(i)Express 2x² – 4x + 1 in the form a(x+ b)² + c and hence state the coordinates of the minimum point, A, on the curve y= 2x² 4x+ 1.
The line x– y + 4 = 0 intersects the curve y= 2x² – 4x + 1 at points P and Q. It is given that the coordinates of Pare (3,7).
(ii)Find the coordinates of
(iii) Find the equation of the line joining Q to the mid-point of AP.
I already get the (i) but hav no idea with no.(ii) and (iii)
Can anyone help me? Thank you
Answer:
(i). [tex]y = 2\, x^2 - 4\, x + 1 = 2\, (x - 1)^2 - 1[/tex]. Point [tex]A[/tex] is at [tex](1, \, -1)[/tex].
(ii). Point [tex]Q[/tex] is at [tex]\displaystyle \left(-\frac{1}{2},\, \frac{7}{2}\right)[/tex].
(iii). [tex]\displaystyle y= - \frac{1}{5}\, x + \frac{17}{5}[/tex] (slope-intercept form) or equivalently [tex]x + 5\, y - 17 = 0[/tex] (standard form.)
Step-by-step explanation:
Coordinates of the ExtremaNote, that when [tex]a(x + b)^2 + c[/tex] is expanded, the expression would become [tex]a\, x^2 + 2\, a\, b\, x + a\, b^2 + c[/tex].
Compare this expression to the original [tex]2\, x^2 - 4\, x + 1[/tex]. In particular, try to match the coefficients of the [tex]x^2[/tex] terms and the [tex]x[/tex] terms, as well as the constant terms.
For the [tex]x^2[/tex] coefficients: [tex]a = 2[/tex].For the [tex]x[/tex] coefficients: [tex]2\, a\, b = - 4[/tex]. Since [tex]a = 2[/tex], solving for [tex]b[/tex] gives [tex]b = -1[/tex].For the constant terms: [tex]a \, b^2 + c = 1[/tex]. Since [tex]a = 2[/tex] and [tex]b = -1[/tex], solving for [tex]c[/tex] gives [tex]c =-1[/tex].Hence, the original expression for the parabola is equivalent to [tex]y = 2\, (x - 1)^2 - 1[/tex].
For a parabola in the vertex form [tex]y = a\, (x + b)^2 + c[/tex], the vertex (which, depending on [tex]a[/tex], can either be a minimum or a maximum,) would be [tex](-b,\, c)[/tex]. For this parabola, that point would be [tex](1,\, -1)[/tex].
Coordinates of the Two IntersectionsAssume [tex](m,\, n)[/tex] is an intersection of the graphs of the two functions [tex]y = 2\, x^2- 4\, x + 1[/tex] and [tex]x -y + 4 = 0[/tex]. Setting [tex]x[/tex] to [tex]m[/tex], and [tex]y[/tex] to [tex]n[/tex] should make sure that both equations still hold. That is:
[tex]\displaystyle \left\lbrace \begin{aligned}& n = 2\, m^2 - 4\, m + 1 \\ & m - n + 4 = 0\end{aligned}\right.[/tex].
Take the sum of these two equations to eliminate the variable [tex]n[/tex]:
[tex]n + (m - n + 4) = 2\, m^2 - 4\, m + 1[/tex].
Simplify and solve for [tex]m[/tex]:
[tex]2\, m^2 - 5\, m -3 = 0[/tex].
[tex](2\, m + 1)\, (m - 3) = 0[/tex].
There are two possible solutions: [tex]m = -1/2[/tex] and [tex]m = 3[/tex]. For each possible [tex]m[/tex], substitute back to either of the two equations to find the value of [tex]n[/tex].
[tex]\displaystyle m = -\frac{1}{2}[/tex] corresponds to [tex]n = \displaystyle \frac{7}{2}[/tex]. [tex]m = 3[/tex] corresponds to [tex]n = 7[/tex].Hence, the two intersections are at [tex]\displaystyle \left(-\frac{1}{2},\, \frac{7}{2}\right)[/tex] and [tex](3,\, 7)[/tex], respectively.
Line Joining Point Q and the Midpoint of Segment APThe coordinates of point [tex]A[/tex] and point [tex]P[/tex] each have two components.
For point [tex]A[/tex], the [tex]x[/tex]-component is [tex]1[/tex] while the [tex]y[/tex]-component is [tex](-1)[/tex].For point [tex]P[/tex], the [tex]x[/tex]-component is [tex]3[/tex] while the [tex]y[/tex]-component is [tex]7[/tex].Let [tex]M[/tex] denote the midpoint of segment [tex]AP[/tex]. The [tex]x[/tex]-component of point [tex]M[/tex] would be [tex](1 + 3) / 2 = 2[/tex], the average of the [tex]x[/tex]-components of point [tex]A[/tex] and point [tex]P[/tex].
Similarly, the [tex]y[/tex]-component of point [tex]M[/tex] would be [tex]((-1) + 7) / 2 = 3[/tex], the average of the [tex]y\![/tex]-components of point [tex]A[/tex] and point [tex]P[/tex].
Hence, the midpoint of segment [tex]AP[/tex] would be at [tex](2,\, 3)[/tex].
The slope of the line joining [tex]\displaystyle \left(-\frac{1}{2},\, \frac{7}{2}\right)[/tex] (the coordinates of point [tex]Q[/tex]) and [tex](2,\, 3)[/tex] (the midpoint of segment [tex]AP[/tex]) would be:
[tex]\displaystyle \frac{\text{Change in $y$}}{\text{Change in $x$}} = \frac{3 - (7/2)}{2 - (-1/2)} = \frac{1}{5}[/tex].
Point [tex](2,\, 3)[/tex] (the midpoint of segment [tex]AP[/tex]) is a point on that line. The point-slope form of this line would be:
[tex]\displaystyle \left( y - \frac{7}{2}\right) = \frac{1}{5}\, \left(x - \frac{1}{2} \right)[/tex].
Rearrange to obtain the slope-intercept form, as well as the standard form of this line:
[tex]\displaystyle y= - \frac{1}{5}\, x + \frac{17}{5}[/tex].
[tex]x + 5\, y - 17 = 0[/tex].
The sum of five consecutive integers is five. What is the product of the five integers?
The first integer = x
The second integer = x + 1
The third integer = x + 2
The fourth integer = x + 3
The fifth integer = x + 4
Since they all add up to 5, let's set them equal to 5
x + x + 1 + x + 2 + x + 3 + x + 4 = 5
Combine like terms
5x + 10 = 5
Subtract 10 from both sides
5x = -5
Divide each side by 5
x = -1
The first integer = x = -1
The second integer = x + 1 = -1 + 1 = 0
The third integer = x + 2 = -1 + 2 = 1
The fourth integer = x + 3 = -1 + 3 = 2
The fifth integer = x + 4 = -1 + 4 = 3
We can check this by adding all the integers together and making sure we get 5
3 + 2 + 1 + 0 - 1 = 5
Answer:
0
Step-by-step explanation:
If the sum of the integers is 5, then the arithmetic mean of the integers is [tex]$\frac{5}{5}=1$[/tex] . In five consecutive integers, the arithmetic mean is equal to the middle integer, which means the integers are -1, 0, 1, 2, 3 and the product is [tex]$\boxed{0}$[/tex].
Hope this helped! :)
Yes I am kinda dumb lol
Answer:
I think it is A
Step-by-step explanation:
just trying to help
Answer: The answer is A
Step-by-step explanation: The other answers show an impossible answer. The rays cannot be crossed from A to T or C to T since they are on the other side of the ray.
( I hope this helps and I hope you understand it doesn’t make you any less smart to have difficulty with answering a question)
(I'm not 100% confident in the answer though)
2 times the square root of 25 plus the square root of negative 16
Answer:
10 + 4 i
fnjeob cbjkocbhv bjwokjnb jnokemnef
Answer:
10+4i
Step-by-step explanation:
2*√25 + √-16.
Given
2*5+4i
Simplify the sqrts. If you are wondering where I got "i" from, "i" is √-1. "i" is short for imaginary number because there is no real number that can be a square root of a negative number. I got the 4 because it is the square root of -16
10+4i
I multiplied the 2 and the 5. This is the simplest form because you cannot add imaginary and real numbers together.
What is the mean of the data?O 5
O 10
O 30
O 50
Answer:
D. 50
Finding out the mean is actually pretty simple. All you have to do is add up the numbers, then divide that by how many numbers there are. For this graph, here's what you do:
35 + 40 + 45 + 50 + 55 + 60 + 65 = 350
Since there are seven numbers, you divide 350 by 7.
350 ÷ 7 = 50
And that's your answer! hope this helps
Is Happyville linear or exponential
Answer:
expoential
Step-by-step explanation:
a) The radius of a circular field is 42 m. Find the perimeter of the field. Also find the length of wire required to fence it with 5 rounds
263.89
263.89*5=1319.45
We are given with the radius of the circular field,
[tex]r = 42 \: m[/tex]
And, we have to find the perimeter of the field. Also, we have to find the length of the wire required to fence it with 5 rounds:
We know,
Perimeter of a circle = 2πr, where r is the radius of the circle, So let's find the perimeter by using this formula.
[tex]perimeter = 2\pi r[/tex]
[tex]perimeter = 2 \times \dfrac{22}{7} \times \: 42 m[/tex]
[tex]perimeter = 2 \times 22 \times 6 \: m[/tex]
[tex]perimeter = \boxed{264 \: m}[/tex]
Now, finding the length of the wire which is equals to 5 × Perimeter of the circular field.
[tex]length \: of \: wire = 264 \: m \times 5[/tex]
[tex]length \: of \: wire = \boxed{1320 \: m}[/tex]
And we are done !!
#CarryOnLearning.
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what is 1/4 to the power of 34???????
Answer:
I believe it is 3.3881318e-21 but you could round that
Step-by-step explanation:
Please help me ASAP
Answer:
Step-by-step explanation:
Base 5 uses 0, 1 , 2, 3, 4
When we add 2+ 1 = 3
When we add 4 +3 = 7 = 5 +2 . 5 will be carried as 1
4 + 3= 12
In subtraction, 32 - 4
From 2 we cannot subtract 4. So, borrow 5 from 3. So, 2+5 = 7.
7-4=3
32 -4 = 23
I have attached the sums
The graph of the ellipse below (passes/does not pass) the______ so the graph (is/is not) that of a function. please help fast
Answer:
does not pass the vertical line test so the graph is not a function
Step-by-step explanation:
evaluate \frac{6x^3}{8x^2}
A model of a plane is made using a scale of 1:5. If the wings of the model are 1.2m long, what is the actual length of the wings on the plane?
Answer:
6 m
Step-by-step explanation:
Multiply your scale by 1.2 m:
1 : 5 = 1.2 m : 6.0 m
The wings on the plane are 6.0 m long.
Solve 2.5 = 6
(answer should start with x=)
Answer: x=3.5
Step-by-step explanation:
Solve for x in simplest form. 3=7/2(3x−4)
Answer:
x = 34/21
Step-by-step explanation:
3 = 7/2 (3x-4)3 = (21/2)x - 28/23 = (21/2)x - 14(21/2)x = 1721x = 34x = 34/21Answer:
x = 34/21
Step-by-step explanation:
3 = 7/2 ( 3 x -4)
We need to perform a multiplication.
The following rule is applied:
In our example, the factors in the new numerator are:
, ,
Notice that all non-fraction factors are placed in the numerator.
The only factor in the new denominator is: 2.
We need to get rid of all the denominators in this equation.
This can be achieved by multiplying both the left and the right side by the Least Common Denominator.
In our example, the LCD is equal to 2.
-5x-y=7 how do I show my work
Step-by-step explanation:
show yourself simplifying into y=mx+b format
Which operation gives a result of 1.39?
Please answer the equation in the attachment.
Answer:
the missing value is 9.33.
Step-by-step explanation:
given,
length (l) = 3x+2
breadth (b) = 4
perimeter(P) = 68 cm
now,
P = 2(l+b)
or, 68 = 2× ( 3x+2+4)
or, 68 = 2× ( 3x+ 6)
or, 68 = 6x+12
or, 68 - 12= 6x
or, 56 = 6x
or, 6x = 56
or, x = 56/6
or, x = 9.33
Find the ratio of the partitioned segments such that C partitions the segment AB where A(-8, -4), B(6, 10), and C(18, 22)
Answer:
how is your day
Step-by-step explanation:
What essential characteristic does a good experiment have?
Answer:
It should have many essential characteristics such as a Control group, a hypothesis, research to back up and defend your hypothesis/claim and it should be replicable by others with similar results to prove your experiments claim.
Hope this helps. :)
Is the ration of 3/4 the same as the fraction of 3/4
Answer:
yes but there might be slightly change in it
Answer:
Yes 3/4 is the same as 3/4.
(I might be wrong)
Step-by-step explanation:
Step One: Rewrite the ratio as a fraction
[tex]3 : 4 = \frac{3}{4}[/tex]
Step Two: Reduce and simplify the fraction
This fraction is already reduced so no further work is needed
Hope this helped you!
which number is a prime 49 51 53 55
53 is a prime number !!
Answer:
53
Step-by-step explanation:
Answer is 53:)
Find the domain and range of f(x)=|x|+ cos x
I need help ASAP
Answer:
Domain All real x.
Range f(x) >= 1.
Step-by-step explanation:
The domain is all real numbers.
|x| is always positive.
At x = 0:
f(x) = |0| + cos 0
= 1 which is the minimum value.
Marlene brought a pitcher filled with 10 cups of water to her volleyball game. The paper cups she brought hold Two-thirds of a cup of water each. How many paper cups can Marlene fill with water?
Which math sentence represents the problem?
Two-thirds divided by 10 = 20/30=1/15
10 divided by two-thirds = 30/2=15
10 divided by three-halves = 20/3] 6 and two-thirds
1/10 divided by two-thirds = 3/20
Marlene needs 15 paper cups to fill with water
An equation is an expression that is used to show the relationship between two or more variables and numbers.
Since each paper cups can hold Two-thirds of a cup of water, hence to hold 10 cups of water:
Number of water cups = 10 cups / (two- thirds) = 10 * 3 / 2 = 15
Marlene needs 15 paper cups to fill with water.
Find out more on equation at: https://brainly.com/question/2972832
Answer: The answer is B
Step-by-step explanation: got it right on e2020
what is this statement Segment AB is congruent to segment AB.
Answer:
reflective
Step-by-step explanation:
This statement shows the reflective property.
Each day for 6 days, the cooks at a school prepared
298 meals for lunch. How many meals did the cooks
prepare in the 6 days?
A
1,748 meals
Nath
B
1,788 meals
2.288 meals
D
596 meals
Answer:
B
Step-by-step explanation:
Because 298 time 6 is 1,788 :)
What set does negative 4 belong to
Irrational numbers
Rational numbers
Real numbers
Whole numbers
Integers
2/3x+2= 2 1/3n
A. 0
B. 1/2
C. -2
D. 2
Answer:
A
Step-by-step explanation:
Does anyone know the answer to this? |21| - |-9|
Answer:
12
Step-by-step explanation:
21-9
12
Answer:
The answer is 12
Step-by-step explanation:
|21| = just 21
|-9| = positive 9
so take that into this:
21 - 9 = 12
An experienced computer technician can set up a network in 12 hours. It would take an intern 20 hours to set up the same network. If they work together, how long will it take to set up the network?
Answer:
X
Step-by-step explanation:
Answer:
ANSWER IS TIME
Step-by-step explanation:
time
10+x=19 need your help
Answer:
9
Step-by-step explanation:
10+x=19
move 10 over the =
x =19-10
x=9
Hey there!
ANSWER: [tex]x=9[/tex]EXPLANATION:[tex]10+x=19[/tex]
To solve, you can just do 19 - 10.
[tex]19-10=9[/tex]
Now let's complete x.
[tex]10+x(9)=19\\10+9=19\\x=9(ANSWER)[/tex]
Hope this helps!
[tex]\text {-TestedHyperr}[/tex]