2 8 difference
.
quotient
formula :
f(x + n)
f(x)] this will cancel
-
n
recall :
example I: find the difference quotient (+ n) + x = + n2
f(x) =
3x -
5x + q
given
:
Step 1 :
find f(x + h)
+(x) =
3x2 -
5x + 9
3(x + n)2 5(x m) + +9
f(x + )
-
=
v)(4)v
= 3(x2 2hx + n2)
+
-
5(x + ) + 9
= 3x2 + 6hx + 342 -
5x + 5h + 9
Step 2 : find the difference
f(x +
h) -
f(x) = 3x + 6hx + 3h2 -
5x -
5n + a -
(3x -
5x + a)
n h
take the variables =
3x2 + 6hx + 3h2 -
5x -
5n + 9 -
3x +5x -
9
Without "n" h
4hx + 342 -
54
see
=
n
- 4
-
5
=
h(bx + 3n -
5)
take out the "h" h
variable
, 2
example :
formula f(x + h) f(x)
-
remember the
:
Y
given :
f(x) = 2x + 7x -
= 2(x + n(2 + 7(x + n)
h
+ m)
= 2(x + h)(x+ n) + T(x
b
= 2(x + 2nx + nz) + 7(x + h)
n
2x2 + 4hx + 24 + 7x + 7n
-
(2x2 + 7x -
1)
=
h
Only
2)
2(x
yxx
"x"
= ( + 44x + 2n + + 2n
- -
h
answer
-
= 4hx + 24 + Th = h(4ux + 22+ 74) = 4x + 22 + 7
h
n
, 2 1 .
rectangular coordinates
i . midpoint formula M(x v) ,
=
A(x 4) B(y 4)
and
points : , ,
example I :
given : A(2 , 5) and (8 3) ,
X , Y x2 42
midpoint formula
:
=853
=
2 8 . =
(3 4) ,
given two points : A (X2 , Y )
=
and B (x .. Y)
ii-distance formula )
-
(x2
"
d (A , B) =
-
x , ) + (x2 -
y .
example I :
find the distance between the point : A(5 1) , and B (7 3) .
X Y(
X242 ,
solution :
(x2 ) (x2 ) V4
148
+
-
+
-
y =
a(A B)
x ,
.
=
,
=8
=
5 7) (1 3)2 -
+ -
= 25 2
237
=
12) ( 2) +
-
, . 2 circles
2
&
-
fixed point : Center (h k) ,
fixed distance radius (r)
-
:
Standard form of the equation of a circle :
formula : x-h)" +
(x -
k) = r2 -
memorize
example : 1 Write an equation of the circle
a) center : (1 ,
4) , radius :
6 b) Center :
(3 0).
,
radius : 29
r
(h , k)
solution : (x n) (y k))
-
+
-
= r
(x n)2 (y k)2 r2
- + - =
(x -
3) (y 0)2 (a)
+ - =
(6)
(x (1)) (x 4) (x 3)
=
yz
+
-
29
-
-
+ =
( + 1) +
(x -
4)2 =
36
example 2 : find the center of the radius
a)(x 1) - + (x -
3) =
25
b)x2 + (y + 4) =
59
( 1)2 + (x -
3)2 (5) =
0) (y - 4)] (259)"
(x
-
=
+
-
(( -h)
ra
(x h)" + (x k)2 -
=
· =
(x k) 1
-
-
+
h 1 =
(hik)
k= 3 Center (1 3) .
radius =
5 Center (0-4) radius : 5
.
quotient
formula :
f(x + n)
f(x)] this will cancel
-
n
recall :
example I: find the difference quotient (+ n) + x = + n2
f(x) =
3x -
5x + q
given
:
Step 1 :
find f(x + h)
+(x) =
3x2 -
5x + 9
3(x + n)2 5(x m) + +9
f(x + )
-
=
v)(4)v
= 3(x2 2hx + n2)
+
-
5(x + ) + 9
= 3x2 + 6hx + 342 -
5x + 5h + 9
Step 2 : find the difference
f(x +
h) -
f(x) = 3x + 6hx + 3h2 -
5x -
5n + a -
(3x -
5x + a)
n h
take the variables =
3x2 + 6hx + 3h2 -
5x -
5n + 9 -
3x +5x -
9
Without "n" h
4hx + 342 -
54
see
=
n
- 4
-
5
=
h(bx + 3n -
5)
take out the "h" h
variable
, 2
example :
formula f(x + h) f(x)
-
remember the
:
Y
given :
f(x) = 2x + 7x -
= 2(x + n(2 + 7(x + n)
h
+ m)
= 2(x + h)(x+ n) + T(x
b
= 2(x + 2nx + nz) + 7(x + h)
n
2x2 + 4hx + 24 + 7x + 7n
-
(2x2 + 7x -
1)
=
h
Only
2)
2(x
yxx
"x"
= ( + 44x + 2n + + 2n
- -
h
answer
-
= 4hx + 24 + Th = h(4ux + 22+ 74) = 4x + 22 + 7
h
n
, 2 1 .
rectangular coordinates
i . midpoint formula M(x v) ,
=
A(x 4) B(y 4)
and
points : , ,
example I :
given : A(2 , 5) and (8 3) ,
X , Y x2 42
midpoint formula
:
=853
=
2 8 . =
(3 4) ,
given two points : A (X2 , Y )
=
and B (x .. Y)
ii-distance formula )
-
(x2
"
d (A , B) =
-
x , ) + (x2 -
y .
example I :
find the distance between the point : A(5 1) , and B (7 3) .
X Y(
X242 ,
solution :
(x2 ) (x2 ) V4
148
+
-
+
-
y =
a(A B)
x ,
.
=
,
=8
=
5 7) (1 3)2 -
+ -
= 25 2
237
=
12) ( 2) +
-
, . 2 circles
2
&
-
fixed point : Center (h k) ,
fixed distance radius (r)
-
:
Standard form of the equation of a circle :
formula : x-h)" +
(x -
k) = r2 -
memorize
example : 1 Write an equation of the circle
a) center : (1 ,
4) , radius :
6 b) Center :
(3 0).
,
radius : 29
r
(h , k)
solution : (x n) (y k))
-
+
-
= r
(x n)2 (y k)2 r2
- + - =
(x -
3) (y 0)2 (a)
+ - =
(6)
(x (1)) (x 4) (x 3)
=
yz
+
-
29
-
-
+ =
( + 1) +
(x -
4)2 =
36
example 2 : find the center of the radius
a)(x 1) - + (x -
3) =
25
b)x2 + (y + 4) =
59
( 1)2 + (x -
3)2 (5) =
0) (y - 4)] (259)"
(x
-
=
+
-
(( -h)
ra
(x h)" + (x k)2 -
=
· =
(x k) 1
-
-
+
h 1 =
(hik)
k= 3 Center (1 3) .
radius =
5 Center (0-4) radius : 5
