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RATIONAL FUNCTIONS

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CURVE SKETCHING

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

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

EXAMPLE 1

a) Domain

x ≠ 4, xe R

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) ( = / ≠) f(–x)

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

Circle the correct answer.

This function is: a) evenb) oddc) neither

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

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

decreasing

This graph is always ____________________

because f '(x) is always ________________

negative

+

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

- 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

Graph #2

Highlight the increasing sections and / or the decreasing sections

decreasing

decreasing

Graph #3

Highlight the concave up sections and concave down sections.

CU

CD

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

0

d) Asymptotes

DNE

DNE

Vertical Asymptote at x = -2

Now lets look at the one-sided limits around

- ∞

- ∞

Horizontal Asymptote at y = 0

+

- 2

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

Interval of increase: x > -2

Interval of decrease: x < -2

- 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)

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

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

EXAMPLE

Since x2 + 3 ≠ 0 xe R

Domain:

yintercept

x-intercepts

(0, 0)

(0, 0)

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.

Asymptotes

x2 + 3 ≠ 0

Therefore there are no vertical asymptotes

Horizontal asymptote at y = 1

min (0, 0)

+

0

Intervals of Increase or Decrease(First Derivative Test)

Increase:x > 0

Decrease:x < 0

6x = 0

x = 0

Local Maximum and Minimum Valuesby second derivative test

positive so (0, 0) is a minimum

+

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

Graph #1

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

y = 1

(0, 0)

Graph #2

Label any turning points with ordered pairs

Highlight the increasing sections and the decreasing sections

increasing

decreasing

(0, 0)

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