Representation Of Rational Functions Graphically.pptx

RATIONAL FUNCTIONS

REVIEW: LEAST COMMON DENOMINATOR Determine the LCD of the following:

1. 5,15 2. 𝑥, 2𝑥 3. 𝑚, 𝑚2 4. 3𝑑, 𝑑 + 1 5. ℎ, 2ℎ − 3

6. 2𝑥 − 1, 𝑥 + 3, 𝑥 − 2 7. 𝑥 − 2, 𝑥 2 −5𝑥 + 6 8. 𝑥 − 2, 5𝑥, 5 9. −5𝑛, −2𝑛, 𝑛2 10.𝑥 − 2, 𝑥 + 3, 𝑥 2 + 𝑥−6

QUIZ Solve for y:

3 𝑦+2

1 𝑦

− =

1 5𝑦

SOLVING RATIONAL INEQUALITIES To solve rational inequalities: (a) Rewrite the inequality as a single rational expression on one side of the inequality symbol and 0 on the other side. (b) Determine over what intervals the rational expression takes on positive and negative values.  i. Locate the x values for which the rational expression is zero or undefined (factoring the numerator and denominator is a useful strategy).  ii. Mark the numbers found in (i) on a number line. Use a shaded circle to indicate that the value is included in the solution set, and a hollow circle to indicate that the value is excluded. These numbers partition the number line into intervals.  iii. Select a test point within the interior of each interval in (ii). The sign of the rational expression at this test point is also the sign of the rational expression at each interior point in the aforementioned interval.  iv. Summarize the intervals containing the solutions.

ASSIGNMENT

Solve for x:

1 𝑥

<

1 𝑥−3

Excluded value/s of rational equation is the value of the variable that will make the denominator equal to zero Examples:

1. 2. 3.

𝑥 2 = 𝑥−3 5 𝑑−3 4 = 2𝑑−1 𝑑+5 3𝑛 − 2 𝑛 −4𝑛−12

2=

1 𝑛

Determine the excluded values of the following rational expression. 2 12𝑥 1.

2. 3. 4. 5.

=

3 13𝑥 − = 3𝑥 𝑥+7 2−𝑐 5𝑐 = 4−𝑐 𝑐+4 𝑥−5 2

2

4 3 = − 5𝑟 + 1 3−5𝑟 3+5𝑟 5 3 6 + = 2 𝑦 −2𝑦−15 𝑦+3 𝑦−5

REPRESENTATION OF RATIONAL FUNCTIONS

A rational function can be presented by table of values To construct a table of values for the given rational function, assign values x and substitute to the function then solve. Examples: 1 f(x)= 𝑥 X

-4

-2

f(x)

-0.25 -0.5

-1

-0.5

-0.25 -0.01 0

0.25

0.5 1

2

4

-1

-2

-4

4

2

0.5

0.25

-100

undefined

1

1 f(x)= , 𝑥−2

 Given a function complete the following table of values. x f(x)

-4

-2

-1

0

1

1.5

2

3

4

10

MATHEMATICAL CONCEPT The value of x that makes the given rational function undefined act as boundary for the values of f(x). The value of f(x) becomes closer to a certain value but will not be equal to it

Determine whether the given table of values represents a rational function 1.

2.

3.

x f(x)

2 3 4 -0.25 -0.33 -0.5

x f(x)

-4 3.75

-3 3.67

x f(x)

-4 -7

-3 -5

-2 3

-2 -3

5 6 7 -1 undefined 1 -1 2

-1 -1

8 0.5

9 10 0.33 0.25

0 1 2 undefined 5 4.5

0 1

1 3

2 5

3 4.33

3 7

4. 5. 6.

x f(x)

-6 6

-5 10

-4.5 -4.1 -4 18 82 undefined

-3.9 -3.5 -78 -14

x -3 f(x) 0

-2 4

-1 0

0 -6

1 -8

2 0

3 24

4 70

x -1 f(x) -5

0 -1

1 3

2 7

3 11

4 15

5 19

6 24

-2 -2

REPRESENTATION OF RATIONAL FUNCTIONS GRAPHICALLY

A rational function can be presented by graph To graph, construct a table of values for the given rational function, then graph. Examples:

1.

1 f(x)= 𝑥

X

-4

-2

f(x)

-0.25 -0.5

-1

-0.5

-0.25 -0.01 0

0.25

0.5 1

2

4

-1

-2

-4

4

2

0.5

0.25

-100

undefined

1

2.

𝑥−1 f(x)= 𝑥+1

3. f x =

𝑥 2 −3𝑥−10 𝑥