f(x) = 2[tex]x^{2}[/tex]+ x − 4 for x ∈R, a ≤ x ≤ a+3. If the range of the function f is −2 ≤ f(x) ≤ 16, find the possible values of a.
Answer:
No answer is possible
Step-by-step explanation:
First, we can identify what the parabola looks like.
A parabola of form ax²+bx+c opens upward if a > 0 and downward if a < 0. The a is what the x² is multiplied by, and in this case, it is positive 2. Therefore, this parabola opens upward.
Next, the vertex of a parabola is equal to -b/(2a). Here, b (what x is multiplied by) is 1 and a =2, so -b/(2a) = -1/4 = -0.25.
This means that the parabola opens upward, and is going down until it reaches the vertex of x=-0.25 and up after that point. Graphing the function confirms this.
Given these, we can then solve for when the endpoints of the interval are reached and go from there.
The first endpoint in -2 ≤ f(x) ≤ 16 is f(x) = 2. Therefore, we can solve for f(x)=-2 by saying
2x²+x-4 = -2
add 2 to both sides to put everything on one side into a quadratic formula
2x²+x-2 = 0
To factor this, we first can identify, in ax²+bx+c, that a=2, b=1, and c=-2. We must find two values that add up to b=1 and multiply to c*a = -2 * 2 = -4. As (2,-2), (4,-1), and (-1,4) are the only integer values that multiply to -4, this will not work. We must apply the quadratic formula, so
x= (-b ± √(b²-4ac))/(2a)
x = (-1 ± √(1-(-4*2*2)))/(2*2)
= (-1 ± √(1+16))/4
= (-1 ± √17) / 4
when f(x) = -2
Next, we can solve for when f(x) = 16
2x²+x-4 = 16
subtract 16 from both sides to make this a quadratic equation
2x²+x-20 = 0
To factor, we must find two values that multiply to -40 and add up to 1. Nothing seems to work here in terms of whole numbers, so we can apply the quadratic formula, so
x = (-1 ± √(1-(-20*2*4)))/(2*2)
= (-1 ± √(1+160))/4
= (-1 ± √161)/4
Our two values of f(x) = -2 are (-1 ± √17) / 4 and our two values of f(x) = 16 are (-1 ± √161)/4 . Our vertex is at x=-0.25, so all values less than that are going down and all values greater than that are going up. We can notice that
(-1 - √17)/4 ≈ -1.3 and (-1-√161)/4 ≈ -3.4 are less than that value, while (-1+√17)/4 ≈ 0.8 and (-1+√161)/4 ≈ 2.9 are greater than that value. This means that when −2 ≤ f(x) ≤ 16 , we have two ranges -- from -3.4 to -1.3 and from 0.8 to 2.9 . Between -1.3 and 0.8, the function goes down then up, with all values less than f(x)=-2. Below -3.4 and above 2.9, all values are greater than f(x) = 16. One thing we can notice is that both ranges have a difference of approximately 2.1 between its high and low x values. The question asks for a value of a where a ≤ x ≤ a+3. As the difference between the high and low values are only 2.1, it would be impossible to have a range of greater than that.
Taehyung was sitting on the ground flying a kite. He had 22 feet of line let out to fly his kite, and the kite was 14 feet in front of him. How high was the kite?
30^∘ Another boy is standing on the roof of a 10 second string be x mIn Δ ABC sin 30^∘ = AC/AB 1/2 = AC/100 AC =
Answer: hope this helps ♡
The kite was 16.9 feet high.
Step-by-step explanation:
Pythagorean theorem
b = [tex]\sqrt{c^{2} - a^{2} }[/tex]
b = [tex]\sqrt{22^{2} - 14^{2} }[/tex]
b = [tex]\sqrt{484 - 196}[/tex]
b = [tex]\sqrt{288}[/tex]
b = 16.9
Julie had 2730 cards and Kim had 3570 cards at first.
Julie gave some of her cards to Kim. In the end, Kim had thrice as many cards as Julie.
How many cards did Julie give Kim.
Answer:
1155
Step-by-step explanation:
Total number of cards is 2730+3570=6300
Since Kim now has 3 times the card of Julie so Julie must have 6300/4=1575.
So, Julie gave 2730-1575=1155
Answer:
1155 cards
Step-by-step explanation:
3(2730-x)=3570+x
8190 - 3x = 3570 + x
4620 - 3x = x
4620 = 4x
1155 = x
1, 1, 1, 1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 89, 89, 89, 89, 89
At what percentile is 34?
Answer:
6 out of the 20 values are below 2. 6/20 = 30%.
Since 30% of the values are below 2, 2 is located at the 30th percentile.
Step-by-step explanation:
whats two-thirds of four-fifths of a dozen
Answer:
6.4Step-by-step explanation:
2/3*4/5*12 = 32/5 = 6.41dozen=12
[tex]\\ \rm\Rrightarrow \dfrac{2}{3}\:of\:\dfrac{4}{5}\:of\:12[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{2}{3}\times \dfrac{4}{5}\times 12[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{96}{15}[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{32}{5}[/tex]
[tex]\\ \rm\Rrightarrow 6.4[/tex]
find the value s of.. a and b if 0 and 2 are the zeros of the polynomial p(x) =2xcube minus 5xsquare plus ax plus b find the third zero of the polynomial
Answer:
a = 2, b = 0 , x = [tex]\frac{1}{2}[/tex]
Step-by-step explanation:
Given that x = 0 and x = 2 are zeros of p(x) , then
p(0) = 0 and p(2) = 0
p(0) = 2(0)³ - 5(0)² + a(0) + b = 0 , that is
0 - 0+ 0 + b = 0, then b = 0
p(2) = 2(2)³ - 5(2)² + 2a + b = 0 , that is
16 - 20 + 2a + 0 = 0
- 4 + 2a = 0 ( add 4 to both sides )
2a = 4 ( divide both sides by 2 )
a = 2
Then
p(x) = 2x³ - 5x² + 2x ← factor out x from each term
= x(2x² - 5x + 2)
= x(2x - 1)(x - 2)
To find zeros equate p(x) to zero
x(2x - 1)(x - 2) = 0
Equate each factor to zero and solve for x
x = 0
x - 2 = 0 ⇒ x = 2
These are the 2 zeros already known (given)
2x - 1 = 0 ⇒ 2x = 1 ⇒ x = [tex]\frac{1}{2}[/tex] ← third zero of p(x)
another one please help thankyouu!!
Answer:
f(-2) =2
Step-by-step explanation:
x = -2
We need to use |x| since -2 <-1
|-2| =2
so f(-2) =2
A giant pie is created in an attempt to break a world record for baking. The pie is shown below: A circle is shown with a central angle marked 38 degrees and the diameter marked 20 feet. What is the area of the slice of pie that was cut, rounded to the nearest hundredth? 22.08 ft2 13.19 ft2 33.14 ft2 28.97 ft2
Answer:
33.14 ft^2
Step-by-step explanation:
First find the area of the circle
The diameter is 20 ft so the radius is 1/2 of the diameter of 20*1/2 = 10 ft
A = pi r^2 = pi ( 10)^2 = 100 pi
The central angle is 38
That is a fraction of the circle, which is 360 degrees
38/360 = 19/180
Multiply the fraction of the circle by the area
19/180 * 100 pi
19/9 * 5 pi
Using 3.14 for pi
33.144444
To the nearest hundredth
33.14 ft^2
The area of the slice of pie that was cut, rounded to the nearest hundredth 33.14 ft^2.
We have to first find the area of the circle.
The diameter is 20 ft so the radius is 1/2 of the diameter of
20*1/2 = 10 ft.
What is the area of the circle?
The area of a circle [tex]A = \pi r^2[/tex]
By using the formula we have,
A= pi ( 10)^2
A = 100 pi
The central angle is 38.
That is a fraction of the circle, which is 360 degrees
38/360 = 19/180
Multiply the fraction of the circle by the area
19/180 * 100 pi
19/9 * 5 pi
Using 3.14 for pi
33.144444
To the nearest hundredth, it is 33.14 ft^2.
To learn more about the circle visit:
https://brainly.com/question/24375372
*WILL MARK BRAINLIEST* The triangle below is isosceles. Find the length of side xx in simplest radical form with a rational denominator.
Answer: 5sqrt2, 10sqrt2/2
Step-by-step explanation:
The isosceles triangle, or 45-45-90 rule states that the hypotenuse is x × square root of 2.
Divide the hypotenuse, 10, by sqrt 2 to get 5sqrt2
However , if you're looking to keep it in fraction form, you will need to rationalize the denominator. Multiply the numerator and denominator to get the fraction, 10sqrt2/2
Kelly answered 3 items on a test incorrectly. This was 6% of the items. How many items were on the test?
Answer:
50
Step-by-step explanation:
6/100=3/y
Let y be the total number of items on the test
[cross multiply]
6×y =3×100
6y = 300
divide both sides by 6
y = 50
Look at points C and D on the graph:
What is the distance (in units) between points C and D? Round your answer to the nearest hundredth. (1 point)
2.83 units
4.00 units
11.31 units
64.00 units
Answer:
11.31 units
Step-by-step explanation:
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
Step 1.
Plug in the points inside the formula where the first point = [tex](x_1,y_1)[/tex] and the second point = [tex](x_2-y_2)[/tex]
Let the first point be C and the second point be D
[tex](-4,-5)\\ (x_1,y_1)[/tex] [tex](4,3)\\ (x_2,y_2)[/tex]
Answer: [tex]d=\sqrt{(4--4)^2+(3--5)^2}[/tex]
Step 2. Simplify the parentheses "( )"
Answer: [tex]d=\sqrt{(8^2)+(8^2)}[/tex]
Step 3. Simplify the exponents "[tex]?^2[/tex]"
Answer: [tex]d=\sqrt{64 +64}[/tex]
Step 4. Add 64 and 64
Answer: [tex]d=\sqrt{128}[/tex]
Step 5. Simplify the square root
Answer: [tex]d=11.31[/tex]
Therefore, the distance between points C and D is 11.31 units
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doomdabomb: all brainliest and thanks are appreciated and would mean a lot to me, thanks!
What information could be added to the following triangles in order to prove them
similar by the angle-angle similarity theorem?
Triangle HIJ is similar to triangle MNO by the angle-angle similarity theorem if ∠O ≅ ∠J.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Two triangle are said to be similar if the ratio of their corresponding sides are in the same proportion and corresponding angles are congruent.
Triangle HIJ is similar to triangle MNO by the angle-angle similarity theorem if ∠O ≅ ∠J.
Find out more on equation at: https://brainly.com/question/2972832
#SPJ1
Help. Step By Step Explanation. Ron and two of his friends ate one fourth each of an eight slice pizza. Find remaining slice of the pizza
Answer:
2 remaining slices
Step-by-step explanation:
It sounds complicated so firstly, let's break it down.
Ron and two of his friends
There are three people involved in this question.
ate one fourth each
These three people ate one fourth of the pizza each
of an eight slice pizza.
From a pizza with eight slices
Find remaining slice of the pizza
The main question : remaining slice of the pizza
So since there were three people and they ate one fourth of the pizza
( One fourth's [tex]\frac{1}{4}[/tex] by the way )
We need to find out how much have they ate.
Simply multiply 3 with [tex]\frac{1}{4}[/tex] .
3 × [tex]\frac{1}{4}[/tex] = [tex]\frac{12}{4}[/tex] × [tex]\frac{1}{4}[/tex]
= [tex]\frac{3}{4}[/tex]
And there you go! Ron and his friends ate [tex]\frac{3}{4}[/tex] of the pizza.
But that's not the end.
The main question : Remaining slice of the pizza
It's an eight slice pizza, so a slice would be : [tex]\frac{1}{8}[/tex]
A full eight slice pizza would be one.
So, to find the remaining slice of pizza, simply subtract [tex]\frac{3}{4}[/tex] from 1.
1 - [tex]\frac{3}{4}[/tex] = [tex]\frac{4}{4}[/tex] - [tex]\frac{3}{4}[/tex]
= [tex]\frac{1}{4}[/tex]
Now there's a remainding of [tex]\frac{1}{4}[/tex] of the pizza.
But the question asked for remaining slices.
So, simply convert the denominator into 8!
( Just in case you don't know what's a denominator )
[tex]\frac{numerator}{denominator}[/tex]
So, simply multiply the denominator by 2 to turn 4 into 8!
And also, whenever the denominator is multiplied/ divided by any number, so do the numerator.
[tex]\frac{1 times 2}{4 times 2}[/tex] = [tex]\frac{2}{8}[/tex]
Your answer's the numerator, 2!
So your final answer should be 2 remaining slices out of the eight sliced pizza.
HOPE THIS ANSWER HELPED :)if a = 4 - √15, find a^4 +1\a^4
Hey! Please answer this question with steps. I will mark you brainliest.
Random answers like 'sdfgh' or wrong answers will be reported
Answer:
[tex](\frac{-3+\sqrt{15} }{4-\sqrt{15} })^{2} -4\\[/tex]
Step-by-step explanation:
[tex]a=4-\sqrt{15}[/tex]
[tex]a^{4} +\frac{1}{a^{4} }\\=a^{2^{2} } +\frac{1}{a^{2^{2} } }\\\\=(x^{2} +\frac{1}{a^{2} } )^{2} -2a^{2} \frac{1}{a^{2} } \\=(a +\frac{1}{a } )^{2} -2a \frac{1}{a}-2\\=(4-\sqrt{15} +\frac{1}{4-\sqrt{15} }) ^{2} -4\\\=(\frac{16-15-4+\sqrt{15} }{4-\sqrt{15} })^{2} -4\\=(\frac{-3+\sqrt{15} }{4-\sqrt{15} })^{2} -4\\[/tex]
please please please help me out somebody!!!!!!!
Find the midpoint of the line segment joining (–4, –2) and (2, 8).
SHOW HOW YOU GOT IT please
Answer:
(-1,3)
Step-by-step explanation:
Using midpoint formula, we have
x=(-4+2)/2=-1
y=(-2+8)/2=3
Which of the following intervals is the graph decreasing?
Answer:
B
Step-by-step explanation:
Values of y are decreasing where x is in the range (-4;02)
Liang bought a basket of apples to make pies for her friends. The basket of apples weighed ppp pounds. Before she had time to make the pies, she ate 3 pounds of apples. There are 17 pounds of apples left to make pies.
Answer: The equation to describe the situation is p-3= 17
And, the total weight of the basket of apples was 20 pounds.
Step by step: Now, to write an equation to describe the situation. And to find the total weight of the basket of apples.
So, the basket of apples total weighed = p pounds
Liang ate apples weighed = 3 pounds.
And the apples left for the pies weighed = 17 pounds.
Now, to write an equation describing the situation:
p-3=17 .
Thus, the equation is p-3=17
Now, to get the total weight of the basket of apples:
p-3=17
adding both sides by 3 we get : p=20
Hence, the basket of apples total weighed = 20 pounds.
Therefore, the equation to describe the situation is p-3=17
And, the total weight of the basket of apples was 20 pounds.
THIS SHOULD HELP U
HAVE A GOOD DAY !
According to the partial construction shown, what is the next step in constructing a line perpendicular to a point on the line?O Place compass at point D, and draw an arc that intersects the arc above overleftrightarrow DF O Place compass at point F, and draw an arc that intersects the arc above overline DF O.Place compass at point P, and draw an arc above overline DF O Place compass at point P, and draw an arc below overline DF
Answer:
Place compass at P and, with compass set to segment AB, draw an arc that intersects line PA 2nd answer.
Step-by-step explanation:
whats a square + b square I don't know thats why I am asking u dude
Answer:
.............................
the value of an article is depreciated form rs.1600 to rs.1024 in 2 years find the rate of depreciation.
Answer:
1600-1024/1600×100
which is 36%
pls mark me as brainliest
To paint a wooden cube, Pinocchio used 4 gram of paint. When it is dry, he cut the cube into 8 equal cubes of smaller size. How much paint additional paint does Pinocchio need to paint the unpainted surfaces of smaller cubes?
Answer:
game1321.netlify.app
Step-by-step explanation:
This is a test sorry I'm trying to see if you can post links in comments
Solve the system by substitution.
y = -3x + 17
2x + 3y = -5
Answer:
(8, -7)
Step-by-step explanation:
Hi there!
We are given this system:
y=-3x+17
2x+3y=5
And we want to solve the system
Let's solve it by substitution.
In the first equation, we are given y=-3x+17, which we can substitute -3x+17 as y into the second equation
2x+3(-3x+17)=5
Now do the distributive property
2x+-9x+51=-5
Combine like terms
-7x+51=-5
Subtract 51 from both sides
-7x=-56
Divide both sides by -7
x=8
Now substitute 8 as the value of x in y=-3x+17 to find the value of y
y=-3(8)+17
y=-24+17
y=-7
The answer is x=8, y=-7. If you need it as a point, it's (8, -7)
Hope this helps!
5. Jill has an old bike. Her parents gave her $40
to repair it. New tires will cost $21. She also
wants to buy a bell for $5 and a basket for $13
How much money will Jill have left?
Answer:
1$ %ffddrjdjjdjdjdjjdjrjduduufufjf
a box of 15 cookies costs 9$ what is the price of 1 coockie
Answer:0.60$
Step-by-step explanation:
15=9$
1=0.60$
Answer:
60 cents
Step-by-step explanation:
Which equation can be used to find the unknown length, b, in this triangle?
Answer: Choice A
4^2 + b^2 = 5^2
This is due to the pythagorean theorem.
What is the additive identity of -17?
Answer:
The additive identity is 0. The sum of any number with the additive identity is the number itself.
I HOPE ITS RIGHT IF NOT THEN SORRYHAVE A GREAT DAY :)
PLS HELP ME ON THIS QUESTION I WILL MARK YOU AS BRAINLIEST IF YOU KNOW THE ANSWER!!
B is correct
Step-by-step explanation:
At the hardware store, you buy a 63-kilogram bag of sand. You also buy 3.75 kilograms of nails. What is the total weight of your purchase?
How do I solve this???
Answer:
6^10
Step-by-step explanation:
We know that a^ -b = 1/ a^b
So 1/ a^-b = a^b
1/6^-10 = 6^10
Answer:
6¹⁰
Step-by-step explanation:
Whenever there is a negative exponent, switch its place between the vinculum (line). In this case, since there is a negative exponent in the denominator, move it up to the numerator and make the exponent positive.
Therefore, the answer is [tex]6^{10}[/tex].
Which equation represents a line which is parallel to y = 0?
Oy=3
= -8
Submit Answer
Oy=1-1
a = 4y
PLS HELP ASAP
Answer:
y = 3
Step-by-step explanation:
Both y = 3 and y = 0 have zero slopes. Therefore, they are parallel.
