(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​

Answers

Answer 1

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 Extrema

Note, 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 Intersections

Assume [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 AP

The 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].

(i)Express 2x 4x + 1 In The Form A(x+ B) + C And Hence State The Coordinates Of The Minimum Point, A,

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39 decillion times 1550000000

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Answer:

6.045e+43

Step-by-step explanation:

hope it helps

Answer:

3900000000000000000000000000000000 multiplied by 1550000000

Step-by-step explanation:

A decillion is a number followed by 32 zeros. This would take very long to solve, so I broke down each number. A calculator cannot accept numbers this large. You could also do 6.045e+43.

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Answer:

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Step-by-step explanation:

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Step-by-step explanation:

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Answers

Answer:

f(4) = 14

Step-by-step explanation:

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2005

Step-by-step explanation:

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Answer:

6.20

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Simon and Andrew see a speed limit sign that says 65 kilometers per hour. They decide to convert this speed limit to miles per hour. They each do the conversion separately and come up with different answers. Their work is shown. Who is correct? Explain.

Answers

Answer:

40.38912735 or about 40 miles per hour (rounded to the nearest whole number)

Step-by-step explanation:

1.) You want to convert kilometers per hour to miles per hour.

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Answers

Answer:

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Step-by-step explanation:

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x=0 solution Hope this helped!

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Answers

Answer:

The 43rd day of spending $45.

Step-by-step explanation:

Let the number of days = x.

If she spends $45 per day, then in x days she spends 45x.

She starts with $2400, so 45x is subtracted from 2400.

2400 - 45x < 500

Subtract 2400 from both sides.

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Divide both sides by -45. Remember to change the direction of the inequality sign when multiplying or dividing by a negative number.

x > 42.2

Answer: The first day is the 43rd day of spending $45.

Jeremiah makes a recipe that calls for 1 1/2 cups of flour and 3/4 stick of butter. If Jeremiah uses 3 sticks of butter, how many cups of flour will he need?

Answers

Answer:

6 cups of flour

Step-by-step explanation:

we know that

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A man (M) is standing halfway between a tree (T) and a lamppost (L). What is the value of x?
T
M
L
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Answers

Answer :

The value of x is 10.

Explanation :

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TM = (2x+8) m

ML = (3x-2) m

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The half distance is TM and half distance is TL.

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[tex]TM=ML[/tex]

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[tex]x=10[/tex]

Hence, The value of x is 10.

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I’m sorry I’m answering this to ask a question

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Step-by-step explanation:Hello! I hope this helped you out :)

What is f(5) for the function f(x)=2x+7

Answers

Answer:

Solution:

f(5)=2×5+7

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Answer:

[tex]f(5)=17[/tex]

Step-by-step explanation:

[tex]f(5)=2(5)+7\\f(5)=10+7\\f(5)=17[/tex]

Reduce the fraction 12/24 to the simplest form. Make sure that it can not be simplified any further. *
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Answers

12 12 1
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Answer is 1/2, because it cannot be simplified any further

The simplest form of the number 12/24 is,

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What is Division method?

Division method is used to distributing a group of things into equal parts. Division is just opposite of multiplications. For example, dividing 20 by 2 means splitting 20 into 2 equal groups of 10.

Given that;

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brainly.com/question/28119824

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Explanation & Process Needed

Answers

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Answers

Answer:it is a translation

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Yes

Step-by-step explanation:

Because a translation moves a shape up, down, or from side to side but it does not change its appearance in any other way. The translation is an example of a transformation. A transformation is a way of changing the size or position of a shape. Every point in the shape is translated at the same distance in the same direction.

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Answers

Answer:

x=−  

12

35

​  

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Step-by-step explanation:

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Answer:

its blurry the photo

Step-by-step explanation:

Answer:

67/24

Step-by-step explanation:

Remember PEMDAS

First you solve the parenthesis which is 11/12.

Then do exponents 2^4=16

For division do left to right so 2/3 ÷ 16 = 1/24

and the other division is 11/12 ÷ 1/3 = 11/4

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67/24 is the answer

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Answer:

x = −1/6,−3

Step-by-step explanation:

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Answer:

456789 to 3 significant figures = 457000

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Answers

We know that the area of a semicircle = 1/2(πr^2)

r = the radius

We also know that the area of a rectangle = xy

x = width

y = length

In our problem,

r = w

x = w

y = w + 5

A(w) = w(w + 5) - 1/2(πw^2)

Let's simplify the right side of the function.

A(w) = w^2 + 5w - 1/2πw^2

Trey received a $15.00 gift card for a photo center. He used it to buy prints that cost 6 cents each. The remaining balance, B (in dollars) on the card after
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B = 15.00 -0.06x
What is the remaining balance on the card if Trey bought 50 prints?

Answers

Answer

I think it's 12.00 left

Step-by-step explanation:

50 * 0.06 = 3.00

3.00 - 15.00 = 12.00

12.00

An object is traveling at a steady speed of 9 4/5 mi/h. ​ How long will it take the object to travel 2 1/5 miles? ​First round to the nearest integer to find the estimated answer. Then find the exact answer.

Answers

Answer:

Time taken by the object = 14 minutes

Step-by-step explanation:

Speed of an object = [tex]9\frac{4}{5}[/tex] miles per hour ≈ 9.8 miles per hour

Distance covered by the object = [tex]2\frac{1}{5}[/tex] miles ≈ 2.2 miles

Formula to calculate the distance in the time 't' is,

Time taken by the object = [tex]\frac{\text{Distance traveled}}{\text{Speed of the object}}[/tex]

                                          = [tex]\frac{2.2}{9.8}[/tex]

                                          = 0.2245 hours

                                          = 13.47 minutes

                                          ≈ 14 minutes

Therefore, distance traveled by the object to cover 2.2 miles = 14 minutes.

which angles are complementary? ​

Answers

Answer:

<LKM and <MKN

<PKO and <LKM

Step-by-step explanation:

<LKM and <MKN = 90°

<PKO and <LKM = 90°

What is true about the relationship between miles and gallons?
gallons 2
4
8
miles
30
60 90
120
K

Answers

Answer:

It is a linear relationship

To get the 10% discount, a shopper must spend no less than $300.

Answers

what are u trying answer

what is the smallest two digit positive integer such that the product of its two digits is one-half the integer

Answers

Let x be the tens and y be the ones digits.

The product of the number is then: 10x + y

The product is 1/2, so you need to multiply the product by 2 and the number:

10x + y = 2(x+y)

10x + y = 2x + 2y

Subtract 1 y from both sides

10x = 2x + y

Subtract 2x from

Both sides

8x = y

X has to be 1 , which would make y 8

The product number becomes 18

Since the product is 1/2, multiply by 2:

18 x 2 = 36

3 x 6 = 18

The answer is 36

HELP ME PLEASE DUE IN 5 MIN

Answers

Answer:

Step-by-step explanation:

im pretty sur ethe answer is 58... i havent done this is 3 yeras so if its wrong sorry

What is the sum of b and 4

Answers

Answer:

b+4

Step-by-step explanation: If you add b to 4, than you get B+4

If 7/35 = 4/y, what is y?

Answers

Answer:

y = 20

Step-by-step explanation:

[tex] \frac{7}{35} = \frac{4}{y} [/tex]

First of all cross multiply

That's

7y = 140

Divide both sides by 7

That's

[tex] \frac{7y}{7} = \frac{140}{7} [/tex]

We have the final answer as

y = 20

Hope this helps you

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