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Section 2.1: Linear Equations and Patterns

When we stack two number lines on top of one another perpendicularly, we get a plane called the Cartesian coordinate plane.















Why Plot Data? We can spot trends and patterns when we plot our data.

Example: The table below monitors the price of a stock over a 6 month period. $$ \begin{array}{c|c} \mbox{Month $x$} & \mbox{Price $y$ in dollars per share} \\ \hline 0 & 6 \\ \hline 1 & 7.80 \\ \hline 2 & 10.20 \\ \hline 3 & 13.15 \\ \hline 4 & 11 \\ \hline 5 & 8.25 \\ \hline 6 & 4.50 \\ \hline \end{array} $$







When we plot the data, the pattern becomes more obvious.



Question: What is the initial price of the stock?

Question: When does the stock reach its peak price?

Question: When would have been the best time to buy this stock?

Question: When would have been the best time to sell this stock?











A Special Kind of Pattern: Lines.

Very often, the patterns we observe are lines.

Example: The speed of a free falling object at low speeds is a line.

Example: Points comparing an individual's height to their arm length, generally fall close to a straight line. Thus, lines often model overall patterns in data.











Arithmetic Sequences. An arithmetic sequence has a constant difference between successive terms.

Example: The sequence -1, 2, 5, 8, 11 is an arithmetic sequence.

Example: The sequence 3, 5, 2, 2, 4 is NOT an arithmetic sequence.

Example: Plot both of these sequences. What do you notice about arithmetic sequences as opposed to non-arithmetic sequences?











Fact: Graphs of arithmetic sequences are lines.