1. BELL-WORK

2. Due tomorrow: PW 11-1 # 1-12(even) PW 11-2 # 1-5,14-18
HW 4.4(f)
Due tomorrow:
PW 11-1 # 1-12(even)
PW 11-2 # 1-5,14-18

3. 13; (5, ½) 5; (4, 2½) 17; (3½, 5) 10; (-1, 1) 6½; (1¾, 2) 12½; (0, 1¼)
HW 4.4(e) Solutions
13; (5, ½)
5; (4, 2½)
17; (3½, 5)
10; (-1, 1)
6½; (1¾, 2)
12½; (0, 1¼)

4. Guiding question:
How are rational functions simplified?

5. Rational Functions
A rational function has a polynomial of at least degree one in the denominator.
Examples: y = 1 x
y =
x2+2
y = 3x + 9
x + 3

6. Simplifying Rational Functions
To simplify rational functions:
factor the numerator,
then factor the denominator, and
then carry out operations as usual.
Simplify: 3x + 9
x + 3 4x
x2 – 9x + 20
3x – 27
81 – x2

7. Simplifying Rational Functions
TB pg 654 Got it 3a, 3b (Simplify only)

8. Multiplying Rational Functions
7 • 8 =
y y2 56 y3
x • x – 2 =
x x – 6
x(x – 2) .
(x + 5)(x – 6)
*Always leave your answer in factored form!

9. Multiplying Rational Functions
3x + 1 • 8x =
x2 – 1
3x + 1 • x
4 (3x – 1)(3x + 1)
= x
4(3x – 1)
= 2x
3x – 1

10. Multiplying Rational Functions
5x • (x2 + 7x + 12) =
3x + 12
= (5x + 1) •(x + 3)(x + 4)
3(x + 4)
= (5x + 1)(x + 3) 3

11. Dividing Rational Functions
a2 + 7a ÷ a =
a – a2 – 36
= (a + 5)(a + 2) ÷ a
a – (a – 6)(a + 6)
= (a + 5)(a + 2) × (a – 6)(a + 6) .
a – 6 a + 5
= (a + 2)(a + 6)

12. Dividing Rational Functions
x2 + 13x ÷ x =
x – x2 – 49
= (x + 5)(x + 8) ÷ x
x – (x – 7)(x + 7)
= (x + 5)(x + 8) × (x – 7)(x + 7).
x – 7 x + 8
= (x + 5)(x + 7)

13. Graphing Rational Functions
On the TI graph y = 1 x
Copy a sketch of the graph to your page

14. Graphing Rational Functions
What is the domain of the function?
All values except x = 0
Notice that the graph gets close to x = 0 but the graph never touches x = 0.
An asymptote is a line that the graph gets closer and closer to but never touches or crosses.
y = 1 has a vertical asymptote at x = 0. x

15. Graphing Rational Functions
What is range of the function?
All values except y = 0
Notice that the graph gets close to y = 0 but the graph never touches y = 0.
y = 1 has a horizontal asymptote at y = 0. x
Note: the excluded values are the asymptotes!

16. Graphing Rational Functions
How will the graph of look?
x + 2
The graph of y = represents a horizontal
x+c
translation of y = 1 x
If c > 0 it is a left-ward translation.
If c < 0 it is a right-ward translation.

17. Graphing Rational Functions
What is the domain of ?
x + 2

18. Graphing Rational Functions
What is the domain of ?
x + 2 x

19. Graphing Rational Functions
Therefore for y = there is a vertical asymptote
x + 2 at
x = -2
What is the range of y = ?
All values except y = 0
Therefore there is a horizontal asymptote at
y = 0.

20. Graphing Rational Functions
Without the TI sketch the graph of y =
x – 5
What is the domain?
All values except x = 5
Therefore there is a vertical asymptote at x = 5
What is the range?
All values except y = 0
Therefore there is a horizontal asymptote at
y = 0.

21. Graphing Rational Functions
How will the graph of y = look? x
The graph of y = 1 + c represents a vertical
translation of y = 1
If c > 0 it is an up-ward translation.
If c < 0 it is a down-ward translation

22. Graphing Rational Functions
What is the domain of y = ? x
All values except x = 0
Therefore there is a vertical asymptote at x = 0
What is the range?
All values except y = 4
Therefore there is a horizontal asymptote at
y = 4.

23. Graphing Rational Functions
Without the TI sketch the graph of y = – 3 ?
x + 4
What is the domain of the graph?
All values except x = -4
Therefore there is a vertical asymptote at x = -4
What is the range?
All values except y = -3
Therefore there is a horizontal asymptote at
y = -3.

24. Who wants to answer the Guiding question?
How are rational functions simplified?