Exponential Functions 4.2 Explorations of growth and decay.

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Exponential Functions 4.2 Explorations of growth and decay

SAT #1 Quick Poll!

SAT #2 Quick Poll!

SAT #3 Quick Poll!

A half-life POD If a substance has a half-life of 25 hours, how much of a 100-gram sample is left after 75 hours? 115 hours?

A POD to start with If a substance has a half-life of 25 hours, how much of a 100-gram sample is left after 75 hours? 115 hours? At 75 hours, there will have been 3 half-lives, decreasing from 100 to 50 to 25 to 12.5 g. At 115 hours, there will have been some fraction of a half-life. We need to use an exponential equation.

Exponential Basics Form: f(x)=ab x a is the initial amount (why?) b is the base (it is also the rate of change-- why?) x is the exponent (which is why they are called exponential functions)– it is often the variable for time.

Graph y=3 x and y=(1/3) x. How do the graphs compare? What are the asymptotes? Intercepts? What do the phrases “exponential decay” and “exponential growth” have to do with these graphs? Could we find inverses for these graphs? The Graphs

Reflected over y-axis The y-intercept for both is (0,1). Both are one-on-one functions. HA for both is the x-axis. No VA for either– no restriction on the domain. One is growth, one is decay. Inverses involve logs. The Graphs

Graph y=3 x and y = log 3 x. Then add y = x to the graph. What do you see? The Graphs

Exponential Graphs Now, graph y=3 x and y=3 -x on the same set of axises on your calculator. How do these graphs compare? What is the transformation from the first to the second? Why does the second one do what it does?

Exponential Graphs Now, graph y=3 x and y=2 x on the same set of axises on your calculator. How do they compare? How do the graphs of y=3 -x and y=2 -x compare? In general, if 1 < c < d, then c x < d x for positive values of x and c x > d x for negative values of x.

Exponential Graphs What would the graph of y=3*2 x look like? Growth or decay? How does it compare to the graph of y=2 x ? What is the transformation? Asymptote? Intercepts? What does the intercept mean?

Exponential Graphs Graph the following and compare: Changey-int asymptote h(x)=2 x f(x)=2 x - 2 g(x)=2 x-2 What sorts of transformations are we looking at?

Exponential Applications Let’s go back to the POD. a = 100 b =.5 x = 115/25 y = 100(.5) 115/25 = 4.12 gr. Does this fit with our other data?

Exponential Applications Find the total amount of dollars for $1000 invested at 6% compounded annually and monthly, after 4 years. (Remember the formulas?)

Exponential Applications Find the total amount of dollars for $1000 invested at 6% compounded annually and monthly, after 4 years. (Remember the formulas?)

Exponential Equations Solve for x: 6 7-x = 6 2x+1 2 x+1 = 4 x

Exponential Equations Solve for x: 6 7-x = 6 2x+1 Same base– set exponents equal to each other 7 – x = 2x + 1 x = 2 2 x+1 = 4 x Different bases– get to common base before setting exponents equal 2 x+1 = 2 2x x + 1 = 2x x = 1

Exponential Equations Find an exponential function of the form y=ab x with a y-intercept (0, 6) and that contains the point (2, 3/32).

Exponential Equations Find an exponential function of the form y=ab x with a y-intercept (0, 6) and that contains the point (2, 3/32). a = 6 x = 2 y = 3/32 Find b and the final equation.