What is a Rational Function? A rational function is a function of the form where p (x) and q (x) are polynomials and q (x)  0. p (x)q (x)p (x)q (x) f (x) = For example - y =

Graphing a Simple Rational Function A rational function is a function of the form where p (x) and q (x) are polynomials and q (x)  0. p (x)q (x)p (x)q (x) f (x) = For instance consider the following rational function: 1x1x y = The graph of this function is called a hyperbola and is shown below. Notice that the graph never crosses the x or y axes. An imaginary line that the graph approaches but usually do not cross is called an asymptote.

Graphing a Simple Rational Function – 4 –2 –1 – 2 – 3 xy – – – – xy The x -axis is a horizontal asymptote. The y -axis is a vertical asymptote. The domain and range are all nonzero real numbers. The graph has two symmetrical parts called branches. For each point ( x, y ) on one branch, there is a corresponding point ( –x, –y ) on the other branch. Notice the following properties for y =

G RAPHING R ATIONAL F UNCTIONS GRAPHS OF RATIONAL FUNCTIONS C ONCEPT S UMMARY Let p (x ) and q (x ) be polynomials with no common factors other than 1.The graph of the rational function has the following characteristics. f (x ) = = a m x m + a m – 1 x m – 1 + … + a 1 x + a 0 b n x n + b n – 1 x n – 1 + … + b 1 x + b 0 p (x )q (x )p (x )q (x ) 1. x - intercepts are the real zeros of p (x ) or when p(x) = 0 2. vertical asymptote at each real zero of q (x ) or when q(x) = 0

3. at most one horizontal asymptote G RAPHING R ATIONAL F UNCTIONS GRAPHS OF RATIONAL FUNCTIONS C ONCEPT S UMMARY f (x ) = = a m x m + a m – 1 x m – 1 + … + a 1 x + a 0 b n x n + b n – 1 x n – 1 + … + b 1 x + b 0 p (x )q (x )p (x )q (x ) To find horizontal asymptotes use the following procedure

G RAPHING R ATIONAL F UNCTIONS GRAPHS OF RATIONAL FUNCTIONS C ONCEPT S UMMARY If upper > lower, the graph has no horizontal asymptote. It has a slant asymptote if the difference is 1. Find it by long division. If upper < lower, the line y = 0 is the horizontal asymptote. If upper = lower, the line y = is the horizontal asymptote. A rational function has an y = Example

Graphing a Rational Function (m < n) The bell-shaped graph passes through (–3, 0.4), (– 1, 2), (0, 4), (1,2), and (3, 0.4). Graph y =. 4 x S OLUTION The numerator has no zeros, so there is no x -intercept. The denominator has no real zeros, so there is no vertical asymptote. The degree of the numerator (0) is less than the degree of the denominator (2), so the line y = 0 (the x-axis) is a horizontal asymptote.

Graphing a Rational Function (m = n) Graph y =. 3x 2 x 2 – 4 The degree of the numerator (2) is equal to the degree of the denominator (2), so the horizontal asymptote is y = = S OLUTION The numerator has 0 as its only zero, so the graph has one x -intercept at (0, 0). The denominator can be factored as (x + 2)(x – 2), so the denominator has zeros at 2 and – 2. This implies vertical asymptotes at x = – 2 and x = 2.

To the left of x = – 2 To the right of x = 2 Between x = – 2 and x = 2 Graphing a Rational Function (m = n) To draw the graph, plot points between and beyond vertical asymptotes. Graph y =. 3x 2 x 2 – –1 0 – – 4 – 3 – xy xy

Graphing a Rational Function (m > n) Graph y =. x 2 – 2x – 3 x + 4 The degree of the numerator (2) is greater than the degree of the denominator (1), so there is no horizontal asymptote. Since the difference in degrees is 1, there is a slant asymptote. Find it by dividing the numerator by the denominator. S OLUTION The numerator can be factored as ( x – 3) and ( x + 1); the x -intercepts are 3 and –1. The only zero of the denominator is – 4, so the only vertical asymptote is x = – 4.

Graph y =. x 2 – 2x – 3 x The slant asymptote is at y = x - 6

Graphing a Rational Function (m > n) To draw the graph, plot points to the left and right of the vertical asymptote. To the left of x = – 4 To the right of x = – 4 Graph y =. x 2 – 2x – 3 x + 4 – 20.6 –19.2 – – 0.75 – –12 – 9 – 6 – y xy xy

Graphing a Rational Function with your calculator Graph y =. x 2 – 2x – 3 x + 4 Y1=(x 2 – 2x – 3)/(x+ 4) Y2= Y3= Begin by entering the function into Y= Notice that the graph has a vertical line at x = -4. This line is not part of the graph – it is simply the graphing calculator’s attempt to connect the two parts of the graph. To remove this line you can go to MODE and choose DOT instead of CONNECTED

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