• REACH ASSESSMENT

• REVIEW SKETCHING Y Intercept (Monday)

Zeroes (Monday)

Asymptotes (Wednesday)

• THURSDAY QUIZ ( Y T D + KA T R I G O N O M E T RY )

Week 3 –Day #s 8-12(week of Reach, Explore, Plan)

VERTICAL/HORIZONTAL/SLANT

MR. POULAKOS

Asymptotes

Asymptotes of Rational Functions,

A Rational Function is:An Asymptote is, essentially, a line that a

function approaches, but never touches or crosses. There are three types:

Vertical Asymptote Horizontal Asymptote Slant Asymptote (future lesson)

The Asymptote is represented on x-y coordinate system as a dashed line “- - - - - - - ”

Why?

- - - - - - - - -

- - - - - - - - - - - - - - - - horizontal

Vertic

al

Vertical asymptotes

The Vertical asymptote is a line that the graph approaches but does not intersect. Vertical asymptotes almost always occur because the denominator of a fraction has gone to zero, but the numerator (top) has not.

For example,

Note that as the graph approaches x=2. From the left, the curve drops rapidly towards negative infinity. This is because the numerator is staying at 4, and the denominator is getting close to 0.

x

=2

Definition

Horizontal Asymptote

The Horizontal asymptote is also a line that the graph approaches but does not intersect

In the following graph of y=1/x, the line approaches the x-axis (y=0) as x gets larger. But it never touches the x-axis. No matter how far we go into infinity, the line will not actually reach y=0, but it will keep getting closer and closer.

This means that the line y=0 is a horizontal asymptote.

The domain for y=1/x is all real numbers except 0

Definition

REMEMBER:

ASYMPTOTES ARE ALWAYS LINES.

THEY ARE LINES THAT A GRAPH APPROACHES BUT DOES NOT TOUCH OR CROSS (DOES NOT INTERSECT)

Finding Asymptotes

Vertical asymptotes

Remember, Vertical asymptotes almost always occur because the denominator of a fraction has gone to 0, but the top has not the zeroes of the denominator.

Therefore, set the denominator to zero and solve for the variable.

For example, x–7=0x=7 is the asymptote.Factor : x2–16=0 (x–4)(x+4) = 0 Solve: a) x–4=0 and b) x+4=0, Therefore, there are 2 asymptotes. a) x = +4 and b) x = –4

Rational Function Vertical Asymptote (s) is/are at …

Vertical asymptotes – Sample Problems

x = 5

x = +4 and x = –4

x = – 4 and x = – 2

Rational Function Vertical Asymptote (s) is/are at …

h

Vertical asymptotes – Sample Problems

𝒇 ( 𝒙 )= 𝟐𝟔(𝒙−𝟑)

g

x = 3

x = +5 and x = –5

x = – 4 and x = 6

Horizontal asymptotes

Finding the Horizontal asymptote(s) are more challenging… Compare the degree of the numerator

(n) to that of the denominator (m). If n<m, then the horizontal asymptote is at y

= 0. If n=m (the degrees are the same), then the

asymptote is at y = 1st coefficient of numerator ÷ 1st coefficient of denominator

If n>m, then there are no Horizontal asymptotes.

Examples follow …

n < mAsymptote is at y=0

n > mNo

Asymptote

Examples -- Horizontal Asymptotes

Assymptotey=0

Assymptotey=0

Assymptotey=0

n = mAsymptote

is at y=an/bm

Asymptote is at…

y = 2/5

y=2

y= 6/4

= 3/2

NO

H A

SY

MP

TO

TE

S

Slant asymptotes

Finding the Slant asymptote(s) is similar to Horizontal in that we look at the degrees. Compare the degree of the numerator (n) to

that of the denominator (m). If n=m+1 (exactly), then there is a slant asymptote

and it is determined by: Performing long division dividend ÷ divisor. The slant asymptote is the quotient without

including the remainder. y=“quotient” f(x)= (x2–x−2) ÷ (x−1) slant: y=x f(x)= (x2+4x+13) ÷ (x+1) slant: y=x+3

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Sketching !

Asymptotes: Summary

For Vertical – set D(x) to zero & solve for x.Horizontal – look at degrees of N(x) & D(x)

If n<m, then the asymptote is y=0 If n=m, then the asymptote y=an ÷ bn (an is the

coeff. of highest degree variable) If n>m, no horizontal asymptote but we might have

a Vertical or Slant asymptote…Slant

If n=m+1 (exactly), then we have a slant asymptote !y=N(x) ÷ D(x) dropping the remainder…

Review !Note: n is the degree of the numerator & m that of the denominator

Sketching

To Sketch Rational Functions hints are in [ ]s:1) Simplify, if possible.2) Find & plot y intercepts [x=0]3) Find & plot the x intercepts [N(x)=0].4) Find & sketch any vertical asymptotes

[D(x)=0]5) Find & sketch any horizontal asymptotes

[“degrees”]6) Find & sketch any slant asymptotes [n=m+1]7) Plot at least one point between and one point

beyond each x-intercept and vertical asymptote.

8) Use smooth curves to complete the graph between and beyond the vertical asymptotes.