rational function

What is an asymptote?

• An asymptoteasymptote is an imaginary line being approached but never touched or intersected by a graph as it goes through infinity.

Example 1: Sketch the graph of a rational function

g xx

( ) 3

2V. A : x = 2

H. A : y = 0

g (0) = - 3/2

g (3) = 3

Example 2: Sketch the graph of

f xx

x x( )

2 2V. A : x = 2, x = -1

H. A : y = 0

f(0) = 0

f(1) = - 1/2

f(-2) = - 1/2

f(3) = 3/4

The square root function is actually the inverse operation of the quadratic function.

xxf )(

x f(x)

0

1

4

9

The square root function is actually the inverse operation of the quadratic function.

xxf )(

x f(x)

0 0

1

4

9

The square root function is actually the inverse operation of the quadratic function.

xxf )(

x f(x)

0 0

1 1

4

9

The square root function is actually the inverse operation of the quadratic function.

xxf )(

x f(x)

0 0

1 1

4 2

9

The square root function is actually the inverse operation of the quadratic function.

xxf )(

x f(x)

0 0

1 1

4 2

9 3

The square root function is actually the inverse operation of the quadratic function.

xxf )(

x f(x)

0 0

1 1

4 2

9 3

xxf )(

x f(x)

0

1

4

9

The square root function is actually the inverse operation of the quadratic function.

xxf )(

x f(x)

0 0

1 1

4 2

9 3

xxf )(

x f(x)

0 0

1

4

9

The square root function is actually the inverse operation of the quadratic function.

xxf )(

x f(x)

0 0

1 1

4 2

9 3

xxf )(

x f(x)

0 0

1 -1

4

9

The square root function is actually the inverse operation of the quadratic function.

xxf )(

x f(x)

0 0

1 1

4 2

9 3

xxf )(

x f(x)

0 0

1 -1

4 -2

9

The square root function is actually the inverse operation of the quadratic function.

xxf )(

x f(x)

0 0

1 1

4 2

9 3

xxf )(

x f(x)

0 0

1 -1

4 -2

9 -3

Remember that it is not possible to get the square root of a negative number, therefore the values that are chosen for x in the table of values must be strategic.

xxf )(

x f(x)0 0-1 Error-4 Error-9 Error

Error

x f(x)0 01 12 1.43 2.8

It is possible however, to use non-perfect squares with this function. You will end uo with an irrational y-value that needs to be rounded off but that’s okay. It just makes it a little more awkward to plot on the graph.

xxf )(

If we play with parameter ‘a’, we can see its effect on the resulting graph.

xxf )( a = 1xxf 2)( a = 2

21

;21

)( axxf xxf )( a = -1

If we play with parameter ‘b’, we can see its effect on the resulting graph.

xxf )( a = 1 xxf 2)( b = 2

21

;21

)( bxxf xxf )( b = -1

The signs of parameters a and b determine which way the square root function moves from its vertex.

xxf )( a = +; b = +

xxf )( a = -; b = +

xxf )( a = -; b = -xxf )( a = +; b = -

THANK YOU !!!

---Ms. Jerlyn Fernandez