Question about dilations & TP form

[sinclair]

Hi, just wondering if someone could help clear up my confusion about turning-point form of quadratics.

For any function of the form f(x)=a(x-b)^2+c, “a” represents a dilation by a factor of 1/a parallel to the x-axis (relative to a function where a=1), if I understand correctly.

For f(x), a transformation to f(qx) will result in a dilation by a factor of 1/q parallel to the x-axis.

Therefore, for functions in this form, a change in the value of “a” should dilate the parabola by the same factor as a transformation by the same value of q would?

But it doesn’t seem like this is true.

e.g. for f(x)=(x-3)²+5
change a-value to 3      
f(x)=3(x-3)²+5      
=3(x²-6x+9)+5      
=3x²-18x+27+5      
=3x²-18x+32

transform by q where q=3
f(3x)=(3x-3)²+5
=9x²-18x+9+5
=9x²-18x+14

Those two functions appear to be dilated by different factors when examined graphically.

[Will Sparks]

a represents the dilation by a factor of a from the x axis, not 1/a.

At my school, we've been taught f(x) = a(bx-h)+k where a is a dilation by factor of a from the x axis, b is a dilation by a factor of 1/b from the y axis and h and k are self explanatory.

You are correct that the transformation from f(x) to f(qx) will result in the dilation by a factor of 1/q, but it is from the y axis (i.e. parallel to the y axis)
The dilation of q from the x axis in this case would be given by qf(x)

[sinclair]

"You are correct that the transformation from f(x) to f(qx) will result in the dilation by a factor of 1/q, but it is from the y axis (i.e. parallel to the y axis)
The dilation of q from the x axis in this case would be given by qf(x)"

Pretty sure that's wrong:

From http://www.mathsisfun.com/sets/function-transformations.html
"You can stretch or compress it in the y-direction by multiplying the whole function by a constant.

Scaling

g(x) = 0.35(x2)

    C > 1 stretches it
    0 < C < 1 compresses it

 
You can stretch or compress it in the x-direction by multiplying x (wherever it appears) by a constant.

Scaling

g(x) = (2x)2

    C > 1 compresses it
    0 < C < 1 stretches it

Note that (unlike for the y-direction), bigger values cause more compression."

[BubbleWrapMan]

I think he meant to say "but it is from the y axis (i.e. parallel to the x axis)"