Pre-Calculus

Next stop: GRAPHING!

17th Century

Shakespeare William Harvey Galileo Copernicus Monteverdi Pascal & Fermat

• And. . .

Rene Descartes

Pronounced “day-cart”

French, 1596-1650 Unified Algebra and

Geometry Founded Analytic

Geometry Huge new idea. . .

Cartesian Plane

Aka “Cartesian Coordinate System”

x-axis and y-axis Four quadrants Points correspond to x-coordinates and y-coordinates

Relation Vs. Function

Relation: any set of ordered pairs Function: correspondence from one set of

numbers (x-values) to another (y-values) such that each x-value corresponds to EXACTLY one y-value

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1.{(2,3),(3,4),(3,5),(−2,3)}

2.{(−1,1),(−2,4),(0,0),(1,1),(2,4)}

3.{(−1,−1),(−2,4),(−1,7),(−2,−1)}

4.{(1,0),(1,2),(1,3),(1,4)}

5.{(1,3),(2,3),(3,3),(4,3)}

Which of these are Functions?

Note: they are all Relations!

Vertical Line Test

Since each x can only have one y (but y’s can have multiple x’s), a FUNCTION passes the VERTICAL LINE TEST

Function! Not a Function!

Graphing Equations

Using A Graphing Calculator

1. Input Function in “Y=“ menu

2. Adjust “Window” as needed

3. Push “Graph”

Trace or Zoom as needed

Using a Graphing Calculator

Graph each pair of functions in the same window. Describe how the graphs are related to each other.

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f (x) = x

g(x) = x −1

€

f (x) = x

g(x) = x + 2 €

f (x) = x

g(x) = x −1

€

f (x) = x

g(x) = x + 2

1.

2.

3.

4.

Using a Graphing Calculator

In addition to graphs, the calculator will give you a table of values for a function.

Input a function (or two) Push 2nd, Graph [TABLE] Need to change something about the table

(starting value, interval, etc)? Push 2nd, Window [TBLSET]

• TblStart: first value displayed• ΔTbl: interval between values in table

Basics of Graphs

Domain• Set of all x-values

Range• Set of all y-values

Intercepts • X-intercepts• Y-intercepts

Basics of Graphs

Increasing: • A function is increasing on an interval if for some

h, f(x+h)>f(x)

Decreasing: • A function is decreasing on an interval if for some

h, f(x+h)<f(x)

In other words, It is increasing if the line is going up It is decreasing if the line is going down (as you move from left to right).

Basics of Graphs

Domain

Range

Intercepts

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x −∞ < x < ∞{ }

(−∞,∞)

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y −∞ < y < 3{ }

(−∞,3]

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(0.2,0);(3.8,0)

(0,−1)

Basics of Graphs

Increasing?• Yes, in the interval

(-∞,2)

Decreasing?• Yes, in the interval

(2, ∞)

Note: Always report increasing or decreasing on INTERVALS of x-values!

Domain?

Range?

Intercepts?

Basics of Graphs

Increasing?

Decreasing?

Your turn!

Piecewise Functions

Function defined by two or more equations over a specified domain

Example: Time vs. Temp graph of heating ice--what happens in each part of graph?

Piecewise Functions

To Graph:• Find points on both sides of “breaks” in

domain• Plot these points!

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f (x) =2x x < −1

x −1 x ≥ −1

⎧ ⎨ ⎩

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g(x) =

x 2 x > 4

2x −1 0 < x ≤ 4

0 x ≤ 0

⎧

⎨ ⎪

⎩ ⎪

Time out for an Application!

You have a cell phone plan. You are charged a flat rate of $65 for the first 500 minutes, and $0.10 for each minute after that. Create a graph to show this situation.

Piecewise Functions

Your turn to practice! Graph the following piecewise

functions:

€

f (x) =3x + 5 x < 0

4x + 7 x ≥ 0

⎧ ⎨ ⎩

g(x) =x + 5 x ≥ −5

−(x + 5) x < −5

⎧ ⎨ ⎩

Even and Odd Functions

Even Functions are symmetric about the y-axis

Examples

Even and Odd Functions

Odd Functions are symmetric about the origin

Examples

Even and Odd Functions

Even Functions

Odd Functions€

f (x) = f (−x)

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−f (x) = f (−x)

To determine if a function is even or odd ALGEBRAICALLY,

Find f(-x).

If it is the SAME as the original function, the function is EVEN.

If it is the OPPOSITE of the original function -f(x), the function is ODD.

Even and Odd Functions

Determine ALGEBRAICALLY if the function is even, odd, or neither.

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1. f (x) = x 3 + x

2.g(x) = x 2 − x

3.h(x) = 2x +1

4. f (x) = 2x 3 − 6x 5

Some practice with functions

Do p. 165-167, #57-62 and 85-94.