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What Makes a Function Linear

Discover what makes a function linear through rate of change analysis. Learn to plot points using a calculator and interpret data matrices. Explore slope-intercept form and the family of linear functions. Beware of non-linear functions!

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What Makes a Function Linear

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  1. What Makes a Function Linear Lesson 1.3

  2. What Is A Line? • Check out this list of definitions or explanations to the question • Define Line • Today we look at what makes a function linear.

  3. Rate of Change • Consider example y = 3x + 5 • We note the rate of change is constant • This means it is a linear function • It graphs as a straight line y=3x + 5

  4. Plotting Points on the Calculator • Use Data Matrix on Calculator • Choose APPS, then 6 Data/Matrixthen NEW

  5. Starting Up • Choose DATA • Give it a variablename for savingin memory

  6. Cursor must be here Enter formula here Entering Data • Enter numeric values in the cells • Enter a formula at the top • Using column name

  7. Viewing Data • Note the results of the formula • We can do further calculations • We can also plot these points

  8. Choose line type Specify the columns for the X and Y values Plotting Data • Choose F2 for Plot SetupScreen • Then F1 for Define

  9. Plotting Data • Goto the Y= Screen to turn off any functions there • Then specify ZoomData • This fits the window to the limits of the data

  10. Plotting Data • Note the graph includes the points we had in the data matrix • It is a line-graph, the points are represented by boxes

  11. Another Example • Consider the following table of values • Note the value of for any two pairs of values

  12. Example From Text $20,000 • See Example 2, pg. 19 • Formula used for depreciation Value of equipment = original value – $4000 * number of years • To generalize:Dependent Qty = startValue + rateOfChange * independentQuantity

  13. Family of Linear Functions • Slope = Rate of Change View Geobebra file which demonstrates the slope-intercept formula And Another y=3x + 5 • Slope = m = 3 • y-intercept = b = 5

  14. Family of Linear Functions • Calculating slope with two ordered pairs (X2, Y2) • (X1, Y1) • Given two ordered pairs, (7,5) and (-3,12). What is the slope of the line through these two points?

  15. Warning • Not all functions which appear linear will actually be linear!! • Consider the set of ordered pairs • Graph them • Decide whether graphis linear • Check slope for differentpairs

  16. Results • Graph appearsstraight • But …rate of change is not a constant

  17. Assignment • Lesson 1.3 • Page 24 • Exercises1 – 5, 7, 9, 13, 15, 19, 21, 23

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