GRAPHING RATIONAL FUNCTIONS. Warm Up 1) The volume V of gas varies inversely as the pressure P on it. If the volume is 240 under pressure of 30. Write.

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Presentation transcript:

GRAPHING RATIONAL FUNCTIONS

Warm Up 1) The volume V of gas varies inversely as the pressure P on it. If the volume is 240 under pressure of 30. Write an equation to represent the relationship. What pressure has to be applied to have a volume of 160?

Investigation Graph the following: 1. Y = 3/X2. Y = 6/X 3. Y = -8/X4. Y = -4/X What do you notice?!?!?

Rational Function There are 2 basic forms of a rational function 1) 2) Where p(x) & q(x) are polynomials

Rational Functions Rational functions are split into parts called BRANCHES

Asymptotes An Asymptote is a line (vertical or horizontal) that the graph approaches but NEVER touches!

Asymptotes

What’s domain? The domain of a function is the set of all possible x-values. Vertical Asymptotes are values not included in the domain

What’s range? The range of a function is the set of all possible y-values. Horizontal Asymptotes are possible values not included in the range

Finding Asymptotes from an equation 1 st Form the Vertical Asymptote is when you set x – b = 0 and solve for x the Horizontal Asymptote is y = c

Identifying Asymptotes Identify the Asymptotes from the following functions. State the Domain and Range. 1.

Identifying Asymptotes Identify the Asymptotes from the following functions. State the Domain and Range. 2.

Identifying Asymptotes Identify the Asymptotes from the following functions. State the Domain and Range. 3.

Second form of a rational function : where p and q are polynomials

Vertical Asymptotes To find a vertical Asymptote, set the bottom equal to zero and solve!

State the VA, and the domain?

Asymptotes What is the Asymptotes? Graph it, what do you notice?!

Holes A HOLE in the graph is when (x – a) is a factor in both the numerator and the denominator. So on the graph, there is a HOLE at 4.

Find VA, Holes and State Domain

WHATS MISSING?!?!? We have been talking A LOT about Vertical asymptotes, but did we forget Horizontal?! Lets take a more in depth look at Horizontal asymptotes when a polynomial is divided by a polynomial.

Horizontal Asymptotes To find a horizontal asymptotes, we focus on the degree of the numerator and the denominator. What’s the degree?

How do we use degrees to find the horizontal asymptote?

Degree of 3 Degree of 5 The degree is bigger on the bottom, so the horizontal asymptote is the line y = 0.

Degree of 6 Degree of 1 The degree is bigger on the top, so there is no horizontal asymptote.

Degree of 1 The degrees are the same, so divide the leading coefficients. The horizontal asymptote is y = 2.

Find it all! Vertical and Horizontal Asymptote and Holes and State Domain and Range