section 1.1 lines 1. - ms....

AP Calculus AB Name____________________________________________

Chapter One: Prerequisites for Calculus Notes

Section 1.1 Lines 1. Connect these four points: (-6, 2), (-2, 6), (10, 4), (2, -2) to form a quadrilateral.

What figure is formed by connecting the midpoints of each side? Prove it.

2. Given: A circle with center at the origin and radius 5.

a. Is the point (3, 4) on the circle? Justify.

(Hint: Equation of circle: 2 2 2 x h y k r

b. Write the equation of the line tangent to the circle at (3, 4)

3. Simplify: 1 21

xx

x

4. Write a formula for the diagonal of a square as a function of its side.

CHICKEN MATH

Match each story with 1 of the graphs. When finished, write a similar story that would correspond to the extra

graph.

a) I took my chicken out of the freezer at noon, and left it on the counter to thaw. Then I cooked it in the oven

when I got home.

b) I took my chicken out of the freezer this morning, and left it on the counter to thaw. Then I cooked it in the

oven when I got home.

c) I took my chicken out of the freezer this morning, and left it on the counter to thaw. I forgot about it, and

grabbed some pizza on my way home from work. When I finally got home, I put the chicken back in the

refrigerator.

d) ______________________________________________________________________________________

________________________________________________________________________________________

________________________________________________________________________________________

_____________________________________________________________________

Warm up: Can you recall these ten basic functions?

Without using a calculator, sketch an accurate graph of each function. State the domain and range of each.

1. f x x( ) 2. f x x( ) 2 3. f x x( ) 3

4. f xx

( ) 1

5. f x x( ) 6. f x ex( )

7. f x x( ) ln 8. f x x( ) sin 9. f x x( ) cos

10. f x x( ) Domain Range Domain Range

1. ____________ ____________ 6. ____________ ____________

2. ____________ ____________ 7. ____________ ____________

3. ____________ ____________ 8. ____________ ____________

4. ____________ ____________ 9. ____________ ____________

5. ____________ ____________ 10. ____________ ____________

Graph Transformations

f(x) to …

f (x) k Vertical shift k units

Up if k > 0

Down if k < 0

f (x k) Horizontal shift k units

Right if k < 0

Left is k > 0

f ( x) Reflection over y-axis

f (x) Reflection over x-axis

f (kx) Horizontal

Stretch if 0 < k < 1

Shrink if k > 1

kf (x) Vertical

Stretch if k > 1

Shrink if 0 < k < 1

1.2 Functions and Graphs Day 1

GRAPHICALLY:

even odd

ALGEBRAICALLY:

1. 3( ) 4f x x x 2.

2( ) 5f x x 3. 3( ) 1f x x

For each function, tell the Domain, Range, and Symmetry (even/odd). It may be helpful to sketch the

graphs. Do these without a calculator and then check with a calculator.

1. ( ) 8f x x 2. 2( ) ( 4)f x x

3. 1

( )7

f xx

4. ( ) 7f x x

Even & Odd Functions (Symmetry)

A function f(x) is an even function if f ( x) f x

odd function if f ( x) f x

for every x-value in the function’s domain

Continue with same directions with your partner.

5. ( ) 7f x x 6. 2( ) 9f x x *7.

2

4f (x)

x 25

Piecewise-defined functions

Graph the functions

1. , [0, )

( ), ( ,0)

xf x

x

2.

2

1, ( , 1)

( ) , [ 1,0]

, (0, )

g x x

x

3. Write a piecewise formula for the following function:

1.2 Functions and Graphs Day 2 Composition of Functions

#1-4 Given: Find:

1. 2g

2. 4 3f

3. 1f g

4. 0f h g

#5-7 Given the graphs of k and q.

y k x y q x

Find:

5. 1k k 6. 2k q 7. 2q k

Composition of Functions

Symbol: __________________________

Not ______________________________

Domain:_______________________________________

Range: _______________________________________

x f x g x h x

-1 2 3 -1

0 0 2 1

1 1 1 3

2 0 -1 5

3 2 -1 7

Find the rule for the composition ( )( )f g x and its domain algebraically, and then check by using the calculator

graphing feature for composition.

1. 2( ) 1f x x , 1

( )1

g xx

2. ( ) 1f x x , 2( ) 2g x x

Find the missing function

f g f(g(x))

3x 3 5x

x

x

cos x 2cos x

1

3x x

If time permits, go back to #1 and #2 and do the problems over again this time finding ( )( )g f x

Section 1.3 Exponential Growth and Decay

Growth or Decay?

1. xy 2 2. xy 3(1.5 ) 3. xy 3(1.5 )

Graph. State the domain, range and intercepts.

4. xy (2 ) 4

Solve with a calculator: Solve without a calculator:

5. xe 8 6. 2

x 34 16

Exponential Functions

Let apositive reals and a 1

The exponential function with base a is xf (x) a

If k > 0, a model for exponential growth: xf (x) k a a 1

If k > 0, a model for exponential decay: xf (x) k a 0 a 1

7. Have you seen a coyote in your backyard? In 2011, there were 5000 coyotes in Chicagoland and they are

increasing at a rate of 9% a year.

a. Write an equation modeling the problem.

b. How many coyotes will there be in 2030?

c. When were there 4350 coyotes? Solve graphically.

8. Suppose the half-life of a radioactive substance is 8 days and that there are 8.8 grams of the substance

initially. How much will be present in 88 days?

9. Fruit flies double every day. How many are there in 10 days?

1.5 Functions and Logarithms

Inverse Functions

List what you remember if two functions are inverses of each other:

______________________________________________________________

______________________________________________________________

Let f(x) = (x – 2)2 + 5; x < 2 (Why the restriction?)

1. Find 1f (x) algebraically.

2. Domain of f(x) = __________________ Domain of f -1

(x) = __________________

Range of f(x) = __________________ Range of f -1

(x) = __________________

3. Let y1 = f(x), y2 = g(x), y3 = f(g(x)), and y4 = g(f(x)) for this activity. What is true of the graphs of y3 and y4

if f and g are inverses of each other?

One-to-One Function

A function is one-to-one on a domain if f (a) f (b) whenever a b

Use horizontal line test

Logarithmic Functions

1. Solve algebraically:

a. 2te 7 b. ln( 5) 4 x c. ln( 1) ln4 4 ln2 y

2. Graph without a calculator. State the domain and range.

y 2ln(x 4)

D:____________________ R:____________________

The base a logarithm function ay log x is the inverse of the base a exponential function xy a ( 0, 1 a a )

Properties of Logarithms

Product rule: log log log a a axy x y

Quotient rule: log log log

a a a

xx y

y

Power rule: log logy

a ax y x

Change of base: ln

logln

a

xx

a

Inverse properties: log , log , 1, 0 a x x

aa x a x a x

log , log , 0 e x x

ee x e x x

1.6 Trigonometric Functions

Trig Unit Circle

cos x sec reciprocalof x

sin y csc reciprocalof y

ytan

x

xcot

y

( , )1 0 ,1 0

,0 1

,0 1

3 1,

2 2

2 2,

2 2

1 3,

2 2

3 1,

2 2

2 2,

2 2

1 3,

2 2

1 3,

2 2

2 2,

2 2

3 1,

2 2

3 1,

2 2

2 2,

2 2

1 3,

2 2

1. Given: y 3sin(2x)

a. State the period __________

b. What is the amplitude? __________

c. Identify the viewing window of its graph.

X: _________________

Y: _________________

2. Given: y .5sin x2




X: _________________

Y: _________________

3. y 1.5sin x




X: _________________

Y: _________________

Trigonometry Homework WS Name___________________________________

Find the exact value of the trigonometric expression. No Calculators!

1. 3

cos4

2. sin 3.

4tan

3

4. sec6

5. cot

4

6. csc2

7. 7

sin4

8.

11cos

6

9.

5tan

6

True or false

10. cos(2 ) 2cosx x 11. cos( ) cos x x 12. 2 2( 3) 9 x x

section 1.1 lines 1. - ms....

Documents