Do Now from 1.2c

Do Now from 1.2cState whether the function is odd, even, or neither. Support

graphically and algebraically.

therefore: EVEN

Find all vertical and horizontal asymptotes.

2

3( )

1g x

x

( ) ( )g x g x

2

( 2)( 3)( )

16

x xp x

x

: 4

: 1

VA x

HA y

I’ve been to the mountaintop...

…and I’ve seen…

Twelve BasicTwelve BasicFunctionsFunctions

We will study these “basic” functions inWe will study these “basic” functions inmuch more depth throughout the year –much more depth throughout the year –

here, we are just introducing them…here, we are just introducing them…(and in particular, their graphs)(and in particular, their graphs)

The Identity Function: f x x

Interesting facts:

This is the only function that acts on every real number by leaving it alone.

There are only two man-made structures visible from space, and we live very close to one of them!

The Squaring Function: 2f x x

Interesting facts:

The graph of this function, called a parabola, has a reflection property that is useful in making flashlights and satellite dishes.

Goats in Morocco often climb trees in order to consume delicious berries.

The Cubing Function: 3f x x

Interesting facts:

The origin is called a “point of inflection” for this curve because the graph changes curvature at that point.

In 1940, a Soviet scientist kept a dog’s head alive without a body for 3 days.

The Reciprocal Function: 1f x

x

Interesting facts:

This curve, called a hyperbola, also has a reflection property that is useful in satellite dishes.

A catfish has over 27,000 taste buds, most of them on the outside of its body.

The Square Root Function: f x xInteresting facts:

Put any positive number into your calculator. Take the square root. Continue to take the square root, and eventually you will always get 1.

No normal piece of paper can be folded in half more than seven times.

The Exponential Function: xf x e

Interesting facts:

The number e is an irrational number (like pi) that shows up in a variety of applications.

No word in the English language rhymes with the word “month.”

(0, 1)

The Natural Logarithm Function: lnf x x

Interesting facts:

This function increases very slowly. If the x-axis and y-axis were both scaled with unit lengths of one inch, you would have to travel more than two and a half miles along the curve just to get a foot above the x-axis.

It cost 7 million dollars to build the Titanic and 200 million to make a film about it.

(1, 0)

The Sine Function: sinf x x

Interesting facts:

This function and the sinus cavities in your head derive their names from a common root: the Latin word for “bay.”

Human fingernails grow faster than human toenails.

The Cosine Function: cosf x x

Interesting facts:

The local extrema of the cosine function occur exactly at the zeros of the sine function, and vice versa.

In many U.S. states, it is illegal to commit suicide (not sure how many arrests have been made…).

The Absolute Value Function: absf x x x

Interesting facts:

This function has an abrupt change of direction (a “corner”) at the origin, while our other functions are all “smooth” on their domains.

All of the clocks in the movie “Pulp Fiction” are stuck on 4:20.

The Greatest Integer Function: intf x x x

Interesting facts:

This function has a jump discontinuity at every integer value of x. Similar-looking functions are called step functions.

Donald Duck comics were banned from Finland because he doesn’t wear pants.

1 2 3–1–2

–2

–1

1

2

The Logistic Function: 1

1 xf x

e

Interesting facts:

There are two horizontal asymptotes, the x-axis and the line y = 1. This function provides a model for many applications in biology and business.

There are more possible “states” of your brain than there are atoms in the known universe.

1

(0, ½)

Identity

Squaring

Cubing

Recip.

SquareRoot Expon.

Nat. Log.Sine

Cosine Abs. Val. GreatestInteger

Logistic

Which of the twelve basic functions:

Are bounded above?Are bounded above?

Identity

Squaring

Cubing

Recip.

SquareRoot Expon.

Nat. Log.Sine

Cosine Abs. Val. GreatestInteger

Logistic

Which of the twelve basic functions:

Are increasing on their entire domain?Are increasing on their entire domain?

Identity

Squaring

Cubing

Recip.

SquareRoot Expon.

Nat. Log.Sine

Cosine Abs. Val. GreatestInteger

Logistic

Which of the twelve basic functions:

Have infinitely many local extrema?Have infinitely many local extrema?

Identity

Squaring

Cubing

Recip.

SquareRoot Expon.

Nat. Log.Sine

Cosine Abs. Val. GreatestInteger

Logistic

Which of the twelve basic functions:

Have a range of all real numbers?Have a range of all real numbers?

Identity

Squaring

Cubing

Recip.

SquareRoot Expon.

Nat. Log.Sine

Cosine Abs. Val. GreatestInteger

Logistic

Which of the twelve basic functions:

Have end behaviorHave end behavior limx

f x

??

Identity

Squaring

Cubing

Recip.

SquareRoot Expon.

Nat. Log.Sine

Cosine Abs. Val. GreatestInteger

Logistic

Which of the twelve basic functions:

Have graphs that are identical except for a horizontal shift?Have graphs that are identical except for a horizontal shift?

Homework: p. 109-110 1-12 all, 13-27 odd