Hello_Kitty's Methods Unsure Questions :(

[hello_kitty]

For these questions I'm not sure what is the question asking, and how would i come about it. Any help would be great These are from the Cambridge Essential Methods Text. Sorry if it all seems a lot, just want to clear this all before school starts 2 days!!

General Queries:
- When sketching hyperbolas and truncas - it is necessary to find the x and y ints right? But what if you are unable to find either the x or y int? even when using the calculator?  i.e.  y= 1/(x-3)^2  + 2  (unable to find the x int)
- If y is underfined in the hyperbola/truncas equation, what do you do on the graph??
- How do I solve y = 3x + 5/x - 2 by hand without using the calc function propfact where the answer is 11/(x-2)  + 3
- Why in equations such as y = (squareroot) -(x-2) + 3   you only need to find the y int?
   and why in equations such as y = -2 (squareroot) x-1    + 3  - why do you only need to find the x int?
- What is actually meant by the term 'power functions' and how are these different to polynomials??
- What does it mean in questions asking 'without dividing' - i.e. without dividing find the remainder when the first polynomial is divided by the second - x^3 + 4x^2 - 5x + 1  (x + 6)    Does it mean just to solve it as P(-6) and sub x = -6??

Questions:

1) For a > b > c and P(x) = (x − a)^2(x − b)(x − c), P(x) < 0 for x ∈
Options: A(−∞, a)     B (−∞, b)    C (−∞, c)    D (c, b)    E (b, a)
Answer:  D

2) The image of the graph of y = x3 under a dilation of factor 2 from the y-axis followed by a reflection in the y-axis and then a translation of 4 units in the negative direction of the y-axis is:
Options:  A y = −x^3/8 - 4     B y = −x^3/2 − 4     C y = −8x^3 − 4     D y = −x^3/2 + 4     E y = x^3/8 + 4
Answer: A

3) The equation x3 + 5x − 10 = 0 has only one solution. This solution lies between:
Options:   A −2 and −1      B −1 and 0       C 0 and 1       D 1 and 2       E 2 and 8
Answer:  D

4) The polynomial P(x) = x^4 + ax^2 − 4 has zeros at x = (squareroot)2 and x= − (squareroot)2. The value of a is:
Options:   A 0      B 2      C −2      D −3      E 3
Answer: A

5) Find the values of a and b for which x^3 + ax^2 − 10x + b is divisible by x^2 + x − 12.
Answer:  a = 3, b = -24


Thanks again guys
hello_kitty xxx

[taiga]

First one

1) The graph of y = 2x^2 - 5x + c will have two intercepts if c is ___ ?
I tried using the discriminant, and im not sure if it is c > 25/8   or c < 25/8

The other options are c = 25/8 , c = 23, c < 25/4

So our discriminant is b^2 - 4ac = 25 -4*2*c = 25-8c

For two solutions we need the discriminant to be more than 0
hence
25-8c>0
-8c > -25
c < 25/8
we change signs because as a rule whenever we multiply or divide by a negative number we change the sign

for example 2 is more than 1 (2 > 1)
multiply by -1 and -2 is not more than -1, hence we swap the signs such that -2 < -1

Hope this explains that question, I'll try answer the other ones a bit later



Don't worry about merging the threads, just put all your questions in this thread and it will be easier

[monkeywantsabanana]

5) state the domain of the graph   4x (6-x)(4-x)
It shows the x ints are 0, 4, 6 but im not sure how to statehte domain?  would it be (negative infinity, infinity) ?

yes (-∞,∞)

[taiga]

Question 2, the 4th one isnt a polynomial.

All the powers of the x values must be positive integers (numbers like 0,1,2,3,4,5,6), so no x's within square roots! (assuming everything is in simplified form.

[hello_kitty]

Thanks Guys!
Ive deleted the ones answered

[luken93]

4) In the expansian of (x + a)^2 (x - 3), the coefficient of x^2 is 3 if a is equal to:
The options are:   -27, -3, 3, 18, -9

Expand:








If the coefficient is 3, then the bracket of


----------

6) Find the equation of the perpendicular bisector of the line segment joining (2,6) and (-3,4) in the form ax + by + d = 0, a,b,d subset of Z (im not sure if the d is suppose to be c..but i think it is a d)

Firstly, find the gradient of the line:



Thus, the perpendicular bisector has a gradient of

Secondly, the midpoint of the line segment is









Don't know if mine or yours is wrong...

[hello_kitty]

For these questions I'm not sure what is the question asking, and how would i come about it. Any help would be great These are from the Cambridge Essential Methods Text. Sorry if it all seems a lot, just want to clear this all before school starts  2 days!!

General Queries:
- When sketching hyperbolas and truncas - it is necessary to find the x and y ints right? But what if you are unable to find either the x or y int? even when using the calculator?  i.e.  y= 1/(x-3)^2  + 2  (unable to find the x int)
- If y is underfined in the hyperbola/truncas equation, what do you do on the graph??
- How do I solve y = 3x + 5/x - 2 by hand without using the calc function propfact where the answer is 11/(x-2)  + 3
- Why in equations such as y = (squareroot) -(x-2) + 3   you only need to find the y int?
   and why in equations such as y = -2 (squareroot) x-1    + 3  - why do you only need to find the x int?
- What is actually meant by the term 'power functions' and how are these different to polynomials??
- What does it mean in questions asking 'without dividing' - i.e. without dividing find the remainder when the first polynomial is divided by the second - x^3 + 4x^2 - 5x + 1  (x + 6)    Does it mean just to solve it as P(-6) and sub x = -6??

Questions:

1) For a > b > c and P(x) = (x − a)^2(x − b)(x − c), P(x) < 0 for x ∈
Options: A(−∞, a)     B (−∞, b)    C (−∞, c)    D (c, b)    E (b, a)
Answer:  D

2) The image of the graph of y = x3 under a dilation of factor 2 from the y-axis followed by a reflection in the y-axis and then a translation of 4 units in the negative direction of the y-axis is:
Options:  A y = −x^3/8 - 4     B y = −x^3/2 − 4     C y = −8x^3 − 4     D y = −x^3/2 + 4     E y = x^3/8 + 4
Answer: A

3) The equation x3 + 5x − 10 = 0 has only one solution. This solution lies between:
Options:   A −2 and −1      B −1 and 0       C 0 and 1       D 1 and 2       E 2 and 8
Answer:  D

4) The polynomial P(x) = x^4 + ax^2 − 4 has zeros at x = (squareroot)2 and x= − (squareroot)2. The value of a is:
Options:   A 0      B 2      C −2      D −3      E 3
Answer: A

5) Find the values of a and b for which x^3 + ax^2 − 10x + b is divisible by x^2 + x − 12.
Answer:  a = 3, b = -24


Thanks again guys 
hello_kitty xxx

[Greatness]

- When sketching hyperbolas and truncas - it is necessary to find the x and y ints right? But what if you are unable to find either the x or y int? even when using the calculator?  i.e.  y= 1/(x-3)^2  + 2  (unable to find the x int)
Well you can usually tell if there is or isnt a y or x intercept. If there isnt one theres no point in finding it is there. So only find the x and y intercepts if there are one

- If y is underfined in the hyperbola/truncas equation, what do you do on the graph??
What do you mean by this? If you had this: y= 1/(x-3)^2  + 2  then the horizontal asyptote is y=2 so thats where y is undefined.

- How do I solve y = 3x + 5/x - 2 by hand without using the calc function propfact where the answer is 11/(x-2)  + 3
Use long division so you get the quotient, remainder and divisor:  R/D +Q

- Why in equations such as y = (squareroot) -(x-2) + 3   you only need to find the y int?
   and why in equations such as y = -2 (squareroot) x-1    + 3  - why do you only need to find the x int?
Again just look at the transformations and do them in your head or roughly graph them first. Then you can see whether or not you need to find the intercepts.

- What is actually meant by the term 'power functions' and how are these different to polynomials??
Im not sure abot this, but i think power funtion means they are in the general form so y=(x-a)^3 + b and polynomial means ax^3 + bx^2 + cx + d. (definitely unsure abotu this lol.)

- What does it mean in questions asking 'without dividing' - i.e. without dividing find the remainder when the first polynomial is divided by the second - x^3 + 4x^2 - 5x + 1  (x + 6)    Does it mean just to solve it as P(-6) and sub x = -6??
It means without using long divisoin, so use the factor theorem.

[hello_kitty]

can anyone have a look at the uploaded file of the questions, was unable to type them up properly..
Thank you!