1. Warm Up
Graph the function
𝑓 𝑥 =− 𝑥

2. Graphing Rational Functions
We have graphed several functions, now we are adding one more to the list!
Graphing Rational Functions

3. Parent Function: 𝒇 𝒙 = 𝟏 𝒙

4. Pay attention to the transformation clues!
f(x) = k a
x – h
(-a indicates a reflection in the x-axis)
vertical translation
(-k = down, +k = up)
horizontal translation
(+h = left, -h = right)
Watch the negative sign!! If h = -2 it will appear as x + 2.

5. Asymptotes
Places on the graph the function will approach, but will never touch.

6. Vertical Asymptote: x = 0 Horizontal Asymptote: y = 0
f(x) = 1 x
Graph:
Vertical Asymptote: x = 0
Horizontal Asymptote: y = 0
No horizontal shift.
No vertical shift.
A HYPERBOLA!!

7. W𝐡𝐚𝐭 𝐝𝐨𝐞𝐬 𝒇 𝒙 =− 𝟏 𝒙 look like?

8. Vertical Asymptote: x = -4 Horizontal Asymptote: y = 0
Graph: f(x) = 1
x + 4
x + 4 indicates a shift 4 units left
Vertical Asymptote: x = -4
No vertical shift
Horizontal Asymptote: y = 0

9. 1 Graph: f(x) = – 3 x + 4 x + 4 indicates a shift 4 units left
Vertical Asymptote: x = -4
–3 indicates a shift 3 units down which becomes the new horizontal asymptote y = -3.
Horizontal Asymptote: y = 0

10. x Graph: f(x) = + 6 x – 1 x – 1 indicates a shift 1 unit right
Vertical Asymptote: x = 1
+6 indicates a shift 6 units up moving the horizontal asymptote to y = 6
Horizontal Asymptote: y = 1

11. You try!!
1. 𝑦= 1 𝑥 +2
2. 𝑦= 1 𝑥+3 −4

12. How do we find asymptotes based on an equation only?

13. Vertical Asymptotes (easy one)
Set the denominator equal to zero and solve for x.
Example: 𝑦= 6 𝑥−3
x-3= x=3
So: 3 is a vertical asymptote.

14. Horizontal Asymptotes (H.A)
In order to have a horizontal asymptote, the degree of the denominator must be the same, or greater than the degree in the numerator.
Examples:
𝑦= 𝑥 2 −3 𝑥 No H.A because 2>1
𝑦= 𝑥 3 −2 𝑥 3 − Has a H.A because 3=3.
𝑦= 𝑥+1 𝑥 Has a H.A because 1<2

15. 3 cases

16. If the degree of the denominator is GREATER than the numerator.
The Asymptote is y=0 ( the x-axis)

17. If the degree of the denominator and numerator are the same:
Divide the leading coefficient of the numerator by the leading coefficient of the denominator in order to find the horizontal asymptote.
Example: 𝑦= 6𝑥 3 3𝑥 3 −2
Asymptote is 6/3 =2.

18. If there is a Vertical Shift
The asymptote will be the same number as the vertical shift.
(think about why this is based on the examples we did with graphs)
Example: 5 𝑥−3 +7
Vertical shift is 7, so H.A is at 7.

19. Practice