sketch the image of f(x) after:
a) a horizontal compression by a factor of 1/2
b) a veritcal expansion by a factor of 3
c) both transformations in parts a and c have been applied
d) write the equations of the three image for a b and c
i think i graphed a b and c right
but i cant seem to get the equations
i thought it was:
a) y = f(2x)
b) y = f(x-3)
c) y = f(2(x-3))
but the back of the book says
a)f(x) = (root(4-(2x-2)squared)
b) f(x) = (3(root(4-(x-2)squared)))
c) f(x) = (3(root(4-(2x-2)squared)))
I agree with you about (a). Part (b) is just y=3*f(x), and (c) is y=3*f(2x)). What you're describing in (b) is a translation of the function of 3 units to the right, not a vertical expansion of it by a factor of 3.
Having said all that, I can't see where the answers at the back of the book come from at all, unless either the function f(x) is some specific function that's given earlier in the question as opposed to a general f(x) that I think we're both assuming it is, or unless "the image of f(x)" means something other than just what f(x) is mapped onto. Does anyone else have any views on this?
To sketch the image of f(x) after different transformations, let's start with the original function f(x). Once we understand the transformations, we can then derive the equations of the transformed functions.
Given the original function f(x), we need to apply the following transformations:
a) Horizontal Compression by a factor of 1/2:
To horizontally compress f(x) by a factor of 1/2, we need to divide the x-values by 1/2 (or multiply them by 2). Therefore, the transformation equation is: f(2x).
b) Vertical Expansion by a factor of 3:
To vertically expand f(x) by a factor of 3, we need to multiply the y-values by 3. Therefore, the transformation equation is: 3f(x).
c) Both Transformations in parts a and b applied:
When both transformations are applied, first we compress horizontally by a factor of 1/2 (f(2x)), and then we vertically expand by a factor of 3 (3f(2x)). Therefore, the transformation equation is: 3f(2x).
Now, let's derive the equations of the transformed functions using the original function f(x):
a) Horizontal Compression by a factor of 1/2:
The equation for the transformed function after horizontal compression by a factor of 1/2 is: f(x) = sqrt(4 - (2x-2)^2)
b) Vertical Expansion by a factor of 3:
The equation for the transformed function after vertical expansion by a factor of 3 is: f(x) = 3 * sqrt(4 - (x-2)^2)
c) Both Transformations in parts a and b applied:
The equation for the transformed function after both transformations are applied is: f(x) = 3 * sqrt(4 - (2x-2)^2)
It seems there might be a mistake in your answer choices in the back of the book, as the correct equations for the transformations are given above.
It's important to understand that the exact form of the transformed function can vary depending on the specifics of the original function f(x), but the general approach of applying the transformation to the original function remains the same.