We start from the parent function
[tex]y=\dfrac{1}{x}[/tex]
If we transform [tex]x\mapsto x-5[/tex] we have
[tex]y=\dfrac{1}{x-5}[/tex]
Every transformation of the form [tex]f(x)\mapsto f(x+k)[/tex] translates the function horizontally, k units to the left if k is positive, k units to the right if k is negative.
So, with this first step, we translate the graph 5 units to the right.
Then, we can transform the function by multiplying it by one half:
[tex]y=\dfrac{1}{x-5}\cdot \dfrac{1}{2} = \dfrac{1}{2(x-5)}=\dfrac{1}{2x-10}[/tex]
Every transformation of the form [tex]f(x)\mapsto kf(x)[/tex] scales the function vertically. The function is squeezed by a factor k if k is between 0 and 1, and it is stretched by a factor k if k is greater than 1.
So, with this second step, we squeeze the function vertically with a factor of 1/2.
Finally, we can transform the function by subtracting three:
[tex]\dfrac{1}{2x-10}-3[/tex]
Every transformation of the form [tex]f(x)\mapsto f(x)+k[/tex] translates the function vertically, k units up if k is positive, k units down if k is negative.
So, with the last step, we translate the graph 3 units down.
To recap: starting with the graph of the parent function, you translate it 5 units to the right, scale it vertically by a factor of 1/2, and then move it down 3 units.
