Week 1 LSP 120 Joanna Deszcz

 Relationship between 2 variables or quantities  Has a domain and a range  Domain – all logical input values  Range – output values that correspond to domain  Can be represented by table, graph or equation  Satisfies the vertical line test:  If any vertical line intersects a graph in more than one point, then the graph does not represent a function.

 Straight line represented by y=mx + b  Constant rate of change (or slope)  For a fixed change in one variable, there is a fixed change in the other variable  Formulas ▪ Slope = Rise Run ▪ Rate of Change = Change in y Change in x

 QR Definition:  relationship that has a fixed or constant rate of change

xy  Does this data represent a linear function?  We’ll use Excel to figure this out

y2- y1 x2-x1 Example: = xy

 Input (or copy) the data  In adjacent cell begin calculation by typing =  Use cell references in the formula  Cell reference = column letter, row number (A1, B3, C5, etc.) ABC 1xyRate of Change =(B3-B2)/(A3-A2)

 If the rate of change is constant (the same) between data points  The function is linear

 General Equation for a linear function  y = mx + b  x and y are variables represented by data point values  m is slope or rate of change  b is y-intercept (or initial value) ▪ Initial value is the value of y when x = 0 ▪ May need to calculate initial value if x = 0 is not a data point

ABC 1xy Rate of Change  Choose one set of x and y values  We’ll use 3 and 11  Rate of change = m  m=2.5  Plug values into y=mx+b and solve for b  11=2.5(3) + b  11=7.5 + b  3.5=b So the linear equation for this data is: y= 2.5x + 3.5

xy xy xy xy

 Select all the data points  Insert an xy scatter plot  Data points should line up if the equation is linear

tP Not all graphs that look like lines represent linear functions! Calculate the rate of change to be sure it’s constant. t=year; P=population of Mexico

tP  Does the line still appear straight?

 Previous examples show exponential data  It can appear to be linear depending on how many data points are graphed  Only way to determine if a data set is linear is to calculate rate of change  Will be discussed in more detail later

Linear Modeling and Trendlines

 Need to plan, predict, explore relationships  Examples ▪ Plan for next class ▪ Businesses, schools, organizations plan for future ▪ Science – predict quantities based on known values ▪ Discover relationships between variables

 Equation  Graph or  Algorithm  that fits some real data reasonably well  that can be used to make predictions

 2 types of predictions  Extrapolations  predictions outside the range of existing data  Interpolations  predictions made in between existing data points  Usually can predict x given y and vise versa

 Be Careful -  The further you go from the actual data, the less confident you become about your predictions.  A prediction very far out from the data may end up being correct, but even so we have to hold back our confidence because we don't know if the model will apply at points far into the future.

 Cell phones.xls Cell phones.xls  MileRecords.xls MileRecords.xls

 5 Prediction Guidelines  Guideline 1  Do you have at least 7 data points? ▪ Should use at least 7 for all class examples ▪ more is okay unless point(s) fails another guideline ▪ 5 or 6 is a judgment call ▪ How reliable is the source? ▪ How old is the data? ▪ Practical knowledge on the topic

Does the R-squared value indicate a relationship? ▪ Standard measure of how well a line fits R2Relationship =1perfect match between line and data points =0no relationship between x and y values Between.7 and 1.0strong relationship; data can be used to make prediction Between.4 and.7moderate relationship; most likely okay to make prediction <.4weak relationship; cannot use data to make prediction

 Verify that your trendline fits the shape of your graph.  Example: trendline continues upward, but the data makes a downward turn during the last few years  verify that the “higher” prediction makes sense  See Practical Knowledge

 Look for Outliers  Often bad data points  Entered incorrectly ▪ Should be corrected  Sometimes data is correct ▪ Anomaly occurred  Can be removed from data if justified

 Practical Knowledge  How many years out can we predict?  Based on what you know about the topic, does it make sense to go ahead with the prediction?  Use your subject knowledge, not your mathematical knowledge to address this guideline