Find Holes and y – intercepts Rational Functions Find Holes and y – intercepts Holt McDougal Algebra 2 Holt Algebra 2

Consider the rational function f(x) = (x – 3)(x + 2) x + 1 The numerator of this function is 0 when x = 3 or x = –2. Therefore, the function has x-intercepts at –2 and 3. The denominator of this function is 0 when x = –1. As a result, the graph of the function has a vertical asymptote at the line x = –1.

For any simplified rational function, what information can you obtain from the numerator? If you set the numerator = to 0, you determine the x-intercepts (zeros, solutions, roots).

Let’s find y-intercepts To find the y-intercept, what do we do? We set the x-value to 0 and solve for y. This is easy…lets look.

Let’s do what we learned yesterday and find the y-intercept. x2 – 3x – 4 X + 2 f(x) = Step 1 – Always factor if possible.

Step 2 – Find the x-intercepts. x2 – 3x – 4 X + 2 f(x) = Step 2 – Find the x-intercepts. (x – 4)(x + 1) X + 2 2 - x = 4 x= -1

Step 3 – Find the vertical asymptotes. x2 – 3x – 4 X + 2 f(x) = Step 3 – Find the vertical asymptotes. x = -2

Step 4 – Find the y-intercept. Let x = 0. x2 – 3x – 4 X + 2 f(x) = Step 4 – Find the y-intercept. Let x = 0. – 4 = -2 + 2 f(x) =

Try another. Find everything. Remember to factor first!! f(x) =

But there is a slant asymptote.

Slant Asymptotes Slant asymptotes occur when the degree of the numerator is exactly one bigger than the degree of the denominator. In this case a slanted line (not horizontal and not vertical) is the function’s asymptote. To find the equation of the asymptote we need to use long division – dividing the numerator by the denominator.

When dividing to find slant asymptotes: Do synthetic division (if possible); if not, do long division! The resulting polynomial (ignoring the remainder) is the equation of the slant asymptote.

EXAMPLE: Finding the Slant Asymptote of a Rational Function Rational Functions and Their Graphs EXAMPLE: Finding the Slant Asymptote of a Rational Function Find the slant asymptotes of f (x) = Solution Because the degree of the numerator, 2, is exactly one more than the degree of the denominator, 1, the graph of f has a slant asymptote. To find the equation of the slant asymptote, divide x - 3 into x2 - 4x - 5: 1 -4 -5 3 -3 1 -1 -8 3 Remainder more

EXAMPLE: Finding the Slant Asymptote of a Rational Function 3.6: Rational Functions and Their Graphs EXAMPLE: Finding the Slant Asymptote of a Rational Function Find the slant asymptotes of f (x) = Solution The equation of the slant asymptote is y = x - 1. Using our strategy for graphing rational functions, the graph of f (x) = is shown. -2 -1 4 5 6 7 8 3 2 1 -3 Vertical asymptote: x = 3 Slant asymptote: y = x - 1

Graph: Notice that in this function, the degree of the numerator is larger than the denominator. Thus n>m and there is no horizontal asymptote. However, if n is one more than m, the rational function will have a slant asymptote. To find the slant asymptote, divide the numerator by the denominator: The result is . We ignore the remainder and the line is a slant asymptote.

1st, find the vertical asymptote. Graph: 1st, find the vertical asymptote. 2nd , find the x-intercepts: and 3rd , find the y-intercept: 4th , find the horizontal asymptote. none 5th , find the slant asymptote: 6th , sketch the graph.

A Rational Function with a Slant Asymptote Graph the rational function Factoring:

A Rational Function with a Slant Asymptote Finding the x-intercepts: –1 and 5 (from x + 1 = 0 and x – 5 = 0) Finding the y-intercepts: 5/3 (because )

A Rational Function with a Slant Asymptote Finding the horizontal asymptote: None (because degree of numerator is greater than degree of denominator) Finding the vertical asymptote: x = 3 (from the zero of the denominator)

A Rational Function with a Slant Asymptote Finding the slant asymptote: Since the degree of the numerator is one more than the degree of the denominator, the function has a slant asymptote. Dividing, we obtain: Thus, y = x – 1 is the slant asymptote.

A Rational Function with a Slant Asymptote Here are additional values and the graph.

Slant Asymptotes and End Behavior So far, we have considered only horizontal and slant asymptotes as end behaviors for rational functions. In the next example, we graph a function whose end behavior is like that of a parabola.

Finding a Slant Asymptote If There will be a slant asymptote because the degree of the numerator (3) is one bigger than the degree of the denominator (2). Using long division, divide the numerator by the denominator.

Finding a Slant Asymptote Con’t.

Finding a Slant Asymptote Con’t. We can ignore the remainder The answer we are looking for is the quotient and the equation of the slant asymptote is

Graph of Example 7 The slanted line y = x + 3 is the slant asymptote

Cwk/Hwk Same worksheet as before. Find y-intercepts and horizontal asymptotes and them to your graphs.