slopes and difference quotient

1. 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.
f(x+h) – f(x)
h
Using the difference
quotient, the slope 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
(2.2, 2.44)
(2, 2)
2 2.2
=
0.44
0.2
0.44
0.2
slope m = 2.2
Slopes and the Difference Quotient

2. The algebra of calculating the slopes of some basic
types of functions are given below.
Slope Algebra
Example B. Simplify the difference quotient.
Make sure the h is cancelled.
f(x+h) – f(x)
h
=
–2(x+h)2 + 3(x+h) + 1 – [–2x2 +3x +1]
h
a. (2nd degree polynomials) f(x) = –2x2 + 3x + 1
–2x2 –4xh –2h2 +3x +3h +1 – [–2x2 +3x +1]
h
=
–4xh –2h2 +3h
h
=
h(–4x –2h +3)
h
= = –4x –2h +3
See 1.2 slide 109.

3. Rational Expressions
–x + h – 1
2
x – 1
2
h
Multiply the top and bottom by (x + h – 1)(x – 1)
to remove fractions in the numerator.
(x + h –1)(x – 1)
(x + h –1)(x – 1)*
=
–2(x – 1) 2(x + h –1)
h
b. (Simple rational function) f(x) x – 1
2
Simplify the difference quotient.
f(x+h) – f(x)
h =
–x + h – 1
2
x – 1
2
h
(x + h –1)(x – 1) =
– 2h
h(x + h –1)(x – 1)
=
–2
(x + h –1)(x – 1)
[ ]
See 1.2 slide 109.=

4. Rational Expressions
h
c. (Simple root function) f(x) = √2x – 3
Simplify the difference quotient.
f(x+h) – f(x)
h =
√2(x + h) – 3 – √2x – 3
Rationalize the numerator to cancel the h in the
denominator.
h
√2x + 2h – 3 – √2x – 3
*
=
√2x + 2h – 3 +√2x – 3
√2x + 2h – 3 +√2x – 3
2x + 2h – 3 – (2x – 3)
h √2x – 3√2x + h – 3 +
1
= 2h
h √2x – 3√2x + h – 3 +
1
=
√2x – 3√2x + h – 3 +
2
*
*

5. Example C.
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 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

6. 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