8.4 Rules For Linear Functions

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Chapter 8, Section 4: Rules for Linear Functions.

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8.4 Rules For Linear Functions

  1. 1. Chapter 8 Section 4 Writing Rules for Linear Functions February 10 th , 2009
  2. 2. Writing Rules from Words <ul><li>Write functions using Function Notation . </li></ul><ul><li>f ( x ) replaces y. That’s all. </li></ul><ul><li>Read f ( x ) as “ f of x ”. </li></ul><ul><li>The Domain Value (x value ) is the INPUT . </li></ul><ul><li>The resulting Range Value (y value ) is the OUTPUT . </li></ul><ul><li>A Function Rule is an equation that describes a function. </li></ul>
  3. 3. Function Rule y = 3 x + 7 f ( x ) = 3 x + 7 output input Read it as “f of x is equal to the product of 3 and x, plus 7”. The Function rule is the Equation that you come up with to compare the Input and the Output.
  4. 4. Real World <ul><li>Jerry works at a local store. Each week he earns $300 salary plus a 3% commission on his sales. </li></ul><ul><li>Write a function rule that relates total earnings to sales. </li></ul><ul><li>Find his earnings for one week if his sales are $2,500. </li></ul>
  5. 5. Function, Part A: <ul><li>Jerry works at a local store. Each week he earns $300 salary plus a 3% commission on his sales. </li></ul><ul><li>A) Write a function rule that relates total earnings to sales. </li></ul><ul><li>Total Earnings = $300 + 3% (of Sales) </li></ul><ul><li>Total Earnings = y = f ( x ) = output </li></ul><ul><li>$300 = the constant </li></ul><ul><li>3% = slope </li></ul><ul><li>(of Sales) = x = input </li></ul>
  6. 6. Function, Part B: <ul><li>Jerry works at a local store. Each week he earns $300 salary plus a 3% commission on his sales. </li></ul><ul><li>B) Find his earnings for one week if his sales are $2,500. </li></ul><ul><li>t(s) = 300 + 0.03s </li></ul><ul><li>t(2,500) = 300 + 0.03s </li></ul><ul><li>t(s) = 300 + 0.03(2,500) </li></ul><ul><li>t(s) = 300 + 75 </li></ul><ul><li>t(s) = $375 a week. </li></ul>S, here, is the input, like f(x). You can make your function notation different from f of x. Here we used t of s. t is dependant on s. So, s can change.
  7. 7. Try This One: <ul><li>Scrumptious Snacks Mix is sold by mail order. It costs $3/lb, plus $4 for shipping and handling. </li></ul><ul><li>Write a function rule for the total cost c ( p ) based on the number of pounds p bought. </li></ul><ul><li>Use your function to find the total cost of 5 lbs of snack mix. </li></ul>
  8. 8. Writing Rules from Tables or Graphs <ul><li>To write a function rule from a table, look for a pattern. </li></ul><ul><li>The formula y = mx + b in function notation is, f ( x ) = mx + b. </li></ul><ul><li>The slope of m is the difference in f ( x )-values difference in x-values </li></ul><ul><li>The y-intercept (b) is the value of f ( x ) when x=0. </li></ul>
  9. 9. Write a rule for the linear function in the table. <ul><li>What is the pattern for x and f ( x )? </li></ul><ul><li>As the x values increase by 2, the f ( x ) values increase by 6. So, m = 6/2, or 3. </li></ul><ul><li>When x = 0, f ( x ) = 1. </li></ul><ul><li>So b = 1. </li></ul><ul><li>f(x) = 3x + 1 is the rule. </li></ul>13 4 7 2 1 0 -5 -2 f ( x ) x
  10. 10. Write the rule for each linear function. f(x) = 2x, because when x = 0, y = 0. f(x) = -2x, because when x = 0, y = 0. f(x) = 2x + 1, because when x = 0, y = 1. 4 2 2 1 0 0 -2 -1 f ( x ) x -12 6 -6 3 0 0 6 -3 f ( x ) x 1 0 -3 -2 -7 -4 -11 -6 f ( x ) x
  11. 11. Use y=mx+b to Find Rules for Graphs. <ul><li>Use the slope-intercept form, f(x)=mx+b, or y=mx+b, when you write a rule for a linear function. </li></ul>Points (0, -1) and (4, 1) Find Slope First. 2/4 or ½ = m Y-Intercepts at? (0, -1) = b = -1 The rule is f(x) = ½x + -1.
  12. 12. Find the Rule <ul><li>Write a rule for the function graphed here. </li></ul><ul><li>Find the Slope, this is m. </li></ul><ul><li>Find the y-intercept, this is b. </li></ul><ul><li>Make your function rule. </li></ul><ul><li>y = -x + 2 </li></ul>
  13. 13. Question? <ul><li>What are advantages you see in using a rule for a function rather than listing function values in a table? </li></ul>
  14. 14. Assignment #4: Pages 406-407: 1-20 all.

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