell us the leading coefficient is 1.5, so the equation is ...
y = 1.5(x² -3x -4)
y = 1.5x² -4.5x -6 . . . . . . matches the first choice
__
Additional comment
We could find the value of the leading coefficient 'a' by evaluating ...
y = a(x -4)(x +1)
for x = 1. We would get ...
-9 = a(1 -4)(1 +1) = -6a ⇒ a = -9/-6 = 1.5
As we saw above, this isn't necessary. We only need to know that its sign is positive. The answer choices tell us the value.
If the x-value of the given point is not between the roots, then we know the sign of the y-value is the sign of the leading coefficient.
For multiple-choice questions, you only need to work enough of the problem to determine which answer choice is correct. You don't necessarily need to work the problem all the way to an answer.
Related Questions
Anthony purchased a $227 stereo system at Wrigley's Department Store. He paid $47 down and agreed to pay the balance in 12 equal monthly payments. The finance charge was 9.6% simple interest. What was the amount of each payment
Answers
Answer:
Calculation:
Total Amount: $227
Down Payment: $47
Interest Rate: 9.6% (0.096)
Number of Payments: 12
Remaining Balance: $227 - $47 = $180
Remaining Balance with Interest: $180 + (9.6/100 * $180) = $197.68
Amount of Payment: $197.68 / 12 = $16.47
Answer: The amount of each payment is $16.47.
Writing equations of lines
Answers
y=mx+b
where m is the slope and b is the y-intercept
To find m, calculate the change in y over the change in x. To do so, choose two points on the graph that land on coordinates of whole numbers. Lets use points (0,6) and (1,1).
the change in y-values = 1-6 = -5
the change in x-values = 1-0 = 1
This makes the slope -5/1 or just -5.
m = -5
Now let’s find the y-intercept, which is easy for us because we already used it as one of our slope points. This is when x=0, so our y-intercept is the point (0,6).
b = 6
Now piece this together:
y = mx+b
y = -5x + 6
That is the equation of this line!
8x÷4=100 What is the value of x in the equation?
Answers
Answer:
Solving 8x ÷ 4 = 100, we have:
2x = 100
x = 50
So the value of x is 50.
Answer:
x = 50
Step-by-step explanation:
To answer this question, we have to isolate the x. To do this, we have to get rid of any other numbers around it by doing the inverse of the operation.
1)
÷ 4 = ×4 to both sides8x ÷ 4 × 4 = 8x100 × 4 = 4008x = 4002)
8x = ÷ 8 to both sides8x ÷ 8 = x400 ÷ 8 = 50x = 50This means that x is 50!
To check our answer is correct we can substitute 50 to x in the question...
(8 × 50) ÷ 4 = 100It is correct!
Hope this helps, have a lovely day! :)
1. State the converse, contrapositive, and inverse of the conditional statement “A positive integer is a prime only if it has no divisors other than 1 and itself”.
Answers
Answer:
Step-by-step explanation:
A contrapositive statement changes "if not p then not q" to "if not q to then, not p." The converse of the conditional statement is “If Q then P.” The contrapositive of the conditional statement is “If not Q then not P.” The inverse of the conditional statement is “If not P then not Q.” if p → q, p → q, then, ∼ q →∼ p ∼ q →∼ p
A Note about Notation: Be careful with the notation a|b
. This does not represent the rational number ab
. The notation a|b
represents a relationship between the integers a
and b
and is simply a shorthand for “ a
divides b
.” "Divides" as in a|b
is a relation (true or false), while "divided by" as in ab
or a/b
is an operation (results in a number).
The definition for “divides” can be written in symbolic form using appropriate quantifiers as follows: A nonzero integer m
divides an integer n
provided that (∃q∈Z)(n=m⋅q).
Find the maximum value of = √ subject to the cost constraint K + 4L = 16.
Estimate the change in the optimal value of Q if the cost constraint is changes to K +
4L = 17
Answers
Answer:
This problem can be solved using the method of Lagrange multipliers. Let's define the Lagrangian function as: L(K, L, λ) = Q - λ(K + 4L - 16) Taking partial derivatives with respect to K, L, and λ, we get: dL/dK = 0 => 1 - λ = 0 => λ = 1 dL/dL = 0 => 1 - 4λ = 0 => λ = 1/4 dL/dλ = K + 4L - 16 = 0 Solving the last equation for K, we get: K = 16 - 4L Substituting this into the Lagrangian function, we get: L = Q - λ(16 - 4L + 4
Jacob drove from Nicktown to Binghamton at
an average speed of 70 mph. If it took him 4
hours to get there, how far is Binghamton from
Nicktown?
A. 17.5 miles
B. 210 miles
C. 280 miles
D. 300 miles
Answers
distance = average speed * time
In this case,
average speed = 70 mph
time = 4 hours
distance = 70 mph * 4 hours
distance = 280 miles
So, Binghamton is 280 miles from Nicktown (Option C).
Are conditional probabilities always dependent?
Answers
Answer:
Two events are said to be independent if one event occurring does not affect the probability that the other event will occur. However, if one event occurring or not does, in fact, affect the probability that the other event will occur, the two events are said to be dependent.
If one event occurs or not, it does, in fact, affect the probability that the other event will occur, and then the two events are said to be dependent.
Explain how to simplify this expression: (7a + 9) + (4a - 7).
Answers
The simplified expression would be 11a + 2.
Simplifying an expressionIn order to simplify the given expression, we first need to combine like terms.
The terms can be combined by adding their coefficients:
(7a + 4a) = 11a(9 - 7) = 2Now we can substitute these combined terms back into the original expression:
(7a + 9) + (4a - 7) = 7a + 4a + 9 - 7
And then simplify further:
= 11a + 2
Therefore, the simplified form of the expression (7a + 9) + (4a - 7) is 11a + 2.
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A school bus has 22 rows of seats, and 4 students can be seated in each row. Students riding in the bus have filled 19 rows of seats, and 1/2 of the remaing seats. How many seats on the bus are empty?
Answers
Thus, remaining seats on the school bus that are empty is found as: 6 seats.
Explain about the fraction of number:The fractional bar is a horizontal bar that divides the numerator and denominator of every fraction into these two halves.
The number of parts by which the whole has already been divided is shown by the denominator. It is positioned in the fraction's lower portion, just below fractional bar.How several sections of both the fraction are displayed or chosen is shown in the numerator. It is positioned well above fractional bar in the upper portion of the fraction.Given data:
Total rows of seats = 22Number of seats in each row = 4Filled rows = 191/2 of the remaining.Total seats in bus = Total rows of seats * Number of seats in each row
Total seats in bus = 22 * 4
Total seats in bus = 88
Filled seats = 19 * 4 = 76
Left seats = 88 - 76 = 12
Now, 1/2 of the 12 seats is filled = 6 seats.
remaining seats = 6
Thus, remaining seats on the school bus that are empty is found as: 6 seats.
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Peter observes that 75% of the birds at his bird feeder are finches. The results from 20-simulation trials of the model showed that finches are a
guaranteed bird type for 80% of the birds.
Based on the results from the simulation trials, what should Peter conclude about his hypothesis if he predicted that birds at his bird feeder woul
continue with the same percentage of finches as before?
A. Reject the hypothesis.
B. Conclude the alternate hypothesis was correct.
C. Conclude that 80% of the birds at the bird feeder should always be finches.
D. Conclude the hypothesis was correct.
Answers
Option A "Reject the hypothesis" is the appropriate conclusion in this case.
As per the question,
Based on the results from the simulation trials, Peter should conclude that his hypothesis should be rejected.
The simulation showed that finches are a guaranteed bird type for only 80% of the birds, which is lower than the observed percentage of 75% finches at Peter's bird feeder.
This indicates that the predicted percentage of finches continuing at the same rate is not supported by the simulation results.
Therefore, option A "Reject the hypothesis" is the appropriate conclusion in this case.
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Tom and John are engaged in buying and selling certain products A and B. Tom BUYS 5 of product A but
SELLS twice as much of product B. John on the other hand SELLS three times what Tom BOUGHT of
product A and BUYS 13 of product B. At the end of the business day, John banks Ksh 110,000/- while
Tom banks Ksh 230,000.
Under the assumption that the sale prices for product A and B are the same for the two men, and the costs prices for the products A and B are also the same for the two men, obtain the following:
The price for product A which was determined as Ksh 42,727.27 and the price for product B as 44,363.63
Question: If there was a mark up of 25% on the cost price and a discount of 15% on the sale price, how
much would each of the partners have banked at the end of the business day?
Answers
Product A is priced at Ksh 55,000, while Product B is priced at Ksh 32,000.
How to calculate Product A and B?Assume that the selling price for both the two items A and B is "S" and the cost price for both is "C". Using this knowledge, we can create two equations that relate the costs and revenues of the two items, one for Tom and one for John:
Tom: 5C - 2(5S) = [tex]P_1[/tex]
John: 3(5C) - 13C = [tex]P_2[/tex]
where [tex]P_1[/tex] and [tex]P_2[/tex] represent Tom's and John's respective profits.
We are aware that John deposits Ksh. 110,000 while Tom deposits Ksh. 230,000 at the end of the business day. Thus, we can say:
[tex]P_1[/tex] = 230,000 - 5C
[tex]P_2[/tex] = 110,000 - 18C
These values are what we obtain when we enter them into the equations for Tom and John:
5C - 2(5S) = 230.000 - 5C
When we simplify these equations, we obtain:
10C - 10S = 230,000
2C = 110,000
Solving for C, we get:
C = 55,000
We may determine S by adding this value back into the original equation:
10(55,000) - 10S = 230,000
Simplifying this equation, we get:
S = 32,000
Therefore, the price for product A is Ksh 55,000 and the price for product B is Ksh 32,000.
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Complete the square to solve the quadratic below. 1=2x^2+7x
Answers
the solutions to the quadratic equation 1 = 2x^2 + 7x are:
[tex]x= \frac{(-7+\sqrt{51} )}{4}[/tex] and [tex]x= \frac{(-7-\sqrt{51})}{4}[/tex]
ToToToToToToToToToToToToToToToToToToToToToToToToToToToToToToToTo solve this quadratic equation by completing the square, we need to rewrite the equation in the form of (x + p)^2 + q = 0, where p and q are constants. This will allow us to solve for x.
First, we can factor out the coefficient of the x^2 term, which is 2:
[tex]2x^2 + 7x - 1 = 0[/tex]
Next, we can isolate the x^2 and x terms on one side of the equation by subtracting 1 from both sides:
[tex]2x^2 + 7x = 1[/tex]
Now, we need to add and subtract a constant term inside the parentheses to create a perfect square trinomial. To determine this constant term, we take half of the coefficient of the x term (which is 7/2) and square it:
[tex](\frac{7}{2} )^{2} = \frac{49}{4}[/tex]
So we add and subtract 49/4 inside the parentheses:
[tex]2x^2 + 7x + \frac{49}{4} - \frac{49}{4} = 1[/tex]
We can simplify the left side by factoring the perfect square trinomial:
[tex]2(x + \frac{7}{4} )^2 - \frac{51}{8} = 0[/tex]
Now, we can solve for x by isolating the perfect square term and taking the square root of both sides:
[tex]2(x + \frac{7}{4} )^2 = \frac{51}{8}[/tex]
[tex](x + \frac{7}{4} )^2 = \frac{51}{16}[/tex]
[tex]x + \frac{7}{4}=±\sqrt{\frac{51}{16} }[/tex]
[tex]x = -\frac{7}{4} ± \sqrt{\frac{51}{16} } /2[/tex]
Simplifying this expression, we get:
[tex]x = (-7 ±\sqrt{51} )/4[/tex]
Therefore, the solutions to the quadratic equation 1 = 2x^2 + 7x are:
[tex]x = (-7 + \sqrt{51} )/4 and x = (-7 -\sqrt{51} )/4.[/tex]
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- w/5 + 9 = 13 what w
Answers
Answer:
w = -20
"The beautiful thing about learning is that no one can take it away from you." :)
?/5+9=13
13-9=4
4x5=20
W=20
Which worksheet helps you figure out how much you can spend and invest?
Answers
Answer:
A monthly budget worksheet cause it shows your income and expenditure, so you are able to spend within your budget and it helps you know your disposable income
Janct was driving home from work at a speed of 40 mph. Realizing she forgot her documents as she arrived home, she then drove back to work at a speed of 60 mph. What is her average speed for the entire trip?
A. 48 mph
B.49 mph
C. 50 mph
D. 52 mph
Answers
Answer: 50 mph
Step-by-step explanation:
average speed = 2 * (speed1 * speed2) / (speed1 + speed2)
In this case,
speed1 = 40 mph (driving home)
speed2 = 60 mph (driving back to work)
average speed = 2 * (40 * 60) / (40 + 60)
average speed = 2 * (2400) / (100)
average speed = 4800 / 100
average speed = 48 mph
So, Janet's average speed for the entire trip is 48 mph (Option A).
(3, -3) and (8, 12)
Answers
Step-by-step explanation:
what did you ask the question? ask it clearly please?? if you ask clearly the question. we also give answer
HELP PLSSSSSSSSSSSSSSSS
Answers
Answer:
H
Step-by-step explanation:
% of 11 year olds is 24
% of 12 year olds is 16
then ratio of
11 year olds : 12 year olds
= 24 : 16 ( divide both parts by 8 )
= 3 : 2
simplify -2(2x - 5) -4x
Answers
Answer: -8x +10
Step-by-step explanation:
-2(2x -5) -4x
1. simplify parthentheses
-4x +10 -4x
add like terms
-8x +10
Ling bought d donuts to share with coworkers. There were 6 donuts in each box. Write an expression that shows how many boxes of donuts Ling bought.
Answers
the expression would be used as follows:
Number of boxes of donuts (N) = 6 x Number of donuts (D)
Ling bought 108 boxes of donuts.
This expression is a mathematical equation that uses multiplication to calculate the number of boxes of donuts Ling bought.
What is mathematical equation?Equations are used to solve problems involving unknowns, such as how much a person owes for groceries or how many people are required to complete a certain task.
The expression that would show how many boxes of donuts Ling bought is:
Number of boxes of donuts (N) = 6 x Number of donuts (D)
In this expression, N represents the number of boxes of donuts Ling bought and D represents the number of donuts. This expression can be used to calculate the number of boxes of donuts Ling bought when the number of donuts is known.
For example, if Ling bought 18 donuts, the expression would be used as follows:
Number of boxes of donuts (N) = 6 x Number of donuts (D)
N = 6 x 18
N = 108
Therefore, Ling bought 108 boxes of donuts.
This expression is a mathematical equation that uses multiplication to calculate the number of boxes of donuts Ling bought. The expression can be used whenever the number of donuts is known and can be used to calculate the number of boxes of donuts Ling bought.
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Given the following functions find and simplify (f-g)(1) f(x)=-x^2-x-3 g(x)=-1
Answers
To find (f-g)(1), we first need to find f(1) and g(1): f(1) = -(1)^2 - 1 - 3 = -5 g(1) = -1 Now we can substitute these values into (f-g)(x) = f(x) - g(x) and simplify: (f-g)(1) = f(1) - g(1) = (-5) - (-1) = -4 Therefore, (f-g)(1) = -4.
How do you solve this?
Answers
Below 10 in.*6 in.*14 in., the triangular prism's volume is 420 cubic inches.
The area of the base must be multiplied by the prism's height in order to determine the volume of a triangular prism.
The base of this triangular prism is a triangle with a base of 6 inches and a height of 10 inches, so its area is:
Area of base = (1/2) × base × height = (1/2) × 6 in × 10 in = 30 in²
The height of the triangular prism is given as 14 inches.
Hence, the triangular prism's volume is:
Volume = Area of base × height = 30 in² × 14 in = 420 cubic inches.
So, the volume of the triangular prism is 420 cubic inches.
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Sneeze: According to a study done by Nick Wilson of Otago University Wellington, the probability a
randomly selected individual will not cover his or her mouth when sneezing is 0.267. Suppose you sit on
a bench in a mall and observe 300 randomly selected individuals’ habits as they sneeze.
Task 1: What is and what is ?
Task 2: What are the mean and standard deviation?
Task 3: What are the conditions that must be met to use the normal approximation to the binomial
distribution? Are they met?
Answers
Thus, the mean and standard deviation for the randomly selected individuals is found as: mean = 80.1 and standard derivation = 7.66.
Explain about the standard deviation:The standard deviation is a measurement of how widely spaced out a set of data is from the mean. The bigger the dispersion or variability, the higher the standard deviation and the greater the magnitude of the value's divergence from its mean. It represents the absolute variability of a distribution.
Given data:
Number of population n = 300
probability p = 0.267
Mean is given as np:
np = 300*0.267
np = 80.1
standard derivation: √np(1 - p)
√np(1 - p) = √80.1(1 - 0.267)
√np(1 - p) = √58.7133
√np(1 - p) = 7.66
Thus, the mean and standard deviation for the randomly selected individuals is found as: mean = 80.1 and standard derivation = 7.66.
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Correct question:
According to a study done by Nick Wilson of Otago University Wellington, the probability a randomly selected individual will not cover his or her mouth when sneezing is 0.267. Suppose you sit on a bench in a mall and observe 300 randomly selected individuals’ habits as they sneeze.
What are the mean and standard deviation?
Compare the following box plots below. Which of the following statements is NOT true?
Class B has a higher median
Class A has greater variability
Both A and B have students who are only children
Over half of the students in class A have two or more siblings
Answers
Comparing the box plots, the statements which are not true are Class A has greater variability and over half of the students in Class A have two or more siblings.
Based on the given box plots, we can make the following observations:
a) Class B has a higher median: True. We can see from the plot that the median line for Class B (represented by the orange line inside the box) is higher than the median line for Class A (represented by the blue line inside the box).
b) Class A has greater variability: False. We can see from the plot that the boxes for both classes have similar lengths, indicating similar variability. However, Class A has a few more outliers compared to Class B, which indicates some extreme values that are far from the median.
c) Both A and B have students who are only children: True. We can see from the plot that both classes have a box plot labelled "0" indicating that some students in each class are only children.
d) Over half of the students in class A have two or more siblings: False. We can see from the plot that the box labelled "2+" in Class A only covers about a third of the box, indicating that less than a third of the students have two or more siblings. Therefore, the statement is not true.
Therefore, the answer is b) Class A has greater variability and d) Over half of the students in Class A have two or more siblings.
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3x + y = 3
x + y = 2
Solve the system of equations.
3x + y = 3
x + y = 2
Solve the system of equations.
A x = 12
, y = 3x = 1 2 , y = 3
B x = 32
, y = 12
x = 3 2 , y = 1 2
C x = 3, y = 12
x = 3, y = 1 2
D x = 12
, y = 32
x = 1 2 , y = 3 2
E x = 52
, y = -92
Answers
Answer: x = [tex]\frac{1}{2}[/tex] , y= [tex]\frac{3}{2}[/tex]
Step-by-step explanation:
Solve the systems by elimination.
10x + 10y = 40
5x + 3y = 8
Answers
Answer:
x = -2, y = 6
Step-by-step explanation:
First, find the opposite of 10 by multiplying 5x+3y=8 by -2 to eliminate 10x.
10x+10y=40
-2(5x+3y=8)
10x+10y=40
-10x-6y=-16 (Subtract 10y - 6y) (Subtract 40 - 16)
4y = 24 (Divide by 4 on both sides)
y = 6
Substitute 6 in the y value in any of the equations.
10x+10(6)=40 (Replaced 10y with 10(6) )
10x+60=40
-60 = -60 (Subtract 60 from both sides)
10x = -20 (Divide by 10 on both sides)
10 10
x = -2
Please help with the question below
please write the answer in a decimal
Answers
Answer
15.71 cm
Explanation
Mary Lou Mason purchased baby bottles for $4.56, baby formula for $12.45, and a pacifier for $2.13. For all purchases she must pay the state sales tax of 6.5 percent and the county tax of 1.5 percent. What is the tax on her purchases? Show all of your work.
Answers
Answer:
$1.53
Step-by-step explanation:
You want to know the total tax on purchases of $4.56, 12.45, and 2.13 if the tax rate is the sum of 6.5% and 1.5%.
TaxThe tax is the product of the tax rate and the total of purchases. The tax rate will be the sum of the state and county tax rates:
6.5% +1.5% = 8% = 8/100
The attached calculation shows the tax is $1.53 after rounding to cents.
The tax rate on Mary Lou Mason's purchases is $1.53.
What is meant by rate?
In mathematics, rate usually refers to the ratio of two quantities that have different units of measurement.
To calculate the total tax on Mary Lou Mason's purchases, we need to first calculate the total cost of her purchases, including both the prices of the items and the sales tax.
The total cost of Mary Lou Mason's purchases before taxes is:
$4.56 + $12.45 + $2.13 = $19.14
To calculate the sales tax, we need to multiply the total cost of the purchases by the sum of the state and county tax rates:
6.5% + 1.5% = 8%
So the sales tax rate is 8%.
To calculate the total tax on Mary Lou Mason's purchases, we multiply the total cost of the purchases by the sales tax rate:
$19.14 x 8% = $1.53
Therefore, the tax on Mary Lou Mason's purchases is $1.53.
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A trapezoid has an
area of 35 mm².
What is the height
of the trapezoid?
8 mm
6 mm
Answers
The height of the trapezoid is 7 mm.
A quadrilateral with one set of parallel opposite sides is referred to as a trapezium.
To find the height of a trapezoid when the area and the lengths of the parallel bases are given, we can use the formula:
h = 2A / ([tex]b_1 + b_2[/tex])
where
h is the height,
A is the area,
[tex]b_1[/tex] and [tex]b_2[/tex] are the lengths of the parallel bases.
As per given information in the question.
Using the given area of 35 mm² and the lengths of the parallel bases,
For [tex]b_1[/tex] = 8 mm and [tex]b_2[/tex] = 6 mm
we get:
h = 2 x 35 / (8 + 6) = 7 mm
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In a class of 20 children, 8 had one sister, 6 had 2 sisters. The mean number of children was 1. how many children had no sisters
Answers
There are no children in the class who have no sisters. This is because the mean number of children is 1, and some children have more than one sibling to Balance out those who have only one.
To solve this problem, we need to use some basic principles of statistics and probability. Let's start by identifying the key information provided in the problem. We know that there are 20 children in the class, and that some of them have sisters. Specifically, 8 children have one sister and 6 children have two sisters. We also know that the mean (average) number of children is 1, which means that on average, each child has one sibling.
To find out how many children had no sisters, we can use the following formula:
Total number of children - (number of children with one sister x 1) - (number of children with two sisters x 2) = number of children with no sisters
Using the values from the problem, we get:
20 - (8 x 1) - (6 x 2) = 0
This means that there are no children in the class who have no sisters. While this may seem counterintuitive, it's important to remember that the mean number of children is 1, which means that some children must have more than one sibling in order to balance out the others who have only one.
In summary, the answer to the question is that there are no children in the class who have no sisters. This is because the mean number of children is 1, and some children have more than one sibling to balance out those who have only one.
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The temperature of one furnace is 1400°C and cools at a rate of 2.5°C per minute. The temperature of another furnace is 1200°C and is heated at a rate of 1.5°C per minute. Write an equation that models when the temperature of the two furnaces will be equal.
Answers
Answer:
1400 - 2.5t = 1200 + 1.5t
t = 50
Step-by-step explanation: