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

Jun 19, 2018

Start from the graph of $\frac{1}{x}$ and apply one transformation at a time, paying attention to what consequences each transformation has on the graph.

#### Explanation:

Start from the graph of $\frac{1}{x}$, and let's apply one transformation at a time to see where we're going:

• Horizontal translation: $f \left(x\right) \setminus \to f \left(x + k\right)$. 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
$\frac{1}{x} \setminus \to \frac{1}{x - 1}$

• Squaring: $f \left(x\right) \setminus \to {f}^{2} \left(x\right)$. 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 $0 \mathmr{and} 1$ smaller, and values above $1$ greater. So far, we have transformed
$\frac{1}{x - 1} \setminus \to \frac{1}{x - 1} ^ 2$

• Vertical stretch: $f \left(x\right) \setminus \to k f \left(x\right)$. 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
$\frac{1}{x - 1} \setminus \to \frac{2}{x - 1} ^ 2$

• Vertical translation: $f \left(x\right) \setminus \to f \left(x\right) + k$. 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
$\frac{2}{x - 1} ^ 2 \setminus \to \frac{2}{x - 1} ^ 2 - 3$