**Year 13 IB Mathematical Studies**

**Functions**

**The Function Machine**

To understand functions well, it's helpful to think of it as a machine that takes one kind of thing and turns it into another kind of thing.

**Function Mappings**

Domain

The set of objects or numbers to be put into a function

Range

The set of objects or numbers that are the output of a function

Different Names for the

Same Thing

Domain

Input

Pre-image

Range

Output

Image

**Function Notation**

**If**

**then**

Everytime we see x, we must replace it with -2 in brackets.

Example 1

Construct a mapping for the function

on the domain

Example 2

Example 3

Example 4

Linear Functions

Linear Function

Any function of the form ,

Example 5

Quadratic Functions

Quadratic Function

Any function of the form ,

The graph of a quadratic function is a parabola. If a > 0, the graph opens up. If a < 0, the graph opens down.

Vertex

The maximum or minimum point on a parabola. Also called turning point.

Axis of Symmetry

A vertical line that passes through the vertex of a parabola. The equation for the axis of symmetry is

.

.

Intercepts

and

x

y

The y-intercept is simply y = c.

The x-intercepts (also called zeros) can be found in two ways:

Factorization

Using technology

x-intercepts:

y-intercept:

-1 and 3

-2

Example 7

Example 8

Using the axis intercepts and vertex, sketch the graph of

Using a GDC, sketch the graph of

Label the vertex and axis intercepts.

Example 6

a) Find the equation of the line of symmetry of .

b) Find the coordinates of the vertex.

**Exponential Functions**

Opening Problem

Would you rather be given 20,000 Kc every day of the month, or be given 1 Kc on the first day, 2 Kc the second day, 4 Kc the third day, and have it continue to double every day for the whole month?

Exponential Function

A function in which the variable appears in the exponent

For example, the function representing the doubling of money from the opening problem was:

where is the number of days after the first day.

This is an exponential function because the variable appears in the exponent.

t

Example 1

Asymptotes

Consider the equation .

Make a table of values and graph.

Notice as x gets smaller, y gets really close to 0. This equation has a at y = 0.

horizontal asymptote

Investigation

Complete Investigation: Exponential Graphs on p. 584 in your textbook.

Exponential Growth

Asymptote

An invisible horizontal or vertical line that a graph approaches but never reaches

Example 2

Exponential Decay

Example 3

Big Idea: Graphs of Exponential Functions

From you investigation, you should have figured out that for :

controls the steepness of the graph

controls the steepness of the graph as well

controls the vertical translation and

is the vertical asymptote

If , then the graph is above the asymptote.

If , then the graph is below the asymptote.

Investigation: Understanding Exponential Functions

Filip recently did a photo shoot for the new men's fragrance from Dolce & Gabanna. When his photo was published the D&G's stock was at €36 per share. Each week after the release of the photo, the stock increased by 3.4%.

Problem:

a) Write an exponential function for the value of D&G's stock t weeks after the release of Filip's photo.

b) What was the value (to the nearest cent) of one share of D&G's stock 7 weeks after ...

i) the first 7 weeks?

ii) the first year?

c) How many days (to the nearest day) after the release of the photo did it take the stock rise to €50?

More Example Ideas:

Growth

Cell division

Compound interest

When Mr. Myers leaves his coffee mug over the weekend and mold grows in it

Periodic Funcitons

The London Eye

Imagine you're riding in the London eye and record your height above ground every three seconds. Since you're traveling in a circle, your heights would repeat as you go around. Graphing your height against time would give you a sine wave.

Periodic Data -

data that repeats itself in cycles

Periodic data appears EVERYWHERE:

Weather patterns

Tides

Car engines revolutions

Lunar phases

Animal populations

Hours of daylight

Can you think of anything else?

We can see the repeating pattern easily in the graphs of periodic data:

Period -

the length of one cycle

Amplitude -

the distance from the max or min to the principal axis

Principal axis

Amplitude =

Quick Check: Periodic Basics

The Sine Curve

The period of y = sin x is 360 degrees.

The maximum is 1.

The minimum is -1.

You need to memorize this.

Investigation

Complete Investigations 1 and 2 on page 564 in your book.

You will need a calculator!

Unfamiliar Functions

We will use things we already know about functions and apply it to unfamiliar functions to solve problems. We will use:

axes intercepts

gradients

turning points (maxima and minima)

asymptotes

Example

Asymptotes

Horizontal Asymptotes

Horizontal asymptotes occur when a function approaches a value from above or below (ie. exponential functions)

Vertical Asymptotes

Vertical asymptotes occur when you have a function with the variable in the denominator (dividing by zero is undefined)

Horizontal asymptote (graph approaches -1 from above AND below)

Vertical asymptote at x=2 (when x=2, f(2) is 2/(2-2) or 2/0, which is undefined

Example

Use your GDC to find...

x-intercepts

2nd-CALC-Zero

maxima and minima

2nd-CALC-maximum/minimum

intersections

2nd-CALC-intersect

Example: Optimization