The question is
For each of the following transformations find the rule and the invariant points (if they exist)
One of them is: a dilation from the x axis of factor 4
I know the rule is
But for the invariant points, I know that x isn't getting moved so that is the point but the answers say
Can someone explain to me what this means?
Only if will the vertical co-ordinate of the point also not move under
since
This means that any point on the graph that lies on the x-axis
remains on the x-axis.
(think of the graph of a line... by applying the dilation you multiply the slope by 4,
but it still crosses the x-axis at the same point,
or think of a sinewave that crosses the x-axis...
after the dilation, the sinewave has 4 times the original amplitude
but crosses the x-axis at the same points as before).
Thanks for the answers! I think I understand
A transformation has rule Find the coordinates of the point which is invariant under this transformation.
So if the point is invariant then you require and
So that means the invariant points are
Is this right or wrong...
Now I'm doing composition of transformations and there's only question I don't understand.
Find the rule for a reflection in the x axis followed by a reflection in the line x=2.
So I know the first part is but the next bit reflection in the line x=2. I don't understand how to do this... Can someone help me out?
mm..
The y coordinate won't change...
But sorry I just can't get my head around this
I'm thinking if x=0 what would the rule be and then altering it to x=2, maybe +2 or -2 somewhere?
Am I getting somewhere? More clues?
If the x point is 2 units on the right of the line x=2 that means x' will be 2 units on the left of x=2 therefore it will be x-4... but if the x point is 1 unit on the right of x=2 then everything changes.
I don't know
Geometrically, to "reflect in a line", you draw the line perpendicular to the given line and mark a point on that perpendicular, on the other side of the line, at the same distance from the line.
Let (x, y) be the original point and (x', y') the reflected point. Then the point on the line, (2, y), must be midway between (x, y) and (x', y'). That means that (x+ x")/2= 2 and (y+ y')/2= y. From the first equation, x+ x'= 4 so x'= 4- x. From the second equation, y+ y'= 2y so y'= 2y- y= y. (x, y) is reflected into (4- x, y).
Ok I got it.
I'm up to Determining Transformations and the book does a horrible horrible job of explaining. HORRIBLE (well at least for me)
For example
Find the single transformation which maps to
The book tells me to assume the composition which maps
But then it says "Rearrange to make the transformation from more obvious"...
Then it somehow gets and
Where did the square on the (x+3) go?
Ahh I don't get it
Firstly, think of the point labelled as being on the curve
Any point on is mapped to a corresponding image point on a different curve.
To distinguish between points on the two curves, the points on
may be labelled
We want to find transformation equations that map and
So we can begin, knowing that the points labelled are on the image curve.
Hence write
We want to arrive at
This is of the form if and if
Hence, the transformation equations are and
You can check this....
and as the "dashes" are only used to distinguish between point co-ordinates on both curves,
the image curve is
Okok I understand. I think of it as basically making x/y equal to something in the new equation then subbing it in, sort of like reducible equations.
I have a few that I'm confused about, these ones have more than one step:
to
This is what I've done...
so...
and
So... and
The answers say: A dilation factor of 2 from the x axis followed by a translation determined by the vector (0,-3) (this should be vertical).
How can it be a dilation factor of 2 from the x axis... that is How come I get +3 and it says -3?
Also..
to
I don't rearrange the equation because I think it is fine... so..
and
Is this correct? What do I do from here
Also I just want to say how much I appreciate the help from you guys, I love you guys soo much!!