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Graphs Mathematical Pictures

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7/9/2013 Graphss 2 How to Visualize Mathematics Graph Visualization of a relationship among variables or data Bar Graph Pie Chart Relational Graph Typical algebraic graph is plot of points in the real plane

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7/9/2013 Graphss 3 Dow Jones Closing Average Daily closing stock prices for a week Graphs in Everyday Life Question: Is there a trend in stock prices ? 8149 8700 8516 8332 M T W Th F Stock Price in Dollars

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7/9/2013 Graphss 4 Average daily high temperature Historical daily data points for July Graphs in Everyday Life Question: What is the general trend over July ? 1102031 90 95 100 105 110 Daily High (F°)

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7/9/2013 Graphss 5 Graphs in Everyday Life Speed-time relationship Travel time as a function of rate (speed) t = d r r t Question: How does travel time relate to speed ?

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7/9/2013 Graphss 6 Graphs in Everyday Life Temperature scale comparison Conversion of Fahrenheit to Centigrade () C = 5 9 F – 32 FF CC 212 100 32 ● ● Question: How do Fahrenheit freezing and boiling points of water relate to Centigrade points ?

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7/9/2013 Graphss 7 Graphs in Everyday Life Let’s Review Functions with graphs but no formula DOW Jones Closing Average Average daily temperatures Functions with a formula and a graph Travel time based on rate of travel Conversion of Fahrenheit to Centigrade Question: Do graphs help our understanding ?

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7/9/2013 Graphss 8 Graphs in Everyday Life Function Characterization What is the common characteristic ? Ordered Pairs – common characteristic of functions (day, average), (day, temperature F ), (student id, name), (speed, travel time), (F, C)(F, C) Question:How do pairs fit graphs ?

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7/9/2013 Graphss 9 Describing Functions Ordered pairs make graphing easy F = { ( -5, 4 ), ( -2, 1 ), ( 2, 7 ), ( 4, 1 ) } a b (-5, 4) (-2, 1) (2, 7) (4, 1) What is the graph ? … and now? Scatterplot Line graph Question:

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7/9/2013 Graphss 10 y Function Graphs f = { (x, y) | y = 2x – 1 } x (-1, -3) (1, 1) (2, 3) (3, 5) (4, 7) The Graph of f

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7/9/2013 Graphss 11 Domain = [ 0, 4 ] Finding Domain and Range Consider: f = { (x, y) | y = 2x – 1, 0 x 4 } x y (0, -1) (4, 7) The Graph of f Range = [ -1, 7 ] Note: For THIS function, the domain and range are closed intervals

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7/9/2013 Graphss 12 Graphs, Sets and Tables Ordered Pairs and Functions Input Output x y ● ● ● (x, y) Ordered Pair Input Output x y f = { y = 2x + 1 } (x, y) (x, y)

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7/9/2013 Graphss 13 Graph and Functions Vertical Line Test No vertical line touches the graph of a function at more than one point Function ? YES Function? NO Function? YES ● ● ● ● ● ● ● ● (x 1, y 1 ) (x 1, y 2 ) x = x 1

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7/9/2013 Graphss 14 Graph and Functions Vertical Line Test Examples Does any vertical line cut the graph more than once ? Function? YES Function? NO Function? YES x y x y x y ● ● ● ● ● ● y = x 2 x = y 2 x = y 2, y > 0

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7/9/2013 Graphss 15 Think about it !

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Write a function rule for a graph EXAMPLE 3 Write a rule for the function represented by the graph. Identify the domain and the range of the function.

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