This week we have learned how to graph reciprocals for both linear functions and quadratic functions.

For graphing reciprocal functions, we have to determine the asymptotes.

The x-axis is a horizontal asymptote and a vertical asymptote is when x has a certain value, a vertical line that the graph approaches but never reaches.

For all reciprocal functions, y cannot be 0 since can never be 0.

When we take the reciprocal of a number, the only numbers that stay the same are -1 and 1. Therefore, when we graph, we find the points that line up to where y=1 and y=-1. These are the invariant points.

Let’s look at an example of a linear function.

First step is to graph the line.

Next is to locate the invariant points.

 

Next is to find the asymptotes. Horizontal asymptote is always y=0.

 

Once you have the line graphed, now you must draw a curved line that goes only through the variant points while approaching the asymptotes but not actually touching it.

 

let’s look at another example but with quadratic functions.

Example:

y = and y =

First is to graph the parabola. Start by factoring to find the x-intercepts.

 

 

Now that you have the x-intercepts, you can add them together and divide it by two to find the axis of symmetry and plug that and a point in standard form to find q for the vertex.

 

When you draw out the parabola, circle the invariant points and draw in the asymptotes.

Draw a curved line through the invariant points.

 

If the parabola has one x-intercept, then it will be divided into 4 zones and will have two invariant points:

 

If the parabola has no x-intercepts, then it will divided in two zones and it won’t have any invariant points or a vertical asymptote: