Rational Functions Transformations and Characteristics

a = 1

h = 3

k= -2

a = 1

h = -3

k = -2

a = 1

h = 3

k = 2

a = 1

h = -3

k = 2

a = 1

h = -3

k = 5

a = 1

h = 5

k = -3

a = -1

h = -3

k = 5

a = -1

h = 5

k = -3

The graph is a reflection of the parent funtion $$y=\frac{1}{x}$$

An equation for the function is $$y=\frac{-1}{x-1}$$

There is a vertical asymptote at

$$x=-1$$

There is a horizontal asymptote at $$y=0$$

The graph is reflected vertically

The graph is shifted left 1

The graph is shifted down 3

The graph is stretched vertically

The domain is $$\left(-\infty,\ -8\right)\cup\ \left(-8,\ \infty\right)$$

The range is $$\left(-\infty,\ 6\right)\cup\ \left(6,\ \infty\right)$$

The graph is stetched vertically

There is a vertical asymptote at $$x=8$$

a = 1

h = -1

k = 2

$$y=\frac{1}{x+1}+2$$

The graph has been shifted up 2 from the parent function

The range is $$\left(-\infty,\ -1\right)\cup\ \left(-1,\ \infty\right)$$

$$y=\frac{1}{x+4}+2$$

$$y=\frac{1}{x-4}+2$$

$$y=\frac{1}{x+2}+4$$

$$y=\frac{1}{x-2}+4$$

There is a vertical asymptote at $$x=3$$

a = -2

h = -3

k = -1

The range is $$\left(-\infty,\ -1\right)\cup\ \left(-1,\ \infty\right)$$

$$y=\frac{1}{x+3}-1$$

$$y=\frac{1}{x-3}-1$$

$$y=-\frac{1}{x+3}-1$$

$$y=-\frac{1}{x-3}-1$$