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

1 Answer
KillerBunny
Jun 19, 2018

Start from the graph of 1/x1x 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/x1x, and let's apply one transformation at a time to see where we're going:

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

  • Squaring: f(x) \to f^2(x)f(x)→f2(x). This kind of transformations preserves the zero of the function, and reflects the negative parts with respect to the xx axis, turning them positive. Also, it makes values between 0 and 10and1 smaller, and values above 11 greater. So far, we have transformed
    1/(x-1) \to 1/(x-1)^21x−1→1(x−1)2

  • Vertical stretch: f(x) \to kf(x)f(x)→kf(x). This kind of transformations stretches the graph vertically. It expands the graph if k>1k>1, and compress it if 0 < k < 10<k<1. If k<0k<0, it reflects the function with respect to the xx axis, and then apply the same logic as above. So far, we have transformed
    1/(x-1) \to 2/(x-1)^21x−1→2(x−1)2

  • Vertical translation: f(x) \to f(x)+kf(x)→f(x)+k. This kind of transformations translates the graph vertically, upwards if k>0k>0, downwards if k<0k<0. In this case, we translate three units down. Finally, we have transformed
    2/(x-1)^2 \to 2/(x-1)^2-32(x−1)2→2(x−1)2−3

Related questions
See all questions in Transformations of the Reciprocal Function
Impact of this question
3239 views around the world
You can reuse this answer
Creative Commons License