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Mathematical Modeling

Learn how to model mathematical functions, interpret linear equations, and make predictions with data using function notation. Explore correlation coefficients, trendlines, and Excel tools for accurate forecasting.

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Mathematical Modeling

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  1. Mathematical Modeling Making Predictions with Data

  2. Function A rule that takes an input, transforms it, and produces a unique output • Can be represented by • a table that maps an input to an output • a graph • an equation involving two variables • Domain – the set of inputs • Range – the set of outputs t ≥ 0 y = 5t + 3 d ≥ 3

  3. Linear Function A function that demonstrates a constant rate of change between two quantities • Can be represented by a line on a coordinate grid • Can be represented by a linear equation involving two variables • Can represent real-life situations • Distance traveled over time • Cost based on number of items purchased

  4. Linear Equation A linear function can be expressed by a linear equation • An equation involving two variables • Independent variable, x • Horizontal axis • Dependent variable, y • Vertical axis • Variables can represent any two related quantities

  5. Linear Equation • Data is often collected in tables • Data is graphed on a coordinate plane as ordered pairs (5, 28) (3, 18) (2, 13) 2 13

  6. Linear Equation 3 y = 5x + 3 5 1 5 =

  7. Function Notation • Functions often denoted by letters such as F, f, G, w, V, etc. • G(t) represents the output value of G at the input number t • Garbage production over time • t is a member of the Domain • t ≥ 0 • G(t) is a member of the Range • G(t) ≥ 427.92 tons • slope m = 20.05 tons/year • Garbage production increases by 20.05 tons/year

  8. Function Notation • Example: d(t) = 5t +3 • Slope, m = 5 ft/s • The toy car moves 5 feet for every second of time • y-intercept, b = 3 ft • The toy car is initially 3 feet from the line at time t = 0 • What is the distance at t = 6 s • d(6) = 5 · (6) + 3 = 33 ft • When will the object be 23 feet from the line? • d(t) = 23 = 5t +3, t = 4

  9. Correlation Coefficient, r • Measure of strength of a linear relation • -1 ≤ r ≤ 1 • r = ±1 is a perfect correlation • r = 0 indicates no correlation • Positive r indicates a direct relationship • As one variable increases, so does the other • Negative r indicates an inverse relationship • As one variable increases, the other decreases • Strength of relationship • r > 0.8 is a strong correlation • r < 0.5 is a weak correlation

  10. Coefficient of Determination, r2 • Measure of how well the line represents the data • 0 ≤ r2≤ 1 • Portion of the variance of one variable that is predictable from the other • Example: r2 = 0.65, 65% of variation in y is due to x. The other 35% is due to other variable(s). • Square of the Correlation Coefficient

  11. Finding Trendlines with Excel • Create table of data • Common practice to re-label years starting with n = 1 • Select data

  12. Finding Trendlines with Excel • Insert Scatterplot

  13. Finding Trendlines with Excel • Format the Scatterplot • Select the scatterplot • Choose the Layout tab • Chart Title • Axis Titles • Gridlines • Legend (delete)

  14. Finding Trendlines with Excel • Format the Scatterplot • Select the scatterplot • Under Chart Tools • Choose Format tab • Select Horizontal (Value) Axis in drop down menu • Choose Format selection • Adjust the axis options • Select Vertical (Value) Axis • Choose Format selection • Adjust the axis options Note that the horizontal axis was formatted to show several years in the future.

  15. Finding Trendlines with Excel • Add Trendline • Select the scatterplot • Under Chart Tools • Choose Layout tab • In the Analysis panel • Choose Linear Trendline • Select Trendline (either within chart or in Current Selection panel) • Forecast • Display Equation • Display R-squared value

  16. Making Predictions • Use the trendline to make predictions • Function notation S(t) = 0.1335t+2.0269 where S(t) = projected sales t = year number (t = calendar year - 2002)

  17. Making Predictions • Use the trendline to make predictions • What is the sales projection for 2015? t = 2015 – 2002 = 13 S(13) = 0.1335(13)+2.0269 = $3.76 million 2003 2015

  18. Making Predictions 1.9731 2002 2003 2017

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