Math Is Fun Forum

Could you please explain what dilation, reflection, translation are in transformations.

There are questions like:

y = 2(x+1)^3 - 6

1) Find the stationary point of inflection
2)State the transformation that the graph y=x^3 has undergone to produce the given graph
3) State the transformation(s) that the graph y=x^3 has undergone to produce the given graph.

SO basically what is a stationary point of inflection??

Then there is the HYPERBOLA
There are questions like: Sketch the graph of
y= (2/x+2) - 4
clearly showing the intercepts with the axes and the position of the asymptotes.

SO how do we find out the asymptotes?

y = (2/5) + (4/1+x)
Also find the asymptotes and the intercepts with the axes.

Then there is the TRUNCUS...

1. y = 4 - ( 2/(3+x) ^2)

Clearly state the position of the asymptotes and the intercepts with the axes (correct to 1 decimal place where appropriate)

Then there is the SQUARE ROOT FUNCTION IN POWER FORM

y = -3/5 √3x-4+2

After this there's the absoolute value function, transformations with matrices, sum difference and prooduct functions, composite functions and functional equations AND MODELLING)...

But right now I must know the way to the questions above. I know it might be long but the teacher wants us to know these urgently as he is going to proceed to more advanced questions. Please could you explain the basic principles and the way to do these. There's so much more questions but I will need to know these. Like the Domain, Range, Point of inflection, x-y intercepts, transformations.... There's a lot of questions that also need rearranging which I have no clue to how to do it. I know it need basic algebra for some but I don't have time to review, I need to learn all at once... So it's like algebra and advanced algebra at once. So please help me out (urgently)

1)
Solve 2log10(x-3) = 10
2log10(x-3) = 10
log10 (x-3) = 5
x-3 = 10^5 = 100,000
x=100,003

2)
Solve inequation 8^(n-2) > 2048 for n
2^(3n-6) > 2^11
3n - 6 > 11
3n > 17
17/3 < n

3)
If 0<θ<π/2 then sin(π-θ) =
Sin (π-θ) = sinπcosθ-cosπsinθ =
0xcosθ-(-1)sinθ= sinθ

4)
function y=-3cos(2x) has amplitude and period respectively by:
period: pi , Amplitude: 3

5)
Given cos(θ) = 2/3 and 3π/2≤θ≤2π, then tan(θ) =
cos(θ)= 2/3 = sec^2(10)= 9/3 = tan^2(θ)=sec^2(θ)-1=5/4
As(θ) is in 4'th quad. tan(θ) is -ve
= tan(θ) = -√5/2

6)
If tanθ= 7/24 where 0<θ<π/2 then find exact values of sinθ, cosθ and sinθ (π+θ)
Since tanθ = 7/24, the side opposite θ is 7 and the side adjacent to θ is 24. The hypotenuse is: √(7^2 + 24^2) = √625 = 25
Sinθ = 7/25, cosθ = 24/25, sin(π+θ) = -7/25

7)
Solve following equation for x over domain,
2cosx + √3 = 0, 0≤θ≤360°
2cos(x) + sqrt(3) = 0
2cos(x) = -sqrt (3)
cos(x) = -sqrt (3)/2
cos(30deg) = sqrt(3)/2 so 30 degrees is reference angle.
cos(x) is negative in Quad 2 and Quad 3.
So 30 degrees in second and third quadrants.
In Quad 2, 180-30 = 150
In Quad 3, 180+3 = 210

8)
Maximal domain for functions with rule g(x) = loge(x-2) =
x-2>0
x>0

9)
When simplified and expressed with positive powers, (2a^2 b^3)(^-1)/b3
1/(2a^2b^6)

10)
The range and amplitude respectively of function of y=-2sin(x)+2
amp = 2, range of y is 0≤y≤4

11)
And.. Could you please do this for me:
9^2x x 27^x
--------------- =? (Multiple choice: 1, 0, 3^2x, 3^x, 3^-x)
3^7x

Thank you.