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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
€
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.
€
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
€
x −∞ < x < ∞{ }
(−∞,∞)
€
y −∞ < y < 3{ }
(−∞,3]
€
(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!
€
f (x) =2x x < −1
x −1 x ≥ −1
⎧ ⎨ ⎩
€
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)
€
−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.
€
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.
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