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Slopes and the Difference Quotient
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In order to discuss mathematics precisely, 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).
In order to discuss mathematics precisely, 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 general point P on the graph is often denoted as (x, f(x)).
In order to discuss mathematics precisely, 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 general point P on the graph is often denoted as (x, f(x)).
x
P=(x, f(x))
In order to discuss mathematics precisely, basic geometric information and formulas concerning graphs are given in function notation.
y= f(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 general point P on the graph is often denoted as (x, f(x)).
In order to discuss mathematics precisely, 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 P.
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 general point P on the graph is often denoted as (x, f(x)).
Let h be a small positive value, so x+h is a point close to x,
In order to discuss mathematics precisely, 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 P.
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 general point P on the graph is often denoted as (x, f(x)).
x
P=(x, f(x))
Note that the f(x) = the height of the point P.
Let h be a small positive value, so x+h is a point close to x,
x+h
In order to discuss mathematics precisely, basic geometric information and formulas concerning graphs are given in function notation.
f(x)
y= f(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 general point P on the graph is often denoted as (x, f(x)).
x
P=(x, f(x))
Note that the f(x) = the height of the point P.
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 discuss mathematics precisely, basic geometric information and formulas concerning graphs are given in function notation.
f(x)
y= f(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 general point P on the graph is often denoted as (x, f(x)).
x
P=(x, f(x))
Note that the f(x) = the height of the point P.
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 discuss mathematics precisely, 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
Given x, the output of a function is denoted as y or as f(x). Hence the coordinate of a general point P on the graph is often denoted as (x, f(x)).
x
P=(x, f(x))
Note that the f(x) = the height of the point P.
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 discuss mathematics precisely, basic geometric information and formulas concerning graphs are given in function notation.
Note that the f(x+h) = the height of the point Q.
Q=(x+h, f(x+h))
f(x) f(x+h)
y= f(x)
Slopes and the Difference Quotient
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 – x1 Δx
Slopes and the Difference Quotient
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 – x1 Δx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) as shown in the picture,
Slopes and the Difference Quotient
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 – x1 Δx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) as shown in the picture,
Slopes and the Difference Quotient
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 – x1 Δx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) as shown in the picture, then the slope of the cord connecting P and Q (in function notation) is
Slopes and the Difference Quotient
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 – x1 Δx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) as shown in the picture, then the slope of the cord connecting P and Q (in function notation) is
Δy m =
f(x+h) – f(x) =
(x+h) – x Δx
Slopes and the Difference Quotient
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 – x1 Δx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) as shown in the picture, then the slope of the cord connecting P and Q (in function notation) is
Δy m =
f(x+h) – f(x) =
(x+h) – x Δx or m =
f(x+h) – f(x) h
Slopes and the Difference Quotient
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 – x1 Δx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) as shown in the picture, then the slope of the cord connecting P and Q (in function notation) is
Δy m =
f(x+h) – f(x) =
(x+h) – x Δx or m =
f(x+h) – f(x) h
This is the "difference quotient" formula for slopes
Slopes and the Difference Quotient
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 – x1 Δx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) as shown in the picture, then the slope of the cord connecting P and Q (in function notation) is
Δy m =
f(x+h) – f(x) =
(x+h) – x Δx or m =
f(x+h) – f(x) h
because f(x+h) – f(x) = difference in heightThis is the "difference quotient" formula for slopes
Slopes and the Difference Quotient
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 – x1 Δx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) as shown in the picture, then the slope of the cord connecting P and Q (in function notation) is
Δy m =
f(x+h) – f(x) =
(x+h) – x Δx or m =
f(x+h) – f(x) h
f(x+h)–f(x) = Δy
because f(x+h) – f(x) = difference in heightThis is the "difference quotient" formula for slopes
Slopes and the Difference Quotient
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 – x1 Δx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) as shown in the picture, then the slope of the cord connecting P and Q (in function notation) is
Δy m =
f(x+h) – f(x) =
(x+h) – x Δx or m =
f(x+h) – f(x) h
f(x+h)–f(x) = Δy
because f(x+h) – f(x) = difference in height andh = (x+h) – x = difference in the x's, as shown.
This is the "difference quotient" formula for slopes.
Slopes and the Difference Quotient
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 – x1 Δx
Let (x1,y1) = P = (x, f(x)) and (x2,y2) = Q = (x+h, f(x+h)) as shown in the picture, then the slope of the cord connecting P and Q (in function notation) is
Δy m =
f(x+h) – f(x) =
(x+h) – x Δx or m =
f(x+h) – f(x) h
h=Δx
f(x+h)–f(x) = Δy
because f(x+h) – f(x) = difference in height andh = (x+h) – x = difference in the x's, as shown.
This is the "difference quotient" formula for slopes.
Slopes and the Difference Quotient
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.
Slopes and the Difference Quotient
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.
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
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
Slopes and the Difference Quotient
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) =
Slopes and the Difference Quotient
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
=
Slopes and the Difference Quotient
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
Slopes and the Difference Quotient
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 =0.44 0.2
Slopes and the Difference Quotient
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
=0.44 0.2
Slopes and the Difference Quotient
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
Slopes and the Difference Quotient
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
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
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
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
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
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
Another version of the difference quotient formula is to use pointsP = (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
Another version of the difference quotient formula is to use pointsP = (a, f(a)) and Q= (b, f(b)). We get
Δy m =
f(b) – f(a) =
b – a Δx
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
Another version of the difference quotient formula is to use pointsP = (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 – a Δx
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
Another version of the difference quotient formula is to use pointsP = (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 – a Δx
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
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
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
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 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
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 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
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 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
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 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
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
Use 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
Slopes and the Difference Quotient
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
Use 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
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
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
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
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
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
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
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
HWGiven 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 = 0.053. 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 + 37. y = 2x2 + 1 8. y = -x2 + 3
Simplify the difference quotient: f(x+h) – f(x) h
of the following functions. (Make sure the factor “h” in the denominator is cleared away in the answer)
9. y = -4x + 3 10. y = mx + b
11. y = 3x2 – 2x +2 12. y = -2x2 + 3x -1
Slopes and the Difference Quotient
13. y = 14. y =
2 x + 3
2x – 1 x – 2
16. y = (3 – x)1/215. y = x1/2
