Pre-Calculus Next stop: GRAPHING!. 17th Century Shakespeare William Harvey Galileo Copernicus ...
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Transcript of Pre-Calculus Next stop: GRAPHING!. 17th Century Shakespeare William Harvey Galileo Copernicus ...
![Page 1: Pre-Calculus Next stop: GRAPHING!. 17th Century Shakespeare William Harvey Galileo Copernicus Monteverdi Pascal & Fermat And...](https://reader036.fdocuments.in/reader036/viewer/2022062517/56649ed55503460f94be5c15/html5/thumbnails/1.jpg)
Pre-Calculus
Next stop: GRAPHING!
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17th Century
Shakespeare William Harvey Galileo Copernicus Monteverdi Pascal & Fermat
• And. . .
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Rene Descartes
Pronounced “day-cart”
French, 1596-1650 Unified Algebra and
Geometry Founded Analytic
Geometry Huge new idea. . .
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Cartesian Plane
Aka “Cartesian Coordinate System”
x-axis and y-axis Four quadrants Points correspond to x-coordinates and y-coordinates
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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!
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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!
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Graphing Equations
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Using A Graphing Calculator
1. Input Function in “Y=“ menu
2. Adjust “Window” as needed
3. Push “Graph”
Trace or Zoom as needed
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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.
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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
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Basics of Graphs
Domain• Set of all x-values
Range• Set of all y-values
Intercepts • X-intercepts• Y-intercepts
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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).
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Basics of Graphs
Domain
Range
Intercepts
€
x −∞ < x < ∞{ }
(−∞,∞)
€
y −∞ < y < 3{ }
(−∞,3]
€
(0.2,0);(3.8,0)
(0,−1)
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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!
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Domain?
Range?
Intercepts?
Basics of Graphs
Increasing?
Decreasing?
Your turn!
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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?
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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
⎧
⎨ ⎪
⎩ ⎪
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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.
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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
⎧ ⎨ ⎩
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Even and Odd Functions
Even Functions are symmetric about the y-axis
Examples
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Even and Odd Functions
Odd Functions are symmetric about the origin
Examples
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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.
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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
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Some practice with functions
Do p. 165-167, #57-62 and 85-94.