How do I find the graph of #y=2/(x-1)^2-3#?

1 Answer
Jun 19, 2018

Answer:

Start from the graph of #1/x# and apply one transformation at a time, paying attention to what consequences each transformation has on the graph.

Explanation:

Start from the graph of #1/x#, and let's apply one transformation at a time to see where we're going:

  • Horizontal translation: #f(x) \to f(x+k)#. Any transformation like this will shift the graph horizontally, to the left if #k>0#, to the right if #k<0#. In this case, we are translating one unit right, because we are transforming
    #1/x \to 1/(x-1)#

  • Squaring: #f(x) \to f^2(x)#. This kind of transformations preserves the zero of the function, and reflects the negative parts with respect to the #x# axis, turning them positive. Also, it makes values between #0 and 1# smaller, and values above #1# greater. So far, we have transformed
    #1/(x-1) \to 1/(x-1)^2#

  • Vertical stretch: #f(x) \to kf(x)#. This kind of transformations stretches the graph vertically. It expands the graph if #k>1#, and compress it if #0 < k < 1#. If #k<0#, it reflects the function with respect to the #x# axis, and then apply the same logic as above. So far, we have transformed
    #1/(x-1) \to 2/(x-1)^2#

  • Vertical translation: #f(x) \to f(x)+k#. This kind of transformations translates the graph vertically, upwards if #k>0#, downwards if #k<0#. In this case, we translate three units down. Finally, we have transformed
    #2/(x-1)^2 \to 2/(x-1)^2-3#