Understanding Exponential Equations and Their Applications in Growth and Decay
This guide explores exponential equations, focusing on the general form (y = ab^x) and the concepts of growth and decay factors. Learn about asymptotes and practice graphing exponential functions with provided integer values. Discover how to write and solve exponential equations given specific points, and delve into real-world applications, such as modeling population growth and depreciation. By mastering these concepts, you can effectively analyze and interpret situations involving exponential changes.
Understanding Exponential Equations and Their Applications in Growth and Decay
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Presentation Transcript
What is an exponential equation? An exponential equation has the general form y=abx
Given the general form y=abx When b > 1, b is the growth factor Growth Factor, Decay Factor • When 0 < b < 1, b is the decay factor
Growth or Decay??? Growth Decay Growth Decay Growth Decay
What is an asymptote? “Walking halfway to the wall” An Asymptote is a line that a graph approaches as x or y increases in absolute value. In this example, the asymptote is the x axis.
Graphing Exponential Functions Complete the table using the integers -3 through 3 for x.
Let’s try one Complete the table using the integers -3 through 3 for x. Then graph the function.
Let’s try one Complete the table using the integers -3 through 3 for x. Then graph the function.
Writing Exponential Equations • Find the exponential equation passing through the points (3,20) and (1,5). Start with the general form. Choose a point. Substitute for x and y using (3, 20) Solve for a Substitute x and y using (1, 5) and a using Division property of exponents
Writing Exponential Equations • Find the exponential equation passing through the points (3,20) and (1,5). Simplify Solve for b Go back to the equation for a; substitute in b and solve for a
Writing Exponential Equations • Find the exponential equation passing through the points (3,20) and (1,5). Going back to the general form, substitute in a and b The exponential equation passing through the points (3,20) and (1,5) is
Let’s Try One • Find the exponential equation passing through the points (2,4) and (3,16). Start with the general form. Choose a point. Substitute for x and y using (2, 4) Solve for a Substitute x and y using (3, 16) and a using Division property of exponents
Writing Exponential Equations Simplify Solve for b Go back to the equation for a; substitute in b and solve for a Going back to the general form, substitute in a and b The exponential equation passing through the points (2,4) and (3,16) is
Putting it all together . . . • Find the equation of the exponential function that goes through (1,6) and (0,2). Graph the function.
Modeling Growth with an Exponential Equation • The growth factor can be found in word problems using b = 1 + r where r = rate or amount of increase. You can substitute your new b into your general equation to find the exponential function.
EX- a guy puts $1000 into a simple 3% interest account. What is the exponential equation? r = rate 3% (write as 0.03) b = 1 + r = 1.03 x = time a = amount put into the account ($1,000)
EX – a colony of 1000 bacteria cells doubles every hour. What is the exponential equation? r = 1 (why not 2?) b = r + 1 = 2 x = time (in hours) a = the original number in the colony (1,000 bacteria ) b = r + 1, where r is the amount of increase. We are increasing by 100% each time something doubles, so r = 1
EX- a $15000 car depreciates at 10% a year. What is the exponential equation? r = - 10% (the car is worth 10% less each year) b = 1 - r = 1 – 0.1 = 0.9 x = time (in years) a = amount put into the account ($15,000)
