1. PARENT FUNCTIONS Constant Function Linear Absolute Value Quadratic
Cubic
Square Root
Rational

2. Constant Function
Domain:
Range:
Parent Equation: f(x) = 2

3. y – intercept: x – intercept: Parent Equation: f(x) = 2
Constant Function
Parent Equation: f(x) = 2
x – intercept:
y – intercept:

4. Table: x y -5 2 3 4 5 6 Parent Equation: x = -5
GRAPH THIS:
Table: x y -5 2 3 4 5 6
Parent Equation: x = -5
Graph Description: Vertical Line

5. Domain: Range: Parent Equation: f(x) = x Linear Function (Identity)
(y = x)
Domain:
Range:
Parent Equation: f(x) = x

6. Linear Function
(Identity)
x – intercept:
y – intercept:

7. Parent Equation: f(x) = x
Linear Function
(Identity)
Table: x y -2 -1 1 2
Parent Equation: f(x) = x
Graph Description: Diagonal Line

8. Absolute Value
Function
Parent Equation:
f(x) = │x │
Domain:
Range:

9. y – intercept: x – intercept: Parent Equation: f(x) = │x - 1│ - 4
Absolute Value
Function
Parent Equation: f(x) = │x - 1│ - 4
x – intercept:
y – intercept:

10. Graph this:
Table: x y -3 2 -2 3 -1 1
Graph Description: “V” - shaped

11. Quadratic Function
Parent Equation:
f(x) = x 2
Domain:
Range:

12. y – intercept: x – intercept: Parent Equation: Quadratic Function
f(x) = x 2
Parent Equation:
x – intercept:
y – intercept:

13. Graph Description: Parabola
Graph This: x y -4 -3 3 -2 4 -1
Graph Description: Parabola

14. Cubic Function
Parent Equation:
f(x) = x 3
Domain:
Range:

15. Cubic Function
x – intercept:
y – intercept:

16. Graph Description: Squiggle, Swivel
Graph This: x y -1 2
Graph Description: Squiggle, Swivel

17. Square Root
Function
Parent Equation:
f(x) = x
Domain:
Range:

18. Square Root
Function
f(x) = x+3 + 2
x – intercept:
y – intercept:

19. Graph this: x y -- 1 2 -1 3
Graph Description: Horizontal ½ of a Parabola

20. Graphing Square-root Functions
Domain:
all nonnegative numbers
Range:
all numbers greater than or equal to k
Domain:
Range: x y 1 4 9

21. Graphing Square-root Functions
Domain:
Range: x y 1 4 9

22. Graphing Square-root Functions
Domain:
Range: x y 1 4 9

23. Graphing Square-root Functions
Domain:
all numbers greater than or equal to h
Range:
all nonnegative numbers
Domain:
Range: x y 1 2 5 10

24. Graphing Square-root Functions
Domain:
Range: x y -1 3 8

25. Graphing Square-root Functions
Domain:
Range: x y 4 5 8 13

26. 7.5 Graphing Square Root and Cube Root Functions, p. 431
State the Domain and Range. x y 1 1 4 2 9 3
Domain:
Range:

27. Graph
State the Domain and Range. x y 1 1 8 2 −1 −1 −8 −2
Domain:
Range:

28. Graph
State the Domain and Range. x y −3 1 −2 4 −1
Domain:
Range:

29. Graph
State the Domain and Range. x y −3 −2 1 1 2
Domain:
Range:

30. Graph
State the Domain and Range. x y 1 2 4 4
Domain:
Range:

31. In general,
To graph, first graph
then shift the graph h units horizontally and k units vertically.

32. Example 2a. Graph
State the Domain and Range.
First, graph
Then, shift the graph left 4 and down 1. x y 1 2 4 4
Domain:
Range:

33. Example 2b. Graph
First, graph
Then, shift the graph right 3 and up 2. x y 1 −2 8 −4 −1 2 −8 4
Domain:
Range:

34. Example 3. Find the domain and range without graphing.

35. RATIONAL FUNCTIONS A rational function is a function of the form:
where p and q are polynomials

36. What would the domain of a rational function be?
We’d need to make sure the denominator  0
Find the domain.
If you can’t see it in your head, set the denominator = 0 and factor to find “illegal” values.

37. The graph of looks like this:
If you choose x values close to 0, the graph gets close to the asymptote, but never touches it.
Since x  0, the graph approaches 0 but never crosses or touches 0. A vertical line drawn at x = 0 is called a vertical asymptote. It is a sketching aid to figure out the graph of a rational function. There will be a vertical asymptote at x values that make the denominator = 0

38. Let’s consider the graph
We recognize this function as the reciprocal function from our “library” of functions.
Can you see the vertical asymptote?
Let’s see why the graph looks like it does near 0 by putting in some numbers close to 0.
The closer to 0 you get for x (from positive direction), the larger the function value will be
Try some negatives

39. Does the function have an x intercept?
There is NOT a value that you can plug in for x that would make the function = 0. The graph approaches but never crosses the horizontal line y = 0. This is called a horizontal asymptote.
A graph will NEVER cross a vertical asymptote because the x value is “illegal” (would make the denominator 0)
A graph may cross a horizontal asymptote near the middle of the graph but will approach it when you move to the far right or left

40. vertical translation,
moved up 3
Graph
This is just the reciprocal function transformed. We can trade the terms places to make it easier to see this.
The vertical asymptote remains the same because in either function, x ≠ 0
The horizontal asymptote will move up 3 like the graph does.

41. Finding Asymptotes VERTICAL ASYMPTOTES
There will be a vertical asymptote at any “illegal” x value, so anywhere that would make the denominator = 0
So there are vertical asymptotes at x = 4 and x = -1.
VERTICAL ASYMPTOTES
Let’s set the bottom = 0 and factor and solve to find where the vertical asymptote(s) should be.

42. HORIZONTAL ASYMPTOTES
We compare the degrees of the polynomial in the numerator and the polynomial in the denominator to tell us about horizontal asymptotes.
1 < 2
degree of top = 1
If the degree of the numerator is less than the degree of the denominator, (remember degree is the highest power on any x term) the x axis is a horizontal asymptote.
If the degree of the numerator is less than the degree of the denominator, the x axis is a horizontal asymptote. This is along the line y = 0. 1
degree of bottom = 2

43. HORIZONTAL ASYMPTOTES
The leading coefficient is the number in front of the highest powered x term.
If the degree of the numerator is equal to the degree of the denominator, then there is a horizontal asymptote at:
y = leading coefficient of top
leading coefficient of bottom
degree of top = 2 1
degree of bottom = 2
horizontal asymptote at:

44. OBLIQUE ASYMPTOTES
If the degree of the numerator is greater than the degree of the denominator, then there is not a horizontal asymptote, but an oblique one. The equation is found by doing long division and the quotient is the equation of the oblique asymptote ignoring the remainder.
degree of top = 3
degree of bottom = 2
Oblique asymptote at y = x + 5

45. SUMMARY OF HOW TO FIND ASYMPTOTES
Vertical Asymptotes are the values that are NOT in the domain. To find them, set the denominator = 0 and solve.
To determine horizontal or oblique asymptotes, compare the degrees of the numerator and denominator.
If the degree of the top < the bottom, horizontal asymptote along the x axis (y = 0)
If the degree of the top = bottom, horizontal asymptote at y = leading coefficient of top over leading coefficient of bottom
If the degree of the top > the bottom, oblique asymptote found by long division.

46. RATIONAL FUNCTIONS
Domain:
Range:

47. 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.

48. 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!!

50. 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

51. Graph: f(x) = – 3 1
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

52. x – 1 indicates a shift 1 unit right
Graph: f(x) = x
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

53. EXAMPLE: Graphing a Rational Function-5 -4 -3 -2 -1 1 2 3 4 5 7 6
Vertical asymptote: x = 2
Vertical asymptote: x = -2
Horizontal asymptote: y = 3
x-intercept and y-intercept -5 -4 -3 -2 -1 1 2 3 4 5 7 6
x = -2
y = 3
x = 2

54. EXAMPLE: Finding the Slant Asymptote of a Rational Function
Find the slant asymptotes of f (x) =
Solution Because the degree of the numerator, 2, is exactly one more than the degree of the denominator, 1, the graph of f has a slant asymptote. To find the equation of the slant asymptote, divide x - 3 into x2 - 4x - 5: 3
Remainder

55. EXAMPLE: Finding the Slant Asymptote of a Rational Function
Find the slant asymptotes of f (x) =
Solution The equation of the slant asymptote is y = x - 1. Using our strategy for graphing rational functions, the graph of f (x) = is shown. -2 -1 4 5 6 7 8 3 2 1 -3
Vertical asymptote:
x = 3
Slant asymptote:
y = x - 1