Rationals - Day 1 (Thursday & Friday)

The last function that we will graph and translate this year is the rational function. This function is created when a variable is located in the denominator of a fraction. The parent function is

And its graph is

Notice that this graph is two separate curves. They do not connect in the middle. In fact, the center of the curves, (0, 0) in this graph, has two asymptotes running through it. An asymptote is a line on the graph that the curves do not cross, but instead they always approach. This means that the distance between the curves and the asymptotes is narrowing, but they do not intersect. On the graph of the parent function, the asymptotes are x = 0 (the y-axis) and y = 0 (the x-axis).

The graph of a rational function can be translate similar to our previous transformations. The general form of the rational function is

In this form, the graph translates according to:

• "a" stretches the graph if its absolute value is larger than 1;
• "a" compresses the graph if its absolute value is between 0 and 1;
• "a" reflects the graph upside down if it is a negative value;
• "h" shifts the graph to the left and right (remember, use the opposite of the sign that is "seen" in the function);
• "k" shifts the graph up and down.

Because of the nature of this graph, the characteristics of the graph are quite different from the characteristics of our previous functions.

• This graph has a center at (h, k) that is used to determine the asymptotes. The previous functions each have a vertex at (h, k).
• This graph has 2 asympotes: one vertical and one horizontal.
• This graph has a domain of all real numbers except the x-value of the vertical asymptote.
• This graph has a range of all real numbers except the y-value of the horizontal asymptote.
• This graph has end behaviors approaching the y-value of the horizontal asymptote.
• This graph is either increasing or decreasing, not both. And there are 2 separate intervals for the graph to be increasing (or decreasing).

Example: Graph the following function.

This graph has a center at (5, 3).

This graph has a vertical asymptote at x = 5.

This graph has a horizontal asymptote at y = 3

This graph is reflected upside down.

To find additional points for the graph, substitute in values for x around 5.

Homework: Students were to complete the notes that were handed out on Thursday. 1st period has already had this homework checked. 3rd and 4th periods will have this checked on Monday.

There will be a quiz on Thursday, March 25th on radical and rational functions.

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