2.5 Reciprocal & Rational Functions

Consider  , for , where .

  and    are the equations of the asymptotes of the graph of . Point   lies on the graph.

Find the values of  and .

Consider the function  defined by  ,   ,   .

Write down the value of .

Write down the equation of the horizontal asymptote to the graph of .

Show that ,  where  and  are constants to be determined.

Sketch the graph of .

Let , for .

For the graph of , find the coordinates of

the -intercept

the -intercept.

For the graph of , find the equation of

the vertical asymptote

Consider the function  defined by   ,  ,   .

Find the coordinates of the points where the graph of   intersects the coordinate axes.

Express as partial fractions.

Hence find the equation of the horizontal asymptote to the graph of .

Consider the function    ,  , 

Find the coordinates of the points where the graph of   intersects the

-axis
 

-axis.

Write down the equations of

to the graph of .

By considering the value of for large positive and large negative values of , sketch the graph of .  Be sure to indicate clearly the points of intersection with the coordinate axes, as well as any asymptotes.

Consider the function ,  ,  .

Find the coordinates of the points where the graph of  intersects the

-axis,

-axis.

Write down the equation of the vertical asymptote to the graph of .

Sketch the graph of . Be sure to indicate clearly the points of intersection with the coordinate axes, as well as any asymptotes.

Let  be a function defined by   ,  ,  .

Write down

the value of

the equation of the vertical asymptote to the graph of

the equation of the oblique asymptote to the graph of .

Show that can be written in the form ,  where ,  and  are constants to be determined.

Use an algebraic method to show that the graph of does not cross the -axis.

Sketch the graph of  and hence write down the range of the function .

Let the function  be defined by , where  is a positive constant. 

Write down the value of .

Sketch the graph of . State the equations of the asymptotes and give the coordinates of the intersections with the coordinate axes in terms of .

Given that the graph of  intersects the line given by , find the set of possible values of a.

Consider the function  defined by The graph of  is shown below. The graph has a maximum point at A and a minimum point at B.

Show that is defined for all .

Hence, find the coordinates of A and B.

Consider the function  defined by , where  and are positive constants and .

In the case that and are distinct:

write down the coordinates of the points where the graph of intersects the axes,

show that the equation of the oblique asymptote is  .

Sketch the graph of in that case that:

,

Clearly indicate where the graph crosses the coordinate axes and any asymptotes or discontinuities. 

Sketch the graph of the function  be defined by  Clearly indicate the coordinates where the graph intersects the axes and state the equation of any asymptotes.

Sketch the graph of the function defined by   Clearly indicate the coordinates where the graph intersects the axes and state the equation of any asymptotes.

The graph of a function  is shown below. The graph crosses the -axis at the points  and . The graph crosses the -axis at . The equation of the vertical asymptote is .

The function  can be written in the form , where  

Given that , find the values of and .

Hence find the equation of the oblique asymptote.

Consider the function  defined by ,  where . The line  is the only vertical asymptote of the graph of  as shown below. The graph crosses through the points  and  where  is a positive constant. The line  is a tangent to the graph of  at the point .

Find the values of  and .

Find the coordinates of .

Consider the function 

Solve

Show that , where  and are constants to be found.

Using the answer to part (b), sketch the graph of . Clearly indicate the coordinates of the points where the graph intersects the axes and state the equations of any asymptotes.

The diagram below shows the graphs of two linear functions and  and a quadratic function .

Sketch the graph of . Clearly indicate where the graph intersects the -axis and the location of any vertical asymptotes.

Sketch the graph of . Clearly indicate where the graph intersects the -axis and the location of any vertical asymptotes.