Rational Function Characteristics

$$f\left(x\right)=-\frac{1}{x-4}+1$$

$$f\left(x\right)=\frac{1}{x-4}+1$$

$$f\left(x\right)=-\frac{1}{x+4}+1$$

$$f\left(x\right)=\frac{1}{x+4}+1$$

x=1

x=-3

y=1

y= -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 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 $$x=8$$

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

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

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

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

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 $$\left(-\infty,\ -1\right)\cup\ \left(-1,\ \infty\right)$$

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

a = -2

h = -3

k = -1

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

x = 2

y = 2

x =-2

y=-2

x = 5

x = 0

x =5 , x = -5

x = 25

y = 12

y = x - 5

y = x-5 + 12/(x+2)

y = 12/(x+2)