EQ: How can we use linear models for non linear data? Straightening Data

River water velocity and Distance from shore (cm/s) .5 22 1.5 23.18 2.5 25.48 3.5 25.25 4.5 27.15 5.5 27.83 6.5 28.49 7.5 28.18 8.5 28.50 9.5 28.63

Linear model analysis

Straighten the data

Writing the model Original model: Transformed model:

Using the model Predict the velocity of the river water at a distance of 5 meters from the shore

Most popular transformations Exponential: Power:

Exponential X Y 1 2 3 4 5 16 7 64 8 128 9 256 10 512

Straighten the data X Log(y) 1 2 0.301 3 0.602 4 0.903 5 1.204 7 1.505 2 0.301 3 0.602 4 0.903 5 1.204 7 1.505 8 1.806 9 2.107 10 2.709

Use the model Predict the y value for an x of 2.5

Find the model Mammal Weight (kg) Heart Rate (BPM) Mouse 0.03 580 At 0.32 320 Rabbit 3.97 170 Monkey 6.55 150 Dog 16 120 Elephant 2500 25

Power Model Exponential Model

Power model Predict the heart rate for humans who weigh 60 kg.

Creating a Transformed data model Determine which model is appropriate Plot the original data Plot x vs log y (exponential model) Plot log(x) vs log(y) (power model) Check residuals for each plot and determine which is best 2. Algebraically transform the model so the original units are used Write the model using the log variables Use inverse operations to eliminate the logs 3. Check the model Plot the original data (x vs y) Find the chosen model (exponential or power) Check the calculator model with your model

Algebra: Transforming exponential models

Savings Accounts Sketch a graph of the original data Comment on why a linear model is NOT appropriate Sketch a graph of the years vs log(money) Determine why an exponential model IS appropriate by sketching the residual plot and commenting on it Find the linear model for the transformed data and algebraically transform it back into the original units Predict the amount of savings after 60 years

Algebra: Transforming Power models

Harley stock Prices Sketch a graph of the original data Comment on why a linear model is NOT appropriate Sketch a graph of the years vs log(money) and then log(years) vs log(money) Determine which model is better by examining the residuals Find the linear model for the transformed data and algebraically transform it back into the original units Predict the amount of savings after 13 years