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8.2 – Properties of Exponential Functions
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Review: 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.
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Graphing y=abx when a<0
Ex: Graph Sketch your prediction of what the graph will look like Where is the asymptote?
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Translating y=abx How does the equation change if we want to move both graphs up 4 units? Predictions? Question: where is the asymptote now? To move the graph up or down, add or subtract units at the end of the equations. No need to use inverses – if you want to go up, add; if you want to go down, subtract.
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Translating y=abx How does the equation change if we want to move both graphs left 4 units? Predictions? To move the graph left or right, add or subtract units to the exponent of the equation. Reminder: use the inverse of how you want the graph to move (e.g. x-4 will move to the right; x+4 will move to the left)
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Let’s try some Graph each function as a translation of y=9(3)x
Make a table of values for each Graph, from -3 to 3
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y =9(3)x+1 y =9(3)x-4-1 y =9(3)x-4
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e is an irrational number, approximately equal to 2.718.
What is base “e” ? e is an irrational number, approximately equal to Exponential functions with a base of e are useful for describing continuous growth or decay. In the graph below, y = e is the asymptote to the graph. y = e
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Graphing ex Using your graphing calculators, graph y=ex. Evaluate e4 to four decimal places. We now need to evaluate where x=4
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2. Press 2nd, Calc and select 1 (value). Press enter
3. We are evaluating when x=4. Enter 4 for x and press enter.
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The value of e4 is about 54.59815 Your turn: evaluate e-3
0.0498 So, what is “e” good for???
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Continuously Compounding Interest
A = Pert A = amount of money in the account P = principal (how much is deposited) r = annual rate of interest t = time (in years)
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Example: Continuously Compounded Interest Problem
You invest $1,050 at an annual interest rate of 5.5%, compounded continuously. How much will you have in the account after 5 years? Start with: A = Pert 1050(e)0.055(5) 1050(e)0.275 1050( ) A = $ P=$1050, r=5.5% = 0.055, t=5 Substitute in for p, r, and t Simplify they power Evaluate e0.275 with your calculator Simplify
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Let’s try one: Suppose you invest $1,300 at an annual interest rate of 4.3%, compounded continuously. How much will you have in the account after three years?
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Suppose you invest $1,300 at an annual interest rate of 4
Suppose you invest $1,300 at an annual interest rate of 4.3%, compounded continuously. How much will you have in the account after three years?
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