Find the vertex of the parabola whose equation is y = x2 - 4x + 6.
Answers
Vertex Formula: [tex]x = \frac{-b}{2a}[/tex]
[tex]y = x^2 - 4x + 6 [/tex]
b = (-4)
a = 1
[tex]x=\frac{-b}{2a}[/tex]
[tex]x=\frac{-(-4)}{2(1)}[/tex]
[tex]x=\frac{4}{2}[/tex]
[tex]x=2[/tex]
So the vertex x point is at 2 so (x,y) = (2,y)
Now lets find the y by inserting 2 where the x is located in the equation:
[tex]y = x^2 - 4x + 6[/tex]
[tex]y = 2^2 - 4(2) + 6[/tex]
[tex]y = 4 - 8 + 6[/tex]
[tex]y = 2[/tex]
Now that we found the y for the vertex, we have (x,y) = (2,2) and (2,2) is our vertex. We can graph the equation to and see (2,2) is correct.
Related Questions
Jim worked 40 regular hours last week, plus 8 overtime houts at the time and a half rate. His gross pay was $1,248.
a.) What was his hourly rate?
b.) What was his hourly overtime rate?
Answers
40 hours regular pay
8 hours at 1.5 = 8*1.5 = 12
40+12 = 52
1248/52 = 24
A) $24 per hour
B) 14*1.5 = $36 per hour for overtime
a)
His hourly rate= $ 24
b)
His hourly overtime rate= $ 36
Step-by-step explanation:Jim worked 40 regular hours last week, plus 8 overtime hours at the time and a half rate.
This means if he is paid $ x per hour working in regular hours.
Then he will be paid $ (1.5x) per hour working overtime.
( Since it is given that: he is paid time and a half rate for overtime)
Now, his gross pay is: $ 1248
i.e.
Amount working at regular hours for 40 hours+ Amount obtained for working 8 hours overtime= $ 1248
i.e.
40x+8(1.5x)=1248
i.e.
40x+12x=1248
i.e.
52x=1248
i.e.
x=24
a)
Hence, his hourly rate is:$ 24
b)
His hourly overtime rate is: $ (24×1.5)
i.e.
His hourly overtime rate is: $ 36
A fair die is rolled 8 times.
a. what is the probability that the die comes up 6 exactly twice?
Answers
there are 8C2 ways of obtaining exactly 2 6's in 8 throws = 28 ways
so required probability is 0.00930272 * 28 = 0.26 to nearest hundredth
The probability of rolling a 6 exactly twice in 8 rolls of a fair die is approximately 33.5%.
The situation describes a binomial distribution where we have:
n = 8 (number of trials)k = 2 (number of successes, i.e., rolling a 6)p = 1/6 (probability of success on a single trial)The binomial probability formula is given by:
[tex]P(X = k) = \binom{n}{k} \times p^k \times (1-p)^{n-k}[/tex]
First, calculate (n choose k) which is given by:
[tex]\binom{n}{k}= n! / [k! \times (n-k)!][/tex]
For our problem:
[tex]\binom{n}{k} = 8! / [2! \times (8-2)!] = 28[/tex]
Next, calculate the probability:
[tex]P(X = 2) = 28 \times (1/6)^2 \times (5/6)^6[/tex]
Evaluating the above:
P(X = 2) = [tex]28 \times (1/36) \times (15625/46656)[/tex] ≈ 0.335
Therefore, the probability that the die comes up a 6 exactly twice in 8 rolls is approximately 0.335 or 33.5%.
The amount of interest you pay does not depend on the method the lender uses to calculate interest.
True or False
Answers
Answer:
FALSE
Step-by-step explanation:
The amount of interest paid depends on the method adopted by the lender. There are several methodologies for applying interest, so that each sector will adhere to the most convenient modality. The interest paid by you will be the result of this choice and may for example have a daily, monthly, semi-annual or annual impact. Obviously all interest charged must be clearly described in a contract and must be considered legal in the state legal apparatus.
For a closed cylinder with radius r cm and height h cm, find the dimensions giving the minimum surface area, given that the volume is 40 cm3.
Answers
The dimensions that give the minimum surface area are a radius of approximately 2.83 cm and a height of approximately 0.64 cm.
Explanation:To find the dimensions of the closed cylinder that give the minimum surface area, we need to express the surface area of the cylinder in terms of one variable.
Let's use the radius, r, as the variable.
The surface area of a closed cylinder is given by the formula: A = 2πr² + 2πrh, where h is the height of the cylinder.
Since we are given that the volume of the cylinder is 40 cm³, we can use the formula for the volume of a cylinder to express h in terms of r: V = πr²h = 40 cm³.
Substituting this expression for h in terms of r into the surface area formula, we get:
A = 2πr² + 2πr(40 / (πr²)) = 2πr² + (80 / r).
To find the dimensions that give the minimum surface area, we need to find the value of r that minimizes A.
To do this, we take the derivative of A with respect to r and set it equal to zero:
A' = 4πr - (80 / r²) = 0. Solving this equation, we find that r = √(20/π) ≈ 2.83 cm.
So, the radius that gives the minimum surface area is approximately 2.83 cm.
We can then use the volume formula to find the corresponding height: h = 40 / (π(2.83)²) ≈ 0.64 cm.
Therefore, the dimensions giving the minimum surface area are a radius of approximately 2.83 cm and a height of approximately 0.64 cm.
Ryan earns $7.50 an hour doing yard work if you were 12 hours a week during an 11 week summer break how much money will Ryan earn in all
Answers
$90 times 11 months = $990
Which line has a mistake in the following steps.
12 - 6(8 / 4)
12 - 6(2)
6(2)
12
Answers
It should read 12 - 12
Then the 4th line is 0
the larger of two numbers is nine more than four times the smaller number the sum of the two numbers is 59 find the two numbers
Answers
Larger number = 4x + 9
sum of the two numbers is 59
x + (4x + 9) = 59
Solve for x, which is the smaller number
x + (4x + 9) = 59
x + 4x + 9 = 59
5x + 9 = 59
5x = 59-9
5x = 50
5x / 5 = 50 / 5
x = 10
Smaller number = 10
Larger Number: Insert 10 into x
4x + 9
4(10) + 9
40 + 9 = 49
Larger number = 49
Smaller number = 10
Smaller + Larger = 59
10 + 49 = 59
Determine whether the random variable is discrete or continuous.
a. the distance a baseball travels in the air after being hitdistance a baseball travels in the air after being hit.
b. the number of textbook authors now sitting at a computernumber of textbook authors now sitting at a computer.
c. the number of points scored during a basketball gamenumber of points scored during a basketball game.
d. the amount of snowfallamount of snowfall.
e. the square footage of a housesquare footage of a house.
Answers
If we measure a quantity, its continuous.
The number of people in a crowd is discrete. So are the # of game points scored.
Since every dimension of a house is a continuous random variable, the square footage is also.
See whether you can now identify which of the above choices is discrete and which is continuous.
Use polar coordinates to find the limit. [if (r, θ) are polar coordinates of the point (x, y) with r ⥠0, note that r â 0+ as (x, y) â (0, 0).] (if an answer does not exist, enter dne.) lim (x, y)â(0, 0) 7eâx2 â y2 â 7 x2 + y2
Answers
[tex]\displaystyle\lim_{(x,y)\to(0,0)}\frac{7e^{-x^2-y^2}-7}{x^2+y^2}[/tex]
Converting to polar coordinates, we take [tex]x^2+y^2=r^2[/tex] with [tex]x=r\cos t[/tex] and [tex]y=r\sin t[/tex]. This yields
[tex]\displaystyle\lim_{(r,t)\to(0,0)}\frac{7e^{-r^2}-7}{r^2}[/tex]
Most important is that the limand is independent of [tex]t[/tex]. Evaluating directly yields [tex]\dfrac00[/tex], so we apply L'Hopital's rule:
[tex]\displaystyle\lim_{r\to0}\frac{7e^{-r^2}-7}{r^2}=\lim_{r\to0}\frac{-14re^{-r^2}}{2r}=-7\lim_{r\to0}e^{-r^2}=-7[/tex]
The factored form of 4a3b5 − 16a5b2 + 12a2b3 is ?
Answers
Answer:
The factored form of [tex]4a^3b^5-16a^5b^2+12a^2b^3[/tex] is [tex]4a^2b^2\left(ab^3-4a^3+3b\right)[/tex].
Step-by-step explanation:
To find the factored form of [tex]4a^3b^5-16a^5b^2+12a^2b^3[/tex] you must:
Apply exponent rule: [tex]a^{b+c}=a^ba^c[/tex]
[tex]a^2b^3=a^2b^2b,\:a^5b^2=a^2a^3b^2,\:a^3b^5=a^2ab^2b^3[/tex]
So, we can write our expression as [tex]4a^2ab^2b^3-16a^2a^3b^2+12a^2b^2b[/tex].
Next, rewrite 12 as [tex]3\cdot \:4[/tex] and -16 as [tex]4\cdot \:4[/tex]
[tex]4a^2ab^2b^3+4\cdot \:4a^2a^3b^2+3\cdot \:4a^2b^2b[/tex]
Factor out common term: [tex]4a^2b^2[/tex]
Therefore,
[tex]4a^3b^5-16a^5b^2+12a^2b^3= 4a^2b^2\left(ab^3-4a^3+3b\right)[/tex]
Given the number, two-fifths, generate its equivalent forms as a fraction, a decimal, and a percent. In addition, give a real world example in which each form might be used
Answers
2/5 (fraction), 0.2 (decimal), and 20% (percent).
You could use percentages for the results of a poll. For example, if you held a poll for which flavour of ice cream is your favorite. Let's say you surveyed 100 kids and 50 kids voted chocolate, 30 vanilla, and 20 strawberry. Then you could say 20% voted for strawberry.
You could use the decimal form for money. For example, let's say you want to buy something that costs $10.20 and you only have 10 dollars. You could say that you would need 20 more cents, or $0.20.
As for the fraction, you could use this for measurements. For example, let's say you have a piece of fabric. You want to cut it up into 5 equal pieces, but only use two of those pieces. You could say that you would only use 2/5 of the fabric.
I hope I helped you out! If anything is wrong, please let me know about it! :)
Find two numbers whose sum is 23 and whose product is a maximum.
Answers
xy = Product eq2
Substitute eq1 into eq2. Lets get eq2 in terms of x.
Product = x(23 - x)Product = - x2 + 23x
we must find the vertex has coordinate (h, k). h = -b / 2a where:a = -1b = 23
h= -23 / -2 = 11.5
x = 11.5
Substitute this value of x into eq1 to find y. y = 23 - xy = 23 - 11.5y = 11.5
The two numbers that will give the largest product possible are 11.5 and 11.5.
The unknown numbers are 11.5 and 11.5
Let the unknown numbers be x and y
If the sum of the numbers is 23, hence;
x + y = 23
x = 23 - y ............. 1
If the product is at maximum, hence;
xy = maximum ......... 2
Substitute equation 1 into 2
(23 - y )y = maximum
23y - y² = maximum
maximum = -y² + 23y
A(y) = -y² + 23y
Since the product A(y) is at maximum, hence dA(y)/dy = 0 as shown:
dA(y)/dy= -2y + 23 = 0
-2y + 23 = 0
2y = 23
y = 23/2
y = 11.5
Since x + y = 23
x = 23 - y
x = 23 - 11.5
x = 11.5
Hence the unknown numbers are 11.5 and 11.5
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16.99 cap; 20% discount
Answers
price after discount = 16.99 - (16.99 x 0.20)
= 16.99 - 3.398
= 13.592 or $13.59
2x^3-18x
factor completely
Answers
One-fourth equals the quotient of a number and 8. Find the number.
Answers
multiply both sides by 8 to isolate the variable, n.
2 = n
The number is 2.
so
n = 8/4 = 2
answer
number is 2
what is the answer to simplify 9 yards: 9 feet
Answers
A city council wants to build a public park. Which of these taxes is it most likely to use to fund its effort?
a. Sales Tax
b. Income Tax
c. Property Tax
Answers
c. Property Tax
Let f(x) = 4x-5 and g(x) = 3x, find (fog)(x)
Please answer the question
Answers
Hope I helped
7.58 rounded to the nearest tenth place
Answers
7.6
hope that helps, God bless!
hope it helps!
In the diagram, g ∥ h, m∠1 = (4x + 36)°, and m∠2 = (3x – 3)°.What is the measure of ∠3?
Answers
The surface area of a snowball melting decreases at a rate of 6cm/min. how is the volume changing when the radius is 3
Answers
[tex]\bf \textit{surface area of a sphere}\\\\ s=4\pi r^2\implies \boxed{\cfrac{s}{4\pi }=r^2} \\\\\\ \cfrac{1}{4\pi }\cdot \cfrac{ds}{dt}=\stackrel{chain~rule}{2r\cfrac{dr}{dt}}\implies \boxed{\cfrac{\frac{ds}{dt}}{8\pi r}=\cfrac{dr}{dt}}\\\\ -------------------------------\\\\[/tex]
[tex]\bf \textit{volume of a sphere}\\\\ V=\cfrac{4\pi r^3}{3}\implies V=\cfrac{4\pi }{3}\cdot r^2\cdot r\implies V=\cfrac{\underline{4\pi} }{3}\cdot \boxed{\cfrac{s}{\underline{4\pi} }}\cdot r \\\\\\ V=\cfrac{1}{3}sr\implies \cfrac{dV}{dt}=\cfrac{1}{3}\left( \cfrac{ds}{dt}\cdot r+s\cdot \cfrac{dr}{dt} \right)\\\\ -------------------------------\\\\[/tex]
[tex]\bf \textit{now, when the radius is 3}\qquad \begin{cases} \frac{ds}{dt}=-6\\\\ s=4\pi 3^2\\ \qquad 36\pi \\\\ \frac{dr}{dt}=\frac{-6}{8\pi 3}\\\\ \qquad -\frac{1}{4\pi } \end{cases} \\\\\\ \cfrac{dV}{dt}=\cfrac{1}{3}\left(-6\cdot 3~~+~~36\pi\cdot -\cfrac{1}{4\pi } \right)\implies \cfrac{dV}{dt}=\cfrac{1}{3}(-18-9) \\\\\\ \cfrac{dV}{dt}=-\cfrac{27}{3}\implies \cfrac{dV}{dt}=-9[/tex]
what is the axis of symmetry for the graph of y-4x=7-x^2
Answers
[tex]\bf y-4x=7-x^2\implies y=7-x^2+4x\implies y=-x^2+4x+7 \\\\\\ \textit{ vertex of a vertical parabola, using coefficients}\\\\ \begin{array}{llll} y = &{{ 1}}x^2&{{ +4}}x&{{ +7}}\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array}\qquad \left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right) \\\\\\ \left(-\cfrac{4}{2(-1)}~~,~~7-\cfrac{4^2}{2(-1)} \right)\quad thus\implies \stackrel{\textit{axis of symmetry}}{x=~~~~-\cfrac{4}{2(-1)}}[/tex]
At the store deli, Rita spent $13.45 on 5 pounds of cheese. How much would it cost to buy 2 pounds of cheese?
Answers
13.45/5 = x/2
x = 13.45*2/5
x = 5.38
The answer will be $5.38
Answer:
The cost would be $ 5.38.
Step-by-step explanation:
Given,
The price of 5 pounds of cheese = $ 13.45,
So, the price of 1 pounds of cheese = [tex]\frac{\text{Price of }5\text{ pounds of cheese}}{5}[/tex]
[tex]=\frac{13.45}{5}[/tex]
[tex]=\$ 2.69[/tex]
Thus, the price of 2 pounds of cheese = 2 × price of each pound of cheese
= 2 × 2.69
= $ 5.38
Which investment has the least amount of risk? standard deviation = $450, expected return = $4,500 standard deviation = $600, expected return = $400 standard deviation = $500, expected return = $800 standard deviation = $400, expected return = $5,000?
Answers
Investment 1: standard deviation = $450, expected return = $4,500
expected return to standard deviation ratio = 4500 / 450 = 10
Investment 2: standard deviation = $600, expected return = $400
expected return to standard deviation ratio = 400 / 600 = 0.67
Investment 3: standard deviation = $500, expected return = $800
expected return to standard deviation ratio = 800 / 500 = 1.6
Investment 4: standard deviation = $400, expected return = $5,000
expected return to standard deviation ratio = 5000 / 400 = 12.5
Therefore, the investment that has the least amount of risk is the invetment with standard deviation = $400, expected return = $5,000
Niko uses 12 marshmallows and 8 graham crackers to make 4 s'mores. Drag marshmallows and graham crackers into the box to show how many Niko needs to make 3 s'mores.
Answers
Answer: Niko needs 9 marshmallows and 6 graham crackers to make 3 s'mores.
Step-by-step explanation:
Let x be the number of marshmallows , y be the number of graham crackers and z be the number of s'mores.
Then according to given question
Niko uses 12 marshmallows and 8 graham crackers to make 4 s'mores
gives the equation 12x+8y=4z
If we divide whole equation by 4 , we get the quantity of x and y to make z such that
[tex]\frac{1}{4}(12x+8y)=\frac{1}{4}(4z)\\\Rightarrow3x+2y=z[/tex]
Now to make 3 s'mores, we need to multiply the equation by 3 such that
[tex]3\times(3x+2y)=3\times\ z\\\Rightarrow\ 9x+6y=3z[/tex]
Thus Niko needs 9 marshmallows and 6 graham crackers to make 3 s'mores.
Tell whether the two rates form a proportion
7 inches in 9 hours; 42 inches in 54 hours
explain how you got the answer/show your work
Answers
Another way I did it was that I divided 9/7 to get 1.2857. I tried the same to 54/42 and I got the same answer. I then used 1.2857 to multiply that by 7 and it ended up giving me 9 and I used that to multiply it by 42 as well to get 54.
The rates of 7 inches per 9 hours and 42 inches per 54 hours do form a proportion, as demonstrated by cross-multiplying the two ratios which resulted in equal cross products (378 inch-hours on both sides).
Explanation:To determine if the two rates form a proportion, we need to compare the ratios of inches to hours for both scenarios. For the first rate, we have 7 inches in 9 hours, and for the second rate, we have 42 inches in 54 hours. A proper proportion exists when two ratios are equivalent.
Let's set up our two ratios:
First ratio: 7 inches / 9 hoursSecond ratio: 42 inches / 54 hoursNow, we will test if these ratios form a proportion by creating a fraction out of each ratio and then cross-multiplying.
(7 inches / 9 hours) = (42 inches / 54 hours)
Cross multiply:
7 inches * 54 hours = 378 inch-hours
42 inches * 9 hours = 378 inch-hours
Since the cross products are equal, the two rates indeed form a proportion.
Jayne has a home-based business putting on children’s parties. She charges $60 to design the party and then $10.00 per child. Write a function rule that relates the total cost of the party to the number of children n.
A f(n) = 10 – 55n
B f(n) = 60 + 10n
C f(n) = 10 + 60n
D f(n) = 10n – 55
Answers
So we know we need to add 60. So 60 +
then $10.00 per child
10 dollars per child mean 10 times # of children so we have:
10n
Put it all together:
60 + 10n
f(n) = 60 + 10n
The answer is f(n) = 60 + 10n.
What's the characteristic of the rule also called?The composite function rule (also called the chain rule) Has taken a look at the characteristic f(x)=(x2 + 1)17. we will think about this function as being. the end result of mixing features. If g(x) = x2 + 1 and h(t) = t17 then the result of.
What is an example of a characteristic rule?A function rule along with value = p + zero.08p is an equation that describes a useful relationship. If p is the charge you pay for an object and 0.08 is the income tax, the feature rule above is the value of the item. in case you are given a desk, usually, you need to cautiously study the desk to peer what the function rule is.
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Determine the missing information in the paragraph proof. Given: Lines a and c intersect at point S, creating 4 angles. Prove: Corresponding angles are congruent. We are given that lines a and c intersect at point S. Translate line a down line c until point S reaches point Q. Call the new line through Q line b. Because translations preserve orientation, lines a and b are ________. Because translations preserve angle measure, ∠RSU ≅ ∠RQU'. For the same reason, ∠RST ≅ ∠RQT', ∠PSU ≅ ∠PQU', and ∠PST ≅ ∠PQT'. Each of the angle pairs are corresponding angles. Therefore, corresponding angles are congruent. parallel perpendicular congruent reflected
Answers
Answer: Parallel.
Step-by-step explanation:
Given: Lines a and c intersect at point S, creating 4 angles.
To Prove: Corresponding angles are congruent.
We are given that lines a and c intersect at point S.
Translate line a down line c until point S reaches point Q.
Call the new line through Q line b.
We know that translations preserve orientation then every point on line b is equidistant from every points on line a.
Therefore, lines a and b are parallel by the definition of parallel lines.
Hence, the complete statement is Because translations preserve orientation, lines a and b are parallel.
Select True or False for each statement.
For a real number a, a + 0 = a.
For a real number a, a + (-a) = 1.
For a real numbers a and b, | a - b | = | b - a |.
For real numbers a, b, and c, a + (b ∙ c) = (a + b)(a + c).
For rational numbers a and b when b ≠ 0, is always a rational number.
Answers
For a real number a, a + 0 = a. TRUE
For a real number a, a + (-a) = 1. FALSE
For a real numbers a and b, | a - b | = | b - a |. TRUE
For real numbers a, b, and c, a + (b ∙ c) = (a + b)(a + c). FALSE
For rational numbers a and b when b ≠ 0, is always a rational number. TRUE
Explanation:For a real number a, a + 0 = a. TRUEThis comes from the identity property for addition that tells us that zero added to any number is the number itself. So the number in this case is [tex]a[/tex], so it is true that:
[tex]a+0=a[/tex]
For a real number a, a + (-a) = 1. FALSEThis is false, because:
[tex]a+(-a)=a-a=0[/tex]
For any number [tex]a[/tex] there exists a number [tex]-a[/tex] such that [tex]a+(-a)=0[/tex]
For a real numbers a and b, | a - b | = | b - a |. TRUEThis is a property of absolute value. The absolute value means remove the negative for the number, so it is true that:
[tex]\mid a-b \mid= \mid b-a \mid[/tex]
For real numbers a, b, and c, a + (b ∙ c) = (a + b)(a + c). FALSEThis is false. By using distributive property we get that:
[tex](a + b)(a + c)=a^2+ac+ab+bc \\ \\ a^2+ab+ac+bc \neq a+(b.c)[/tex]
For rational numbers a and b when b ≠ 0, is always a rational number. TRUEA rational number is a number made by two integers and written in the form:
[tex]\frac{u}{v} \\ \\ v \neq 0[/tex]
Given that [tex]a \ and \ b[/tex] are rational, then the result of dividing them is also a rational number.
Answer:
A) True
B) False
C) True
D) False
E) True
Step-by-step explanation:
We are given the following statements in the question:
A) True
For every real number, a, a + 0 = a. 0 is known as the additive identity.
B) False
For a real number a, a + (-a) = 0.
C) True
For a real numbers a and b, [tex]|a-b| = |b-a|[/tex]
D) False
For real numbers a, b, and c, a + (b ∙ c) = (a + b)(a + c).
Counter example: For a = 2, b = 1, c = 3
[tex]a + (b.c) = (a + b)(a + c)\\2 + (1.3) \neq (2+1)(2+3)\\5\neq 15[/tex]
E) True
For rational numbers a and b, b is not equal to zero, [tex]\frac{a}{b}[/tex] is always a rational number.
Daisy is shipping a package to her daughter, Molly. If the box’s dimensions are equal, and the length is 3 feet long, what is the volume of the box in cubic feet?
A. 1
B. 3
C. 9
D. 27
Answers
length=width=height = 3 ft. The volume is (3 ft)^2 = 27 ft^3.