Find the image of (3,-4) corresponding to y=f(x) if it undergoes the following transformations.
a) 3f(x-2)+6
b) -f(-x+4)-2
c) f(3(x+2))
To find the image of a point (3,-4) corresponding to the function y=f(x) under different transformations, we need to apply the given transformations to the coordinates of the point. Let's go through each transformation step by step:
a) 3f(x-2)+6:
To apply this transformation, we need to substitute the x-coordinate of the given point (3,-4) into the equation f(x-2), and then multiply the result by 3 and add 6.
Step 1: Substitute the x-coordinate into f(x-2):
f[(3)-2] = f(1)
Step 2: Multiply the result by 3:
3 * f(1)
Step 3: Add 6:
3f(1) + 6
Therefore, the image of (3,-4) under the transformation 3f(x-2)+6 is given by the coordinates (3f(1)+6, -4).
b) -f(-x+4)-2:
To apply this transformation, we need to substitute the x-coordinate of the given point (3,-4) into the equation -f(-x+4), and then subtract 2.
Step 1: Substitute the x-coordinate into -f(-x+4):
- f[-(3)+4] = - f(1)
Step 2: Subtract 2:
- f(1) - 2
Therefore, the image of (3,-4) under the transformation -f(-x+4)-2 is given by the coordinates (-f(1)-2, -4).
c) f(3(x+2)):
To apply this transformation, we need to multiply the x-coordinate of the given point (3,-4) by 3(x+2).
Step 1: Multiply the x-coordinate by 3(x+2):
3 * (3+2) = 3 * 5 = 15
Therefore, the image of (3,-4) under the transformation f(3(x+2)) is given by the coordinates (15, -4).
In summary:
a) The image of (3,-4) under the transformation 3f(x-2)+6 is (3f(1)+6, -4).
b) The image of (3,-4) under the transformation -f(-x+4)-2 is (-f(1)-2, -4).
c) The image of (3,-4) under the transformation f(3(x+2)) is (15, -4).