For the function f(x) = (x − 2)2 + 4, identify the vertex, domain, and range.
a. The vertex is (–2, 4), the domain is all real numbers, and the range is y ≥ 4.
b. The vertex is (–2, 4), the domain is all real numbers, and the range is y ≤ 4.
c. The vertex is (2, 4), the domain is all real numbers, and the range is y ≤ 4.
d.The vertex is (2, 4), the domain is all real numbers, and the range is y ≥ 4.
Answer:
The vertex is (2, 4), the domain is all real numbers, and the range is y ≥ 4.
Step-by-step explanation:
The equation of the parabola is [tex]f(x)=(x-2)^2+4[/tex]
The vertex form of the parabola is given by
[tex]f(x)=a(x-h)^2+k[/tex], here (h,k) is the vertex.
Comparing given equation with the vertex form of the parabola, we get
h = 2, k = 4
Hence, the vertex of the parabola is (h,k) = (2,4)
Now, domain is the set of x values for which the function is defined. The given function is defined for all real values of x.
Hence, domain is all real numbers.
Range is the set of y values for which the function is defined.
Since, here a = 1>0 hence it is a upward parabola and the vertex is the minimum point of this parabola.
Since, vertex is (2,4) hence, y values never less than 4.
Hence, range is y ≥ 4.
D is the correct options.
What is the third term of a sequence that has a first term of 3 and a common ratio of 3?
1/3
9
27
Answer:
The answer is 27
Step-by-step explanation:
The third term of the given geometric sequence is 27, calculated by using the geometric sequence formula with the first term of 3 and a common ratio of 3.
Explanation:The third term of a sequence that has a first term of 3 and a common ratio of 3 is calculated using the formula for geometric sequences, which is a_n = a_1 \(\cdot\) r^{(n-1)}, where a_n is the nth term, a_1 is the first term, and r is the common ratio. To find the third term (when n=3), we substitute the given values into the formula to get: 3 \(\cdot\) 3^{(3-1)}.
Calculating that gives us 3 \(\cdot\) 3², which equals 3 \(\cdot\) 9, so the third term is 27.
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Joy is collecting rocks. she has 3 jars and 4 buckets. there are 2 rocks in each jar and 3 rocks in each bucket. how many rocks does joy have?
18 is the answer folks.
write 0.08 as a decimal and fraction
0.08 is already a decimal.
To make 0.08 into a fraction, look at the image attached.
The delivery van arrives at an office every day between 3 PM and 5 PM. The office doors were locked between 3:15 PM and 3:35 PM. What is the probability that the doors were unlocked when the delivery van arrived?
A. 5/6
B. 5/12
C. 1/3
D. 1/6
The probability that the office doors were unlocked when the delivery van arrived is 5/6. The correct option is A. 5/6.
To determine the probability that the office doors were unlocked when the delivery van arrived, we first need to understand the time intervals:
The delivery van arrives between 3 PM and 5 PM, which is a total of 2 hours or 120 minutes. The office doors were locked between 3:15 PM and 3:35 PM, which is a total of 20 minutes.The total available time for the delivery van to arrive is 120 minutes, out of which the doors were locked for 20 minutes.
Now, we'll calculate the probability that the doors were unlocked:Unlocked time = Total time - Locked time = 120 minutes - 20 minutes = 100 minutesProbability = Unlocked Time / Total TimeProbability = 100 / 120 = 5/6.The measure of angle 5 is 80 degrees. The measure of angle 2 is one-eighth the measure of angle 5. What is the measure of angle 4
An object is dropped from a height of 1700 ft above the ground. The function h=-16t^2+1700 gives the object’s height h in feet during free fall at t seconds.
a. When will the object be at 1000ft above the ground?
b. When will the object be 940ft above the ground?
c. What are a reasonable domain and range for the function h?
To calculate when the object would be at a particular height, we use the given equation, input the known height, and solve for the time (t), using the quadratic formula. The time when the object is at 1000ft is approximately 6.6 seconds and at 940ft it's approximately 6.9 seconds. The domain of the function is 0 <= t <= 10.3 and the range is 0 <= h <= 1700.
Explanation:This question involves the application of the quadratic equation, in the context of Physics. You would use the given function h=-16t^2+1700 to solve for t when h is set to the desired height.
For height 1000 ft, you would solve the equation -16t^2 + 1700 = 1000. After simplifying, you get -16t^2 = -700, then t^2 = 43.75, which leads to t = square root of 43.75. Thus t = approximately 6.6 seconds.For height 940 ft, you use the same methodology yielding t = approximately 6.9 seconds.The domain for this function would be all real values of t such that 0 <= t <= 10.3, because the object starts falling at t=0 and hits the ground (height = 0) at about 10.3 seconds. The range would be all real values of h such that 0 <= h <= 1700, because the object's height is initially 1700 ft and reduces to 0 as it hits the ground.Learn more about Quadratic Functions here:
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The object will be at 1000ft and 940ft above the ground approximately 6.65 seconds and 6.87 seconds after being dropped, respectively.
The domain of the function representing this situation is t ≥ 0, and the range is 0 ≤ h ≤ 1700.
Explanation:To find out when the object will be at certain heights, we can set h equal to those heights and solve for t.
a. When will the object be at 1000ft above the ground?
1000 = -16t^2 + 1700
16t^2 = 1700 - 1000
16t^2 = 700
t^2 = 700/16
t = √(700/16)
t ≈ 6.65 seconds
b. When will the object be 940ft above the ground?
940 = -16t^2 + 1700
16t^2 = 1700 - 940
16t^2 = 760
t^2 = 760/16
t = √(760/16)
t ≈ 6.87 seconds
c. What are a reasonable domain and range for the function h?
The domain of the function, which stands for time, would be t ≥ 0 as we do not consider negative time.
The range of the function, corresponding to the height, would be 0 ≤ h ≤ 1700 as we know the object's maximum height is 1700 ft and it can't go below ground level (0 ft).
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The shorter tree is 10.5 ft. tall, and casts a shadow that is 11.25 feet long.if the taller tree casts a 17.5 ft. shadow,how tall is the tree?
By using proportions based on the concept of similar triangles, the height of the taller tree is calculated to be approximately 16.3 feet tall.
Explanation:The question asks us to compare the heights and shadows of two trees to determine the height of the taller tree given the height and shadow length of the shorter tree and the shadow length of the taller tree. Since both trees will cast shadows in the same proportions if they're under the same conditions of light, we can use similar triangles to find the height of the taller tree.
The shorter tree is 10.5 ft tall and casts a shadow that is 11.25 feet long. If the taller tree casts a 17.5 ft shadow, we can set up a proportion as follows:
Height of the shorter tree / Shadow of the shorter tree = Height of the taller tree / Shadow of the taller tree
10.5 ft / 11.25 ft = Height of the taller tree / 17.5 ft
From the proportion, we can solve for the height of the taller tree:
Height of the taller tree = (10.5 ft / 11.25 ft) × 17.5 ft
Height of the taller tree = 16.333... ft
So, the taller tree is approximately 16.3 feet tall.
what fraction is equivalent to the expression 4/8 + 2/8
If f(x) = 9x + 1, find f–1 (f (3))
Answer:
The value of [tex]f^{-1}(f(3))[/tex] is 3.
Step-by-step explanation:
Here, the given function is,
[tex]f(x)=9x+1-----(1)[/tex]
For finding [tex]f^{-1}(f(3))[/tex] we have to find out the inverse of f(x),
The steps are as follows,
Step 1 : Replace f(x) by y,
[tex]y=9x+1[/tex]
Step 2 : Switch x and y,
[tex]x=9y+1[/tex]
Step 3 : Isolate y in the left side,
[tex]-9y=1-x[/tex]
[tex]\implies y = \frac{x-1}{9}[/tex]
Step 4 : replace y by [tex]f^{-1}(x)[/tex],
[tex]f^{-1}(x)=\frac{x-1}{9}------(2)[/tex]
Now,
[tex]f^{-1}(f(3))=f^{-1}(9\times 3+1)[/tex] ( From equation (1) )
[tex]=f^{-1}(28)[/tex]
[tex]=\frac{28-1}{9}[/tex] ( From equation (2) )
[tex]=\frac{27}{9}[/tex]
[tex]=3[/tex]
Vector a~ points south. vector b~ points west. what is the direction of their cross product a~ × b~ ?
Final answer:
The cross product a~ × b~, where a~ points south and b~ points west, is directed downwards. This direction is determined using the right-hand rule and is perpendicular to the original plane formed by a~ and b~.
Explanation:
If vector a~ points south and vector b~ points west, their cross product a~ × b~ can be determined using the right-hand rule. By pointing the fingers of your right hand towards the direction of a~ (south) and then rotating your wrist to point your fingers towards b~ (west), your thumb will point downwards. Therefore, the direction of a~ × b~ is downwards, which is into the ground if you are standing in the Northern Hemisphere.
The cross product of two vectors is always perpendicular to the plane that contains the original vectors. So in this case, since a~ and b~ are orthogonal and lie in the horizontal plane, their cross product a~ × b~ will be orthogonal to this horizontal plane. As the vectors move from a~ to b~ in an anticlockwise direction, the resultant vector points downward, opposite to the upward direction that would be indicated by b~ × a~.
Choose the correct simplification of the expression (3x2y3z4)(2x5y2z3).
6x7y5z7
6x10y6z12
5 x7y5z7
5x10y6z12 ...?
yep, your answer is A.
the equation y equals 0.25 x describes a proportional relationship between X and Y what is the constant of proportionality
What 2-dimensional shape can be rotated about the y-axis to create a cylinder which has a smaller diameter than height?
Henry has 2/3 of a bag of popcorn he eats half of the popcorn during the movie what fraction of a bag of popcorn does henry eat during the movie
How much money invested at 5% compounded continuously for 3 years will yield $820
The inverse of the function f(x) = 1/2x + 10 is shown.
h(x) = 2x – ?
What is the missing value?
Answer:
The Answer is 20
Step-by-step explanation:
if ya know ya know
What is ↓ when n = 8?
(n-5)•6
------------
n÷4
simplify the ratio 12:2 ...?
Select the term that best describes the statement.
All lines are straight or a triangle has four sides.
Please help! After five years of earning interest at an annual rate of 4%, an investment has earned $1,200 in interest. Determine the amount of the initial investment. Show all work for full credit.
There are 23 coins in a bank. All the coins are dimes and quarters. The total value of the coins is $3.80. How many dimes are there? How many quarters?
Write the system of equations that would be used to solve this problem.
Let q = quarter
Let d = dime
Here is your system:
q + d = 23
0.25q + 0.10d = 3.80
Take it from here.
Represent the ratio 6 : 18 in two other ways ...?
What is an equation that shows that two ratios are equivalent?
An equation that shows two ratios are equivalent is a proportion. Proportions can be applied in various fields like economics to show equivalent satisfaction based on price paid for goods, or in unit conversion to represent equivalent quantities in different units.
Explanation:An equation showing that two ratios are equivalent is a proportion. A proportion is a mathematical statement that two ratios are equal. For example, 2/4 = 1/2 is a proportion because the two ratios are equivalent. Other forms of this equation could be obtained by rearranging terms, showing the direct and indirect proportions.
In a more practical context, ratios are used to compare the relationship between different quantities. For example, we could state the ratio of the prices of two goods should be equal to the ratio of the marginal utilities. When we divide the price of good 1 by the price of good 2, this should equal the marginal utility of good 1 divided by the marginal utility of good 2. This indicates that the utility or satisfaction obtained is equivalent based on the price paid.
Also, proportions can be used as a unit conversion factor where the ratio of two equivalent quantities expressed with different measurement units can lead to a conversion factor. For example, the lengths of 2.54 cm and 1 in. are equivalent, thus providing a ratio that could be used for conversion.
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Factor the expression.
k^2 - 18k + 81
Answer:
The answer is (k-9)(k+9)
Step-by-step explanation:
Which expression is equivalent to x1/2y5/2 in simplified form?
Shanti has just joined a DVD rental club. She pays a monthly membership fee of $4.95, and each DVD rental is $1.95. If Shanti's budget for DVD rentals in a month is $42, how many DVDs can Shanti rent in her first month if she doesn't want to go over her budget? ________ DVDs
After deducting the monthly membership fee of $4.95 from Shanti's budget, $37.05 remains for DVD rentals. At $1.95 per DVD rental, Shanti can rent 19 DVDs while staying within her budget.
To calculate how many DVDs Shanti can rent without going over her monthly budget of $42, we first need to deduct her monthly membership fee from her total budget. This leaves us with the amount she can spend on renting DVDs.
The monthly membership fee is $4.95. After subtracting this fee from her total budget, we have:
$42 - $4.95 = $37.05 available for DVD rentals.
Each DVD rental costs $1.95. To find out how many DVDs Shanti can rent, we divide the amount available for rentals by the cost per DVD rental:
$37.05 / $1.95 = 19 DVDs (rounded down since you cannot rent a fraction of a DVD)
Therefore, Shanti can rent 19 DVDs in her first month and stay within her budget of $42.
Find the simplified form of the expression. Give your answer in scientific notation. (9 x 10 5)(6 x 10 -7) A. 5.4 x 10 -34 B. 1.5 x 10 -1 C. 5.4 x 10 -1 D. 1.5 x 10 -34
5x^2-3y^2-20x+6y+2=0 what is the value of a
Shannon manages a crew of painters. a homeowner paid the crew $560 to paint a garage. shannon received $164 of the money, and she shared the rest equally among the other three members of her crew. how much money did each of the three crew members receive for the job?
Each of the three crew members received $132 for the job.
Step-by-step explanation:Total amount paid to Shannon = $560
Share received by Shannon = $164
So, money left for other 3 members = [tex]560-164=396[/tex] dollars
As Shannon shared the left money ($396) equally among the other three members of her crew, so each person got :
[tex]\frac{396}{3}[/tex] = $132
Therefore, each of the three crew members received $132 for the job.