Here are the key steps to find the instantaneous rate of change using a graphing calculator: 1. Graph the function over the appropriate domain. 2. Use the arrow keys to move the cursor to the point where you want to find the instantaneous rate of change. 3. Press the TRACE button and select the tangent option. 4. The calculator will display the slope of the tangent line, which is the instantaneous rate of change at that point. 5. For example, if finding the IROC at x=1 for the function f(x) = x3, you would: a) Graph f(x) = x3 b) Use arrows to move cursor

1. A power function is one of the form where
y = xn
Where n is any real number constant.
A power function is the simplest type of polynomial function
Example:
y= x y = x2 y = x3

2. Degree Constant
term ( a0 )
F(X) =3x5 + 2x3 + x2 -1
Leading variable
coefficient
• n must be a whole number

3. ODD-DEGREE POWER FUNCTIONS HAVE POINT SYMMETRY ABOUT THE
ORIGIN, (0,0)

4. Bracket Inequality Number line In words
interval
(a,b) a <x<b X is
  greater
a b than x
than a and
less than
b
(a,b] a <x ≤ b   X is
a b greater
than a and
less than
or equal
to b

5. Key features of the Y=xn ; n = Y=xn; n = even
graph odd
Domain {x € R } {x € R }
Range {y€R} { y € R/ y ≥ 0 }
End behaviour as x y ∞ y
∞
End behaviour as x y- ∞ y ∞
∞
II I
III IV

6. VIDEO TIME ! ^_^
Local maximum
Absolute minimum
Local minimum

7. Graphs of polynomial functions with odd degree
Negative leading
Positive leading
coefficient
coefficient
 odd-degree polynomials have at least one x-intercepts, up to a max of
nx-intercepts.
 The domain of all odd-degree polynomials is {xε R} and the range is
{ yε R}
 Odd degree functions have NO maximum point * minimum point
 Odd-degree polynomials may have point symmetry

8. Graphs of Polynomial Functions with Even Degree
Positive leading coefficient Negative leasing coefficient
 Even-degree polynomials may have from zero to a maximum
of n x-intercepts, where n is the degree of the function
 The domain of all even-degree polynomials is {xε R}
 Even-degree polynomials may have line symmetry.

10.  Using Y X – Intercepts
( Zero‟s )
 The degree of the function
 Sign of leading coefficient
 Graphic Calculator

11. X intercept(s) : Y=0 Y intercept(s) : X=0
F(x)=(X–1)(X+1)(X–3)
Y=0 X=0
X–1=0 Y=X–1
=1 = -1
X+1=0 Y=X+1
= -1 = 1
X–3=0 Y=X–3
=3 = -3

12. F(x)=(X–1)(X+1)(X–3)
X3 – 3X2 – X + 3
(Degree: 3; Three possible
solutions)

13. Odd Degree Polynomials
Even Degree Polynomials
POSITIVE NEGATIVE

14. X3 – 3X2 – X + 3
Positive Leading Coefficient
Degree of 3 = Odd

15. Steps
 Clear Your Memory ( Just In Case )
 Press “ ( Y = )“
 Key in formula
 Press “ Graph “

17. y= a f [k(x+b)] + c

18. „-‟ means reflection in the x-axis
a1 means vertical stretch of factor a
0a1 means vertical compression of factor a
k1 means horizontal compression of factor 1/k
0k1 means horizontal stretch of factor 1/k
b0 means horizontal shift b unit left
b0 means horizontal shift b unit right
c0 means vertical shift c unit up
c0 means vertical shift c unit down
Note: f(-x) means a reflection in the y-axis
-f(x) means a reflection in the x-axis

19. Reflection over the x-axis
-f(x) reflects f(x) over the x-axis.

20. Reflection over the y-axis
f(-x) reflects f(x) over the y-axis.

21. a
Vertical Stretch or Compress
a[f(x)] stretches/compresses f(x) vertically

22. k
Horizontal Stretch or Compress
f(kx) stretches/compresses f(x) horizontally

23. b
Slide to the right or left
f(x +/- b) translates f(x) horizontally

24. c
Slide upward or downward
[f(x)+/-] c translates f(x) vertically

25. f(x)=x 2
f(x)= 4(1/3 x)2 -
7

28.  Measure of how quickly one quantity (the
dependent variable) changes with respect to
another quantity (the independent variable)
 Types of rate of change:-
◦ Average
◦ Instantaneous

29. Average Rate Of Change
 Change that takes place over an interval
Instantaneous Rate Of
Change
 Change that takes place in an instant

30.  Rate of Change
- Average Rate of Change
- Instantaneous Rate of Change
 Slope = Gradient
 Secant = Line that connects two points on a
curve
 Points [Example: P (x,y)]

31.  Refers to the slope of the secant between the
points.
 Average rate of change = ∆y/∆x
= change in
y/change in x
= y2-y1
x2-x1

32. Time (t) in hours ; x 0 1 2 3 3.5
Distance d(t) in km; y 50 98 156 200 256
• Average velocity= slope/gradient=m= ∆y/∆x
Average rate of change = (y2-y1)/(x2-x1)
Time ∆d ∆t ∆d/∆t = Average
ROC
0t1 d(1) – d(o) = 48km 1-0=1 48 km/h
1t2 d(2) – d(1) = 58km 2-1=1 58km/h
2t3 d(3) – d(2) = 44km 3-2=1 44km/h
3t4 d(3.5) – d(3) = 3.5-3=0.5 112km/h
56km

33.  John drops a ball from the cliff of a hill of 150m.
After t seconds it is d meters above the ground,
where d(t) = 90– 4t2; 0t5
a) Calculate the average rate of change of the distance
of the ball above the ground between the times t =
1 and t = 4
Average velocity = d(4) – d(1) / 4-1
=26-86 /4-1
= - 20m/s
b) Find the average rate of change of the distance of
the ball above the ground between t=1 and t=2.5
Average velocity = d(2.5) – d(1) / 2.5 – 1
= 65-86/2.5-1
= -14m/s

34. Amount Of Money
1000
800
600
400 Amount Of Money
200
0
2 4 6 8 10 12

35.  Find the average rate of change of the amount of
money in the account:
i) Month 2 to month 4
ii) Month 4 to month 8
iii) Month 8 to month 10
iv) Month 8 to month 12
i) A.R.O.C = 400-200/ 4-
= 50
ii) A.R.OC = 800-400/ 8-4
= 100
iii) A.R.O.C = 800-800/10-8
=0
iv) A.R.O.C = 500-800/ 12-8
= -75

36.  How can you tell that the A.R.O.C is
positive/negative by examining
a) the table of values ?
- POSITIVE because as x increases, y increases
- NEGATIVE because as x increases, y
decreases
b) the graph ?
- POSITIVE because the graph is increasing
from left to right
- NEGATIVE because the graph is decreasing
from left to right

37.  How can you tell the rate of change is constant or
non-constant by examining
a) The table of values ?
- CONSTANT= the change in the dependent variables
is the same for each one unit increase in the
independent variable.
- NON- CONSTANT = the change in the dependent
variables varies for each one unit increase in the
independent variable.
b) The graph ?
- CONSTANT = Graph is linear
- NON- CONSTANT= Graph is not linear
c) The average rate of change ?
- CONSTANT = A.R.O.C is the same
- NON- CONSTANT = A.R.O.C varies

39. -As a point Y becomes very close to a tangent point
X, the slope of the secant line becomes closer to
(approaches) the slope of the tangent line.
- An arrow is used to denote the word
“approaches”.
-Example: As Y X, the slope of secant XY the
slope of the tangent at X.
- Thus, the A.R.O.C between X and Y becomes closer to
the value of the instantaneous rate of change at X.

40.  From a graph
- Draw a tangent line on the graph and
estimating the slope of that tangent from the
graph

41.  From the table of values method
x y Slope of secant between X= 0.8
0.6 5.0
X= 1.0
= 4.9-5.3/1.0-0.8
0.8 5.3
= -2
1.0 4.9
Slope of secant between X= 1.0
X= 1.3
1.3 3.0
= 3.0- 4.9/1.3-1.0
= -6.3
1.5 3.53
Instantaneous rate of change at
X= 1.0= a+b/2
Therefore,
= -2+(-6.3)/2
= -4.15

42.  From an equation method
- Looking for a trend as the slopes of secants
get closer and closer to the slope of the
tangent.
- Example:
A ball is tossed up in the air so that it‟s position s in meters and
at time t in second, is given by:
s(t) = -5t2 + 30t + 2
Interval ∆s ∆t Average rate of change
1t2 S(2) – s(1) = 15 2-1 = 1 15
1t1.5 S(1.5) –s(1) = 1.75 1.5 -1 = 0.5 17.5
1t1.1 S(1.1) – s(1) = 1.1-1 = 0.1 19.5
1.95
1t1.0 S(1.01) – s(1) = 1.01 – 1 = 19.95
1 1.995 0.01
Therefore, the I.R.O.C at t=1, approximately 20m/s.

43.  Using the graphing calculator (Tangent
Operation)
- Page 515 in the text book