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Exponential Relationships

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Exponential Relationships

  1. 1. Math relationships <ul><li>GOAL OF THIS BOOK: GROWING, GROWING, GROWNG </li></ul><ul><ul><li>Are ALL math relationships linear? </li></ul></ul><ul><li>Why or why not? </li></ul><ul><li>Give an example of a linear one. </li></ul><ul><ul><li>What would the graph look like? the table? </li></ul></ul><ul><li>Give an example of a nonlinear one. </li></ul><ul><ul><li>What would the graph look like? the table? </li></ul></ul><ul><li>Would the equations look the same? Why OR why not? </li></ul><ul><li>Review: What does a linear equation look like? </li></ul><ul><li>y=mx + b y=b + mx </li></ul><ul><li>m is the slope, constant rate </li></ul><ul><li>b is the y-intercept </li></ul><ul><li>x could represent books, months, inches </li></ul>
  2. 2. Who would you rather be? 4 $54.00 $122.00 3 $18.00 $82.00 2 $6.00 $42.00 1 $2.00 $2.00 0 Jill Jack Month
  3. 3. Comparing Jack and Jill <ul><li>How much money do they each start with? </li></ul><ul><li>What do we call the starting amount? </li></ul><ul><li>How much are they increasing by each month? </li></ul><ul><li>Is it a constant rate? Are you adding or subtracting the same amount each time? </li></ul><ul><li>Which one of these is linear? Nonlinear? </li></ul>
  4. 4. Let’s look at each person First, Jack… <ul><li>Jack starts with $2.00 then adds $40.00 every month </li></ul><ul><li>Total money = $2.00 plus $40.00 every month </li></ul><ul><li>t = 2.00 + 40 m </li></ul><ul><li>Linear since you add the same amount each time </li></ul>
  5. 5. Now Jill… <ul><li>Jill also starts with $2.00 then you multiply by 3 for every month </li></ul><ul><li>Total money = $2.00 then multiply by 3 for every month </li></ul><ul><li>t = 2.00(3 x ) </li></ul><ul><li>It is NOT linear since you do NOT add the same amount </li></ul><ul><li>It is Exponential since you are MULTIPLYING and the equation has an exponent in it </li></ul>
  6. 6. Exponential Relationships <ul><li>Exponential growth : </li></ul><ul><li>pattern of change that increases over time </li></ul><ul><li>Each value is multiplied by the previous value by a constant factor which is called the growth factor </li></ul><ul><li>How is this different from linear equations? </li></ul><ul><li>Exponential graphs start growing slowly at first and then quickly increase </li></ul><ul><li>How is this different from linear graphs ? </li></ul>
  7. 7. Exponential growth equations <ul><li>y = b(g x ) </li></ul><ul><li>b = is the starting amount </li></ul><ul><li>it also is the y-intercept </li></ul><ul><li>g = is the growth factor; what you are multiplying by each time </li></ul><ul><li>x = is the time interval such as hours, days, years </li></ul><ul><ul><li>How is this equation different from linear equations? </li></ul></ul>
  8. 8. Exponential Decay <ul><li>pattern of change that decreases over time </li></ul><ul><li>each value is multiplied by the previous value by a constant factor which is called the decay factor </li></ul><ul><li>You can find it by dividing each successive y-value by the previous y-value </li></ul><ul><li>y = b(g x ) same equation but the decay factor will be <1 </li></ul><ul><li>The graph starts out decreasing slowly then decreases quickly </li></ul><ul><li>Will the graph ever go pass the x-intercept? </li></ul><ul><li>1 - % decrease as a rate (decimal) OR ask yourself what % is remaining? </li></ul><ul><li>Examples in real-life : carbon dating to find the age of an object/organism; decay of radioactive substances; populations of endangered species </li></ul>

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