Describe how the graph of each of the following functions can be obtained from the graph of y = f(x). a. y = -2f(x – 1) + 3 b. y = 2f(-3x + 12)
First, lets consider the general case: `y=af(b(x-h))+k` with parent function `y=f(x)` .
a: This parameter performs a vertical dilation/compression (stretch or shrink vertically). Also, if a<0, then the parent function is reflected across the horizontal axis.(If the function is periodic, a affects the amplitude)
b: This parameter performs a horizontal dilation/compression. In addition, if b<0 the parent function is reflected across the vertical axis.(If the function is periodic, b affects the period and frequency)
h: This parameter performs a horizontal translation (shifts the function left/right)
k: This parameter performs a vertical translation (shifts up/down)
(1) `y=-2f(x-1)+3` . Here a=-2,b=1,h=1,k=3.
Thus the parent function f(x) is reflected across the x-axis, stretched by a factor of 2, shifted right 1 unit, and shifted up 3 units.
Example: If `y=x^2` is the parent function then `y=-2(x-1)^2+3` is the transformed function. The original is a parabola with vertex (0,0) and opening up. The transformed graph is a parabola, vertex at (1,3), opening down. Also the transformed graph is stretched vertically -- the original points (1,1) and (2,4) get mapped to (2,1) and (3,-5) respectively.
(2) `y=-2f(-3x+12)` or `y=-2f(-3(x+4))`
The parent function is reflected across the x-axis and streched vertically by a factor of 2. Then the graph is reflected over the y-axis, and compressed horizontally by a factor of 3. The resulting graph is then shifted 4 units left.
** Note that the vertical dilation is a stretch if |a|>1 and a compression if |a|<1; but the horizontal dilation is a compression if |b|>1, and a stretch if |b|<1. See link below to try an interactive version to see these transformations. **