1. definition
Solving
equations
zeros & y
intercepts
Graph
Composite
operations
Properties
inverse
Solving
inequations
x
x x
1
x
c
xlogc
xsin
xcos

2. definition Templates: f(x) = a f(b(x – h)) + k
Function f(x) = af(b(x – h)) + k Simplified template
Absolute value
Square root
Rational
Linear
Exponential
Logarithmic
Trigonometric
k)hx(ba)x(f 
k)hx(ba)x(f 
k
)hx(b
a
)x(f 


k))hx(b(a)x(f 
kca)x(f )hx(b
 
k)hx(bloga)x(f c 
k)hx(bsina)x(f 
k)hx(bcosa)x(f 

3. definition Templates: f(x) = a f(b(x – h)) + k
Function f(x) = af(b(x – h)) + k Simplified template
Absolute value
Square root
Rational
Linear
Exponential
Logarithmic
Trigonometric
k)hx(ba)x(f 
k)hx(ba)x(f 
k
)hx(b
a
)x(f 


k))hx(b(a)x(f 
kca)x(f )hx(b
 
k)hx(bloga)x(f c 
k)hx(bsina)x(f 
k)hx(bcosa)x(f 
  khxba)x(f 
  khxba)x(f 
  k)hx(ba)x(f 
k
hx
b
a
)x(f 


kmx)x(f 
  kca)x(f
hxb


)hx(blog)x(f c 
k)hx(bcosa)x(f 
k)hx(bsina)x(f 

4. Function Domain Range
khxa)x(f 
khxa)x(f 
k)hx(a)x(f 
k
hx
a
)x(f 


kax)x(f 
kca)x(f x

)hx(blog)x(f c 
k)hx(bcosa)x(f 
k)hx(bsina)x(f 
Properties

5. Function Special stuff
khxa)x(f 
khxa)x(f 
k)hx(a)x(f 
k
hx
a
)x(f 


kax)x(f 
kca)x(f x

)hx(blog)x(f c 
k)hx(bcosa)x(f 
k)hx(bsina)x(f 
Properties

6. Function Domain Range Special stuff

a+: [k,∞
a- : -∞,k]
V = (h, k), 2 slopes = +/- a, piecewise
linear
[h, ∞ a+: [k,∞
a- : -∞,k]
V = (h, k), inverse of quad, two templates
-∞, h]
h k
Asymptotes x = h, y = k, two curves,
asymptotes intersect at (h, k)
  Constant slope

a+: ]k,∞
a- : -∞,k[
Asymptote at y = k, c > 0
b+: ]h, ∞
b-: -∞, h[
 Asymptote at x = h, c > 0 log properties, e

Period, frequency, amplitude, loo, phase
shift….
khxa)x(f 
khxa)x(f 
k)hx(a)x(f 
k
hx
a
)x(f 


kax)x(f 
kca)x(f x

)hx(blog)x(f c 
k)hx(bcosa)x(f 
k)hx(bsina)x(f 
Properties
]ak,ak[ 

7. Function Possible graphs and parameters
a > 0 a < 0
a > 0 a < 0
a > 0 a < 0
a > 0
c > 1
a > 0
c < 1
a < 0
c > 1
a < 0
c < 1
b > 0
c > 1
b > 0
c < 1
b < 0
c > 1
b < 0
c < 1
khxa)x(f 
khxa)x(f  k)hx(a)x(f 
k
hx
a
)x(f 


kax)x(f 
kca)x(f x

)hx(blog)x(f c 
k)hx(bcosa)x(f k)hx(bsina)x(f 
GraphGraph y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
a > 0a > 0
a > 0a > 0
a < 0 a < 0
a < 0 a < 0
y
x
y
x

8. Function Possible graphs and parameters
a > 0 a < 0
a > 0 a < 0
a > 0 a < 0
a > 0
c > 1
a > 0
c < 1
a < 0
c > 1
a < 0
c < 1
b > 0
c > 1
b > 0
c < 1
b < 0
c > 1
b < 0
c < 1
khxa)x(f 
khxa)x(f  k)hx(a)x(f 
k
hx
a
)x(f 


kax)x(f 
kca)x(f x

)hx(blog)x(f c 
k)hx(bcosa)x(f k)hx(bsina)x(f 
GraphGraph
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
a > 0a > 0
a > 0a > 0
a < 0 a < 0
a < 0 a < 0
y
x
y
x

9. Function
Obvious when no
solution? # of zeros # y ints
khxa)x(f 
khxa)x(f 
k)hx(a)x(f 
k
hx
a
)x(f 


kax)x(f 
kca)x(f x

)hx(blog)x(f c 
k)hx(bcosa)x(f 
k)hx(bsina)x(f 
Solving equations x and y
intercepts

10. Function
Obvious when no
solution? # of zeros # y ints
No: | something | = - 0, 1, 2 1
No:  0, 1 0, 1
Yes: 5 = 0 0, 1 0, 1
Always a solution 1 1
Meh: 0, 1 1
Always a solution 1 0, 1
Yes: sin x >1 or < -1 0, ∞ 1
khxa)x(f 
khxa)x(f 
k)hx(a)x(f 
k
hx
a
)x(f 


kax)x(f 
kca)x(f x

)hx(blog)x(f c 
k)hx(bcosa)x(f 
k)hx(bsina)x(f 
Solving equations x and y
intercepts
something
x
c

11. Solving equations
Same steps all year!
   somethingkhxbfa 

12. 0121x4 
Solving equations
31x 
31x  31x 
4x  2x 
0103x42 
53x4 
253x4 
5.5x 

13. 83
1x
5


Solving equations
11
1x
5


5)1x(11 
511x11 
11
6
x


02423 2x
 
82 2x

32x
22 
32x 
5x 

14. Solving equations
15)1x(log5 2 
3)1x(log2 
1x2 3

875.0x 
01x3cos2 
2
1
x3cos 


 n,n2
4
3
x3 

 n,n2
4
5
x3




 n,
3
n2
12
3
x 



 n,
3
n2
12
5
x

15. Solving equations
Isolate basic
function
Same steps all year!
   somethingkhxbfa 
   somethingnewhxbf 

16. 0121x4 
Solving equations
31x 
31x  31x 
4x  2x 
0103x42 
53x4 
253x4 
5.5x 

17. 83
1x
5


Solving equations
11
1x
5


5)1x(11 
511x11 
11
6
x


02423 2x
 
82 2x

32x
22 
32x 
5x 

18. Solving equations
15)1x(log5 2 
3)1x(log2 
1x2 3

875.0x 
01x3cos2 
2
1
x3cos 


 n,n2
4
3
x3 

 n,n2
4
5
x3




 n,
3
n2
12
3
x 



 n,
3
n2
12
5
x

19. Solving equations
Isolate basic
function
Same steps all year!
   somethingkhxbfa 
   somethingnewhxbf 
Isolate basic
function
elsesomething)hx(b 

20. 0121x4 
Solving equations
31x 
31x  31x 
4x  2x 
0103x42 
53x4 
253x4 
5.5x 

21. 83
1x
5


Solving equations
11
1x
5


5)1x(11 
511x11 
11
6
x


02423 2x
 
82 2x

32x
22 
32x 
5x 

22. Solving equations
15)1x(log5 2 
3)1x(log2 
1x2 3

875.0x 
01x3cos2 
2
1
x3cos 


 n,n2
4
3
x3 

 n,n2
4
5
x3




 n,
3
n2
12
3
x 



 n,
3
n2
12
5
x

23. Solving equations
Isolate basic
function
Same steps all year!
   somethingkhxbfa 
   somethingnewhxbf 
Isolate basic
function
elsesomething)hx(b 
)s(somethingfinalx Solve
linear equation