Melody

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UsernameMelody
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 #8
avatar+91780 
+1

I am quite sure EP will be correct but I will take a look:

 

Let b and c be constants such that the quadratic \(-2x^2 +bx +c\)   

has roots \(3+\sqrt{5}\)  and \(3-\sqrt{5}\).

Find the vertex of the graph of the equation   

\(y=-2x^2 + bx + c\)

 

The roots of a quadratic   \( ax^2+bx+c\)  is the answers to   \( ax^2+bx+c=0\)

And the answers to this are given by the quadratic equation

 

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\\ x=\frac{-b}{2a}\pm\frac{\sqrt{b^2-4ac}}{2a}=3\pm\sqrt5 \)

 

This means that  \(x=\frac{-b}{2a}\) = 3      must be exactly halfway between the two roots.!!

 

So the axis of symmetry is x=3 and the vertex lies on this line so the x value of the vertex is 3

The y value will be    \(y=-2*3^2+bx+c = -18+3b+c \)

 

This is fine but you need to find the vleu of b and c       frown

 

From the equation above I can see that 

\(\frac{-b}{2a}=3 \qquad and \qquad \pm\frac{\sqrt{b^2-4ac}}{2a}=\pm\sqrt5\\ a=-2\qquad so\\ \frac{-b}{-4}=3 \qquad and \qquad \pm\frac{\sqrt{b^2+8c}}{-4}=\pm\sqrt5\\ b=12 \qquad and \qquad \pm\frac{\sqrt{b^2+8c}}{4}=\pm\sqrt5\\ b=12 \qquad and \qquad \frac{b^2+8c}{16}=5\\ b=12 \qquad and \qquad b^2+8c=80\\ b=12 \qquad and \qquad 144+8c=80\\ b=12 \qquad and \qquad 18+c=10\\ b=12 \qquad and \qquad c=-8\\\)

 

so

 

\(y= -18+3b+c\\ y= -18+3*12+-8\\ y=-18+36-8\\ y=10 \)

 

So the vertex is   (3,10)

 

Which is exactly the same answer has given you. He has done it a number of different ways but stlill all the answers are the same!!

 

Thanks EP :)

Melody 1 hour ago
 #3
avatar+91780 
0
Melody Feb 17, 2018
 #1
avatar+91780 
+2

1.A telegraph has x arms and each arm is capable of (x-1) distinct positions , including the position of rest. The total no. of signals that can be made is?

Rosala, This is just      x(x-1)

 

 

2. How many natural numbers are there fro 1 to 1000 which have none of their digits repeated?

i've tried this question a no. of times but it just doesnt see to be clicked in my mind. i would be really appreciate if anyone can explain it to me in detail.

There are 9 one digit ones

There are no four digit ones

So how many 2 digit ones are there.

The tens digit can be 1 to 9 that is 9 choices, now you used one digit but you can use the zeros so there are 9 possible digits left for the units digit

So that is 9*9=81

AND how many three digit ones are then  9*9*8 = 648

Altogether there are  9+81+648 = 738

 

3. Number of different natural numbers which are smaller than two hundred million and using only the digits 1 or 2 is :

 

less than      200,000,000   only containing the digits 1 and 2

 

1 digit     2

2 digit     2*2=2^2

3 digit      2^3

4 digit       2^4

5 digit      2^5

6 digit      2^6

7 digit      2^7

8 digit      2^8

9 digit      2^8       The biggest value digit must be 1

 

So what do we get when we add these up

 

\(2+2^2+2^3+2^4+2^5+2^6+2^7+2^8+2^8\\ =(2+2^2+2^3+2^4+2^5+2^6+2^7+2^8)+2^8\\ \qquad \text{The brackets is the sum of a GP, a=2 and r=2\;\;n=8}\\ =\frac{a(r^n-1)}{r-1}\;\;\;+2^8\\ =\frac{2(2^8-1)}{2-1}\;\;\;+2^8\\ =2^9-2+2^8\\ =2^9+2^8-2\\ =2^8(2+1)-2\\ =3\times 2^8-2 \)

 

 

i. (3).2^8 - 2 

 

ii. (3).2^8 - 1 

 

iii. 2 (2^9 - 1) 

 

iv) None 

 

I have a few more question like the questions above, and i just dont seem to get the correct answers. 

 

i have tried the questions above many times but i just cant understand a few things , i would be glad if anyone could explain to me in detail. Thank you very much. 

 

i really appreciate the time you'd take out for my answers! 

Melody Feb 17, 2018
 #2
 #2
avatar+91780 
0

Hi all,

Supermanaccz has asked me to come back an talk about this ancient question.

Thanks for your interest Supermanaccz    laugh

 

Firstly the question is not displaying properly because the coding used on this site has changed a lot in the past 3 and a bit years.

So I will try to put the question up how it may originally have been. I believe a diagram like the one below would have been included in the original question.  Thie is the graph of   y=f(x).

I have drawn it in a program called GeoGebra. GeoGebra is a great tool. I found it a little hard to use at first (especially for more complicated things) but if you persist with it it will become much easier. Geogebra is a free download. I use it often.

 

Now to the actual question:

 

The graph of  is shown below. Assume the domain of  is  [-4,4] and that the vertical spacing of grid lines is the same as the horizontal spacing of grid lines. 

Part (a): The points  (a, 4) and  (b, -4)  are on the graph of  y=f(2x)    Find  a  and  b

From the graph you can see that 2 points on the f(x)  graph are (-4,4) and (4,-4)

-------------------------------------

Here is what i said before:

Part a       y=f(2a)

f(-4)=4   but    f(2a)=-4    so   2a=-4     so   a=-2

f(4)=-4   but    f(2b)=4     so    2b=4      so    b=2

----------------------------------------

But in the next post I will do an entirley different example, perhaps you will be able to see how many graphs are related to parent graphs. In this case f(x) is the parent function and f(2x) it the 'offspring'  Multiplying the x by 2 will cause the graph to have half the width (from the y axis). 

Part (b): Find the graph of  y=2x         Verify that your points from part (a) are on the graph. 

Part (c): The points (c,4)  and (d,-4)  are on the graph of y=f(2x-8)     Find  c   and  d

Part (d): Find the graph of  y=f(2x-8)   Be sure to verify that your points from part (c) are on the graph both algebraically and geometrically. 

 

Melody Feb 15, 2018