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CURVE SKETCHING. RATIONAL FUNCTIONS. RATIONAL FUNCTIONS EXAMPLE 1 a) Domain Since x 2 + x – 2 ≠ 0 ( x + 2)( x – 1) ≠ 0 x ≠ – 2 x ≠ 1 x e R b) x -intercepts y ­ - intercept c) Tests for symmetry Circle the correct answer.

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RATIONAL FUNCTIONS

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Rational functions

CURVE SKETCHING

RATIONAL FUNCTIONS


Rational functions

RATIONAL FUNCTIONS EXAMPLE 1

a) Domain Since x2 + x – 2 ≠ 0 (x + 2)(x – 1) ≠ 0 x ≠ – 2 x ≠ 1 xe R

b) x-intercepts y­-intercept

c) Tests for symmetry

Circle the correct answer.

This function is: a) evenb) oddc) neither

d) Asymptotes

f (x) ( = / ≠) f(–x)

The graph (is / is not) symmetrical to the y-axis.

– f (x) ( = / ≠) f(–x)

The graph (is / is not) symmetrical to the y-axis.

Vertical asymptote at x = –2

“Hole” at

Horizontal asymptote at y = 1


Rational functions

Now let’s look at the one-sided limits around

  • Label the x and y- intercepts with ordered pairs

  • Label the “hole” with an ordered pair

  • Draw in the asymptotes and write their equations

  • show how the graph behaves left and right of the vertical asymptote


Rational functions

EXAMPLE 1

a) Domain

x ≠ 4, xe R

b) x-intercepts

y­-intercept

DNE


Rational functions

c) Tests for symmetry

f (x) ( = / ≠) f(–x)

The graph (is / is not) symmetrical to the y-axis.

– f(x) =

– f(x) ( = / ≠) f(–x)

The graph (is / is not) symmetrical to the origin.

Circle the correct answer.

This function is: a) evenb) oddc) neither


Rational functions

d) Asymptotes

DNE

DNE

Vertical Asymptote at x = 4

Now lets look at the one-sided limits around x = 4

- ∞

0

Horizontal Asymptote at y = 0


Rational functions

e) First Derivative Test: Find intervals of increase and decrease.

decreasing

This graph is always ____________________

because f '(x) is always ________________

negative


Rational functions

+

4

f) Concavity: This can change when the second derivative is undefined if

The graph is concave down over the interval x < 4

The graph is concave up over the interval x > 4


Rational functions

  • Graph #1

  • Using ordered pairs,

    • label the x and y-intercepts,

    • draw in any asymptotes and label with their equations

    • show how the graph behaves left and right of the vertical asymptote

y = 0

- ∞

x = 4


Rational functions

Graph #2

Highlight the increasing sections and / or the decreasing sections

decreasing

decreasing


Rational functions

Graph #3

Highlight the concave up sections and concave down sections.

CU

CD


Rational functions

a) Domain

x ≠ -2, x e R

EXAMPLE 2

b) x-intercepts y­-intercept

DNE

c) Tests for symmetry

f (x) ( = / ≠) f(–x)

The graph (is / is not)

symmetrical to the y-axis.

– f(x) ( = / ≠) f(–x)

The graph (is / is not)

symmetrical to the origin.

Circle the correct answer.

This function is: a) evenb) oddc) neither


Rational functions

0

d) Asymptotes

DNE

DNE

Vertical Asymptote at x = -2

Now lets look at the one-sided limits around

- ∞

- ∞

Horizontal Asymptote at y = 0


Rational functions

+

- 2

e) First Derivative Test: Find intervals of increase and decrease.

Interval of increase: x > -2

Interval of decrease: x < -2


Rational functions

- 2

f) Concavity: This can change when the second derivative is undefined if

The graph is always ________________

(CU / CD)

because f "(x) is always _________________

(positive / negative)


Rational functions

x = -2

y = 0

- ∞

- ∞

  • Graph #1

  • Using ordered pairs, label the x and y-intercepts,

  • draw in any asymptotes

  • show how the graph behaves left and right of the vertical asymptote


Rational functions

Graph #2

Highlight the increasing sections

and the decreasing sections

decreasing

increasing

Graph #3

Highlight the concave up sections

and concave down sections.

CD

CD


Rational functions

EXAMPLE

Since x2 + 3 ≠ 0 xe R

Domain:

y­intercept

x-intercepts

(0, 0)

(0, 0)


Rational functions

Tests for symmetry

  • f(–x) =

  • f (x) (= / ≠) f(–x)

  • The graph (is / is not) symmetrical to the y-axis.

– f(x) = =

– f(x) ( = / ≠) f(–x)

The graph (is / is not) symmetrical to the origin.


Rational functions

Asymptotes

x2 + 3 ≠ 0

Therefore there are no vertical asymptotes

Horizontal asymptote at y = 1


Rational functions

min (0, 0)

+

0

Intervals of Increase or Decrease(First Derivative Test)

Increase:x > 0

Decrease:x < 0

6x = 0

x = 0


Rational functions

Local Maximum and Minimum Valuesby second derivative test

positive so (0, 0) is a minimum


Rational functions

+

CD

-1

CU

1

CD

Concavity and Points of Inflection

The graph is concave down over the intervals

x < -1 or x > 1

The graph is concave up over the interval

-1 < x < 1


Rational functions

Graph #1

Using ordered pairs, label the xand y-intercepts, draw in any asymptotes

y = 1

(0, 0)


Rational functions

Graph #2

Label any turning points with ordered pairs

Highlight the increasing sections and the decreasing sections

increasing

decreasing

(0, 0)


Rational functions

Graph #3

Label the inflection points with ordered pairs.

Highlight the concave up sections and concave down sections.

(1, ¼ )

(-1, ¼ )

CD

CD

CU


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