The document discusses the concept of slope and the difference quotient formula for calculating slope. It defines a function f(x) and points P(x,f(x)) and Q(x+h, f(x+h)) on the graph of f(x). The slope of the cord connecting points P and Q is given by the difference quotient (f(x+h) - f(x))/h. An example problem calculates this slope for the specific points P(2,2) and Q(2.2,2.44) on the parabola y=x^2 - 2x + 2.

1. Slopes and the Difference Quotient

2. In order to be precise, basic geometric information
and formulas concerning graphs are given in
function notation.
Slopes and the Difference Quotient

3. Given x, the output of a function is denoted as y or
as f(x).
In order to be precise, basic geometric information
and formulas concerning graphs are given in
function notation.
Slopes and the Difference Quotient

4. In order to be precise, basic geometric information
and formulas concerning graphs are given in
function notation.
Slopes and the Difference Quotient
Given x, the output of a function is denoted as y or
as f(x). Hence the coordinate of a generic point
P(x, y) on the graph is often denoted as (x, f(x)).

5. Given x, the output of a function is denoted as y or
as f(x). Hence the coordinate of a generic point
P(x, y) on the graph is often denoted as (x, f(x)).
x
P=(x, f(x))
In order to be precise, basic geometric information
and formulas concerning graphs are given in
function notation.
y= f(x)
Slopes and the Difference Quotient

6. In order to be precise, basic geometric information
and formulas concerning graphs are given in
function notation.
x
P=(x, f(x))
y= f(x)
f(x)
Note that the f(x) = the height of the point at x.
Slopes and the Difference Quotient
Given x, the output of a function is denoted as y or
as f(x). Hence the coordinate of a generic point
P(x, y) on the graph is often denoted as (x, f(x)).

7. Let h be a small positive value,
so x+h is a point close to x,
In order to be precise, basic geometric information
and formulas concerning graphs are given in
function notation.
x
P=(x, f(x))
y= f(x)
f(x)
Note that the f(x) = the height of the point at x.
Slopes and the Difference Quotient
Given x, the output of a function is denoted as y or
as f(x). Hence the coordinate of a generic point
P(x, y) on the graph is often denoted as (x, f(x)).

8. x
P=(x, f(x))
Note that the f(x) = the height of the point at x.
Let h be a small positive value,
so x+h is a point close to x,
x+h
In order to be precise, basic geometric information
and formulas concerning graphs are given in
function notation.
f(x)
y= f(x)
Slopes and the Difference Quotient
h
Given x, the output of a function is denoted as y or
as f(x). Hence the coordinate of a generic point
P(x, y) on the graph is often denoted as (x, f(x)).

9. x
P=(x, f(x))
Note that the f(x) = the height of the point at x.
Let h be a small positive value,
so x+h is a point close to x,
x+h
then f(x+h) is the output for x+h,
and (x+h, f(x+h)) represents the
corresponding point, say Q,
on the graph.
In order to be precise, basic geometric information
and formulas concerning graphs are given in
function notation.
f(x)
y= f(x)
Slopes and the Difference Quotient
h
Given x, the output of a function is denoted as y or
as f(x). Hence the coordinate of a generic point
P(x, y) on the graph is often denoted as (x, f(x)).

10. x
P=(x, f(x))
Note that the f(x) = the height of the point at x.
Let h be a small positive value,
so x+h is a point close to x,
x+h
then f(x+h) is the output for x+h,
and (x+h, f(x+h)) represents the
corresponding point, say Q,
on the graph.
In order to be precise, basic geometric information
and formulas concerning graphs are given in
function notation.
Q=(x+h, f(x+h))
f(x)
y= f(x)
Slopes and the Difference Quotient
h
Given x, the output of a function is denoted as y or
as f(x). Hence the coordinate of a generic point
P(x, y) on the graph is often denoted as (x, f(x)).

11. x
P=(x, f(x))
Note that the f(x) = the height of the point at x.
Let h be a small positive value,
so x+h is a point close to x,
x+h
then f(x+h) is the output for x+h,
and (x+h, f(x+h)) represents the
corresponding point, say Q,
on the graph.
In order to be precise, basic geometric information
and formulas concerning graphs are given in
function notation.
Note that the f(x+h) = the height of the point at x + h.
Q=(x+h, f(x+h))
f(x)
f(x+h)
y= f(x)
Slopes and the Difference Quotient
h
Given x, the output of a function is denoted as y or
as f(x). Hence the coordinate of a generic point
P(x, y) on the graph is often denoted as (x, f(x)).

12. Recall that if (x1, y1) and (x2, y2)
are two points then the slope m
of the line connecting them is
Δy
m =
y2 – y1
= x2 – x1Δx
Slopes and the Difference Quotient
Δy
Δx

13. Recall that if (x1, y1) and (x2, y2)
are two points then the slope m
of the line connecting them is
Δy
m =
y2 – y1
= x2 – x1Δx x
P=(x, f(x))
x+h
Q=(x+h, f(x+h))
Slopes and the Difference Quotient
Δy
Δx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))
be two points on the graph of y = f(x) as shown,

14. Recall that if (x1, y1) and (x2, y2)
are two points then the slope m
of the line connecting them is
x
P=(x, f(x))
x+h
Q=(x+h, f(x+h))
Δy
m =
y2 – y1
= x2 – x1Δx
Δy
m = Δx
Slopes and the Difference Quotient
ΔyΔy
Δx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))
be two points on the graph of y = f(x) as shown, then
the slope of the cord connecting P and Q is

15. Recall that if (x1, y1) and (x2, y2)
are two points then the slope m
of the line connecting them is
x
P=(x, f(x))
x+h
Q=(x+h, f(x+h))
Δy
m =
y2 – y1
= x2 – x1Δx
Δy
m =
f(x+h) – f(x)
=Δx
Slopes and the Difference Quotient
ΔyΔy = f(x+h) – f(x)
Δx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))
be two points on the graph of y = f(x) as shown, then
the slope of the cord connecting P and Q is

16. Recall that if (x1, y1) and (x2, y2)
are two points then the slope m
of the line connecting them is
x
P=(x, f(x))
x+h
Q=(x+h, f(x+h))
Δy
m =
y2 – y1
= x2 – x1Δx
Δy
m =
f(x+h) – f(x)
= (x+h) – xΔx
Slopes and the Difference Quotient
ΔyΔy = f(x+h) – f(x)
Δx =h
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))
be two points on the graph of y = f(x) as shown, then
the slope of the cord connecting P and Q is

17. Recall that if (x1, y1) and (x2, y2)
are two points then the slope m
of the line connecting them is
x
P=(x, f(x))
x+h
Q=(x+h, f(x+h))
Δy
m =
y2 – y1
= x2 – x1Δx
Δy
m =
f(x+h) – f(x)
= (x+h) – xΔx or m = f(x+h) – f(x)
h
Slopes and the Difference Quotient
Δy = f(x+h) – f(x)
Δx =h
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))
be two points on the graph of y = f(x) as shown, then
the slope of the cord connecting P and Q is

18. Recall that if (x1, y1) and (x2, y2)
are two points then the slope m
of the line connecting them is
x
P=(x, f(x))
x+h
Q=(x+h, f(x+h))
Δy
m =
y2 – y1
= x2 – x1Δx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h))
be two points on the graph of y = f(x) as shown, then
the slope of the cord connecting P and Q is
Δy
m =
f(x+h) – f(x)
= (x+h) – xΔx or m = f(x+h) – f(x)
h
f(x+h) – f(x) = Δy and h = (x+h) – x = Δx.
This is the "difference quotient“ version of the
slope–formula using the the function notation with
Slopes and the Difference Quotient
Δy = f(x+h) – f(x)
Δx =h

19. Example A:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h,f(x+h))
with x = 2 and h = 0.2. Draw the graph and the points.
Slopes and the Difference Quotient

20. Example A:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h,f(x+h))
with x = 2 and h = 0.2. Draw the graph and the points.
Slopes and the Difference Quotient
y = x2 – 2x + 2

21. Example A:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h,f(x+h))
with x = 2 and h = 0.2. Draw the graph and the points.
Slopes and the Difference Quotient
(2, 2)
2
y = x2 – 2x + 2

22. Example A:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h,f(x+h))
with x = 2 and h = 0.2. Draw the graph and the points.
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2.
Slopes and the Difference Quotient
(2.2, 2.44)
(2, 2)
2 2.2
y = x2 – 2x + 2
h=0.2

23. Example A:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h,f(x+h))
with x = 2 and h = 0.2. Draw the graph and the points.
f(x+h) – f(x)
h
Use the difference quotient,
the slope of the cord is
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2
Slopes and the Difference Quotient
(2.2, 2.44)
(2, 2)
2 2.2
y = x2 – 2x + 2
h=0.2

24. Example A:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h,f(x+h))
with x = 2 and h = 0.2. Draw the graph and the points.
f(x+h) – f(x)
h
Use the difference quotient,
the slope of the cord is
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2
f(2.2) – f(2)=
Slopes and the Difference Quotient
(2.2, 2.44)
(2, 2)
2 2.2
y = x2 – 2x + 2
h=0.2

25. Example A:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h,f(x+h))
with x = 2 and h = 0.2. Draw the graph and the points.
f(x+h) – f(x)
h
Use the difference quotient,
the slope of the cord is
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2
f(2.2) – f(2)=
Slopes and the Difference Quotient
(2.2, 2.44)
(2, 2)
2 2.2
y = x2 – 2x + 2
h=0.2
f(2.2)–f(2)

26. Example A:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h,f(x+h))
with x = 2 and h = 0.2. Draw the graph and the points.
f(x+h) – f(x)
h
Use the difference quotient,
the slope of the cord is
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2
f(2.2) – f(2)
0.2
=
Slopes and the Difference Quotient
(2.2, 2.44)
(2, 2)
2 2.2
y = x2 – 2x + 2
h=0.2
f(2.2)–f(2)

27. Example A:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h,f(x+h))
with x = 2 and h = 0.2. Draw the graph and the points.
f(x+h) – f(x)
h
Use the difference quotient,
the slope of the cord is
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2
f(2.2) – f(2)
0.2
=
Slopes and the Difference Quotient
(2.2, 2.44)
(2, 2)
2 2.2
y = x2 – 2x + 2
h=0.2
f(2.2)–f(2)
0.2

28. Example A:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h,f(x+h))
with x = 2 and h = 0.2. Draw the graph and the points.
f(x+h) – f(x)
h
Use the difference quotient,
the slope of the cord is
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2
f(2.2) – f(2)
0.2
=
=
2.44 – 2
0.2
Slopes and the Difference Quotient
(2.2, 2.44)
(2, 2)
2 2.2
y = x2 – 2x + 2
h=0.2
f(2.2)–f(2)
0.2

29. Example A:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h,f(x+h))
with x = 2 and h = 0.2. Draw the graph and the points.
f(x+h) – f(x)
h
Use the difference quotient,
the slope of the cord is
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2
f(2.2) – f(2)
0.2
=
=
2.44 – 2
0.2 =
0.44
0.2
Slopes and the Difference Quotient
(2.2, 2.44)
(2, 2)
2 2.2
y = x2 – 2x + 2
h=0.2
f(2.2)–f(2)
0.2

30. Example A:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h,f(x+h))
with x = 2 and h = 0.2. Draw the graph and the points.
f(x+h) – f(x)
h
Use the difference quotient,
the slope of the cord is
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2
f(2.2) – f(2)
0.2
=
=
2.44 – 2
0.2
= 2.2
=
0.44
0.2
Slopes and the Difference Quotient
(2.2, 2.44)
(2, 2)
2 2.2
y = x2 – 2x + 2
h=0.2
f(2.2)–f(2)
0.2

31. Example A:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (x, f(x)) and (x+h,f(x+h))
with x = 2 and h = 0.2. Draw the graph and the points.
f(x+h) – f(x)
h
Use the difference quotient,
the slope of the cord is
We want the slope of the cord connecting the points
whose x-coordinates are x = 2 and x + h = 2 + h = 2.2
f(2.2) – f(2)
0.2
=
=
2.44 – 2
0.2
= 2.2
=
0.44
0.2
Slopes and the Difference Quotient
(2.2, 2.44)
(2, 2)
2 2.2
y = x2 – 2x + 2
h=0.2
f(2.2)–f(2)
0.2
slope m = 2.2

32. b. Given f(x) = x2 – 2x + 2, simplify the formula for the
slope of the cord connecting the points (x, f(x)) and
(x+h, f(x+h)).
Slopes and the Difference Quotient

33. b. Given f(x) = x2 – 2x + 2, simplify the formula for the
slope of the cord connecting the points (x, f(x)) and
(x+h, f(x+h)).
We are to simplify the difference quotient formula
with f(x) = x2 – 2x + 2
Slopes and the Difference Quotient

34. b. Given f(x) = x2 – 2x + 2, simplify the formula for the
slope of the cord connecting the points (x, f(x)) and
(x+h, f(x+h)).
f(x+h) – f(x)
h
We are to simplify the difference quotient formula
with f(x) = x2 – 2x + 2
=
(x+h)2 – 2(x+h) + 2
h
Slopes and the Difference Quotient

35. b. Given f(x) = x2 – 2x + 2, simplify the formula for the
slope of the cord connecting the points (x, f(x)) and
(x+h, f(x+h)).
f(x+h) – f(x)
h
We are to simplify the difference quotient formula
with f(x) = x2 – 2x + 2
=
(x+h)2 – 2(x+h) + 2 – [ x2 – 2x + 2]
h
Slopes and the Difference Quotient

36. b. Given f(x) = x2 – 2x + 2, simplify the formula for the
slope of the cord connecting the points (x, f(x)) and
(x+h, f(x+h)).
f(x+h) – f(x)
h
We are to simplify the difference quotient formula
with f(x) = x2 – 2x + 2
=
(x+h)2 – 2(x+h) + 2 – [ x2 – 2x + 2]
h
2xh – 2h + h2
h
=
Slopes and the Difference Quotient

37. b. Given f(x) = x2 – 2x + 2, simplify the formula for the
slope of the cord connecting the points (x, f(x)) and
(x+h, f(x+h)).
f(x+h) – f(x)
h
We are to simplify the difference quotient formula
with f(x) = x2 – 2x + 2
=
(x+h)2 – 2(x+h) + 2 – [ x2 – 2x + 2]
h
2xh – 2h + h2
h
= 2x – 2 + h.=
Slopes and the Difference Quotient

38. *Another version of the difference
quotient formula is to use points
P = (a, f(a)) and Q= (b, f(b)),
b. Given f(x) = x2 – 2x + 2, simplify the formula for the
slope of the cord connecting the points (x, f(x)) and
(x+h, f(x+h)).
f(x+h) – f(x)
h
We are to simplify the difference quotient formula
with f(x) = x2 – 2x + 2
=
(x+h)2 – 2(x+h) + 2 – [ x2 – 2x + 2]
h
2xh – 2h + h2
h
= 2x – 2 + h.=
Slopes and the Difference Quotient

39. *Another version of the difference
quotient formula is to use points
P = (a, f(a)) and Q= (b, f(b)), we get
Δy
m =
f(b) – f(a)
= b – aΔx
b. Given f(x) = x2 – 2x + 2, simplify the formula for the
slope of the cord connecting the points (x, f(x)) and
(x+h, f(x+h)).
f(x+h) – f(x)
h
We are to simplify the difference quotient formula
with f(x) = x2 – 2x + 2
=
(x+h)2 – 2(x+h) + 2 – [ x2 – 2x + 2]
h
2xh – 2h + h2
h
= 2x – 2 + h.=
Slopes and the Difference Quotient

40. *Another version of the difference
quotient formula is to use points
P = (a, f(a)) and Q= (b, f(b)), we get
a
P=(a, f(a))
b
Q=(b, f(b))
Δy
m =
f(b) – f(a)
= b – aΔx
b. Given f(x) = x2 – 2x + 2, simplify the formula for the
slope of the cord connecting the points (x, f(x)) and
(x+h, f(x+h)).
f(x+h) – f(x)
h
We are to simplify the difference quotient formula
with f(x) = x2 – 2x + 2
=
(x+h)2 – 2(x+h) + 2 – [ x2 – 2x + 2]
h
2xh – 2h + h2
h
= 2x – 2 + h.=
Slopes and the Difference Quotient

41. *Another version of the difference
quotient formula is to use points
P = (a, f(a)) and Q= (b, f(b)), we get
a
P=(a, f(a))
b
Q=(b, f(b))
Δy
m =
f(b) – f(a)
= b – aΔx
b-a=Δx
f(b)–f(a) = Δy
b. Given f(x) = x2 – 2x + 2, simplify the formula for the
slope of the cord connecting the points (x, f(x)) and
(x+h, f(x+h)).
f(x+h) – f(x)
h
We are to simplify the difference quotient formula
with f(x) = x2 – 2x + 2
=
(x+h)2 – 2(x+h) + 2 – [ x2 – 2x + 2]
h
2xh – 2h + h2
h
= 2x – 2 + h.=
Slopes and the Difference Quotient

42. Example B:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (a, f(a)) and (b,f(b)) with a = 3
and b = 5.
Slopes and the Difference Quotient

43. Example B:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (a, f(a)) and (b,f(b)) with a = 3
and b = 5.
We want the slope of the cord connecting the points
whose x-coordinates are a = 3 and b = 5
Slopes and the Difference Quotient

44. Example B:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (a, f(a)) and (b,f(b)) with a = 3
and b = 5.
f(b) – f(a)
b – a
Using the formula, the slope of
the cord is
We want the slope of the cord connecting the points
whose x-coordinates are a = 3 and b = 5
Slopes and the Difference Quotient

45. Example B:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (a, f(a)) and (b,f(b)) with a = 3
and b = 5.
Using the formula, the slope of
the cord is
We want the slope of the cord connecting the points
whose x-coordinates are a = 3 and b = 5
f(5) – f(3)
5 – 3
=
Slopes and the Difference Quotient
f(b) – f(a)
b – a

46. Example B:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (a, f(a)) and (b,f(b)) with a = 3
and b = 5.
Using the formula, the slope of
the cord is
We want the slope of the cord connecting the points
whose x-coordinates are a = 3 and b = 5
f(5) – f(3)
5 – 3
=
=
17 – 5
2
Slopes and the Difference Quotient
f(b) – f(a)
b – a

47. Example B:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (a, f(a)) and (b,f(b)) with a = 3
and b = 5.
Using the formula, the slope of
the cord is
We want the slope of the cord connecting the points
whose x-coordinates are a = 3 and b = 5
f(5) – f(3)
5 – 3
=
=
17 – 5
2
= 6=
12
2
Slopes and the Difference Quotient
f(b) – f(a)
b – a

48. Example B:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (a, f(a)) and (b,f(b)) with a = 3
and b = 5.
Using the formula, the slope of
the cord is
We want the slope of the cord connecting the points
whose x-coordinates are a = 3 and b = 5
f(5) – f(3)
5 – 3
=
=
17 – 5
2
= 6
(5, 17)
(3, 5)
3 5=
12
2
Slopes and the Difference Quotient
f(b) – f(a)
b – a

49. Example B:
a. Given f(x) = x2 – 2x + 2, find the slope of the cord
connecting the points (a, f(a)) and (b,f(b)) with a = 3
and b = 5.
Using the formula, the slope of
the cord is
We want the slope of the cord connecting the points
whose x-coordinates are a = 3 and b = 5
f(5) – f(3)
5 – 3
=
=
17 – 5
2
= 6
(5, 17)
(3, 5)
3 5=
12
2
12
2
slope m = 6
Slopes and the Difference Quotient
f(b) – f(a)
b – a

50. b. Given f(x) = x2 – 2x + 2, simplify the difference
quotient slope of the cord connecting the points
(a, f(a)) and (b, f(b)).
(b, f(b))
(a, f(a))
a b
f(b)-f(a)
b-a
Slopes and the Difference Quotient

51. b. Given f(x) = x2 – 2x + 2, simplify the difference
quotient slope of the cord connecting the points
(a, f(a)) and (b, f(b)).
f(b) – f(a)
b – a
We are to simplify the 2nd form of the difference
quotient formula with f(x) = x2 – 2x + 2
(b, f(b))
(a, f(a))
a b
f(b)-f(a)
b-a
Slopes and the Difference Quotient

52. b. Given f(x) = x2 – 2x + 2, simplify the difference
quotient slope of the cord connecting the points
(a, f(a)) and (b, f(b)).
f(b) – f(a)
b – a
We are to simplify the 2nd form of the difference
quotient formula with f(x) = x2 – 2x + 2
=
b2 – 2b + 2 – [ a2 – 2a + 2]
b – a
(b, f(b))
(a, f(a))
a b
f(b)-f(a)
b-a
Slopes and the Difference Quotient

53. b. Given f(x) = x2 – 2x + 2, simplify the difference
quotient slope of the cord connecting the points
(a, f(a)) and (b, f(b)).
f(b) – f(a)
b – a
We are to simplify the 2nd form of the difference
quotient formula with f(x) = x2 – 2x + 2
=
b2 – 2b + 2 – [ a2 – 2a + 2]
b – a
=
b2 – a2 – 2b + 2a
b – a
(b, f(b))
(a, f(a))
a b
f(b)-f(a)
b-a
Slopes and the Difference Quotient

54. b. Given f(x) = x2 – 2x + 2, simplify the difference
quotient slope of the cord connecting the points
(a, f(a)) and (b, f(b)).
f(b) – f(a)
b – a
We are to simplify the 2nd form of the difference
quotient formula with f(x) = x2 – 2x + 2
=
b2 – 2b + 2 – [ a2 – 2a + 2]
b – a
=
b2 – a2 – 2b + 2a
b – a
= (b – a)(b + a) – 2(b – a)
b – a
(b, f(b))
(a, f(a))
a b
f(b)-f(a)
b-a
Slopes and the Difference Quotient

55. b. Given f(x) = x2 – 2x + 2, simplify the difference
quotient slope of the cord connecting the points
(a, f(a)) and (b, f(b)).
f(b) – f(a)
b – a
We are to simplify the 2nd form of the difference
quotient formula with f(x) = x2 – 2x + 2
=
b2 – 2b + 2 – [ a2 – 2a + 2]
b – a
=
b2 – a2 – 2b + 2a
b – a
= (b – a)(b + a) – 2(b – a)
b – a
= (b – a) [(b + a) – 2]
b – a
(b, f(b))
(a, f(a))
a b
f(b)-f(a)
b-a
Slopes and the Difference Quotient

56. b. Given f(x) = x2 – 2x + 2, simplify the difference
quotient slope of the cord connecting the points
(a, f(a)) and (b, f(b)).
f(b) – f(a)
b – a
We are to simplify the 2nd form of the difference
quotient formula with f(x) = x2 – 2x + 2
=
b2 – 2b + 2 – [ a2 – 2a + 2]
b – a
=
b2 – a2 – 2b + 2a
b – a
= (b – a)(b + a) – 2(b – a)
b – a
= (b – a) [(b + a) – 2]
b – a
= b + a – 2
(b, f(b))
(a, f(a))
a b
f(b)-f(a)
b-a
Slopes and the Difference Quotient

57. HW
Given the following f(x), x, and h find f(x+h) – f(x)
1. y = 3x+2, x= 2, h = 0.1 2. y = -2x + 3, x= -4, h = 0.05
3. y = 2x2 + 1, x = 1, h = 0.1 4. y = -x2 + 3, x= -2, h = -0.2
Given the following f(x), simplify Δy = f(x+h) – f(x)
5. y = 3x+2 6. y = -2x + 3
7. y = 2x2 + 1 8. y = -x2 + 3
Simplify the difference quotient
f(x+h) – f(x)
h
of the following functions
9. y = -4x + 3 10. y = mx + b
11. y = 3x2 – 2x +2 12. y = -2x2 + 3x -1
13 – 16, simplify the difference quotient
f(b) – f(a)
b – a
of the given functions.
Slopes and the Difference Quotient