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# RATIONAL FUNCTIONS - PowerPoint PPT Presentation

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

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

This function is: a) even b) odd c) 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

• 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

a) Domain

x ≠ 4, xe R

b) x-intercepts

y­-intercept

DNE

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.

This function is: a) even b) odd c) neither

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

Highlight the increasing sections and / or the decreasing sections

decreasing

decreasing

Highlight the concave up sections and concave down sections.

CU

CD

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.

This function is: a) even b) odd c) neither

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

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

Highlight the increasing sections

and the decreasing sections

decreasing

increasing

Graph #3

Highlight the concave up sections

and concave down sections.

CD

CD

Since x2 + 3 ≠ 0 xe R

Domain:

y­intercept

x-intercepts

(0, 0)

(0, 0)

• 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.

x2 + 3 ≠ 0

Therefore there are no vertical asymptotes

Horizontal asymptote at y = 1

+

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

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

y = 1

(0, 0)

Label any turning points with ordered pairs

Highlight the increasing sections and the decreasing sections

increasing

decreasing

(0, 0)

Label the inflection points with ordered pairs.

Highlight the concave up sections and concave down sections.

(1, ¼ )

(-1, ¼ )

CD

CD

CU