How do you describe the transformation of #f(x)=sqrt(1/2x)-4# from a common function that occurs and sketch the graph?

Question

Madelyn Anderson · Accepted Answer

We can write the given equation: # f(x)=sqrt(1/2x) -4  # In the form: # (f(x)+4) = sqrt(x/2) # We should now recognise this as the graph of the function # y=sqrt(x) #  graph{sqrt(x) [-5, 10, -8, 8]}  Which is: 1) scaled by a factor of #1/2# in the #x#-direction to give #y=sqrt(x/2)#
 graph{sqrt(x/2) [-5, 10, -8, 8]}  2) translated #4# units down to give #(y+4)=sqrt(x/2)#
 graph{sqrt(x)-4 [-5, 10, -8, 8]}

Lincoln Anderson · Answer

undefined The function $f(x) = \sqrt{\frac{1}{2}x} - 4$ can be described as a transformation of the square root function. The transformation involves a horizontal compression by a factor of 2, a reflection about the y-axis, and a vertical translation downward by 4 units.

To sketch the graph, start with the basic square root function $y = \sqrt{x}$. Apply the transformations:

1. Horizontal compression by a factor of 2: This makes the graph wider. Points on the graph move closer to the y-axis.
2. Reflection about the y-axis: This flips the graph to the left side of the y-axis.
3. Vertical translation downward by 4 units: This shifts the entire graph downward by 4 units.

Plotting these transformations on the graph of the square root function will give you the graph of $f(x)$.