An exponential function is in the general form Show
We know the points #(-1,8)# and #(1,2)#, so the following are true:
Multiply both sides of the first equation by #b# to find that
Plug this into the second equation and solve for #b#:
Two equations seem to be possible here. Plug both values of #b# into the either equation to find #a#. I'll use the second equation for simpler algebra. If#b=1/2#:
Giving us the equation: #color(green)(y=4(1/2)^x# If#b=-1/2#:
Giving us the equation: #y=-4(-1/2)^x# However! In an exponential function, #b>0#, otherwise many issues arise when trying to graph the function. The only valid function is
The general form for an exponential equation is f(x) = a*bx. To find the equation, you need to find the values of a and b. 1. To find b, take the ratio of f at the two known points, f(3)/f(2)): 270/90 = a*b3/a*b2 = b 2. So far we have f(x) = a*3x. To find the value of a, plug in one of the known points. I'll use (2,90): f(2) = a*32 90 = a*9 90/9 = a 10 = a The full equation is f(x) = 10*3x
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Patrick B. answered • 05/02/19 Math and computer tutor/teacher (x1,y1) and (x2,y2) are the given points Then y1 = a*b^(x1) ---> a = y1 * b^(-x1) y2 = a*b^(x2) this is a non-linear system Ex. The exponential function y = B^x passes through (0,3) (2,12) 3 = a*b^0 3 = a So y = 3*b^x Then 12 = 3*b^2 4 = b^2 b = 2 So the exponential function is y = 3*2^x Still looking for help? Get the right answer, fast.ORFind an Online Tutor Now Choose an expert and meet online. No packages or subscriptions, pay only for the time you need. Learning Outcomes
In the previous examples, we were given an exponential function which we then evaluated for a given input. Sometimes we are given information about an exponential function without knowing the function explicitly. We must use the information to first write the form of the function, determine the constants a and b, and evaluate the function. How To: Given two data points, write an exponential model
Example: Writing an Exponential Model When the Initial Value Is KnownIn 2006, 80 deer were introduced into a wildlife refuge. By 2012, the population had grown to 180 deer. The population was growing exponentially. Write an algebraic function N(t) representing the population N of deer over time t. Try ItA wolf population is growing exponentially. In 2011, 129 wolves were counted. By 2013 the population had reached 236 wolves. What two points can be used to derive an exponential equation modeling this situation? Write the equation representing the population N of wolves over time t. Example: Writing an Exponential Model When the Initial Value is Not KnownFind an exponential function that passes through the points [latex]\left(-2,6\right)[/latex] and [latex]\left(2,1\right)[/latex].
Try ItGiven the two points [latex]\left(1,3\right)[/latex] and [latex]\left(2,4.5\right)[/latex], find the equation of the exponential function that passes through these two points. Q & ADo two points always determine a unique exponential function? Yes, provided the two points are either both above the x-axis or both below the x-axis and have different x-coordinates. But keep in mind that we also need to know that the graph is, in fact, an exponential function. Not every graph that looks exponential really is exponential. We need to know the graph is based on a model that shows the same percent growth with each unit increase in x, which in many real world cases involves time. How To: Given the graph of an exponential function, write its equation
Example: Writing an Exponential Function Given Its GraphFind an equation for the exponential function graphed below. Try ItFind an equation for the exponential function graphed below. Investigating Continuous GrowthSo far we have worked with rational bases for exponential functions. For most real-world phenomena, however, e is used as the base for exponential functions. Exponential models that use e as the base are called continuous growth or decay models. We see these models in finance, computer science, and most of the sciences such as physics, toxicology, and fluid dynamics. A General Note: The Continuous Growth/Decay FormulaFor all real numbers t, and all positive numbers a and r, continuous growth or decay is represented by the formula [latex]A\left(t\right)=a{e}^{rt}[/latex] where
If r > 0, then the formula represents continuous growth. If r < 0, then the formula represents continuous decay. For business applications, the continuous growth formula is called the continuous compounding formula and takes the form [latex]A\left(t\right)=P{e}^{rt}[/latex] where
How To: Given the initial value, rate of growth or decay, and time t, solve a continuous growth or decay function
Example: Calculating Continuous GrowthA person invested $1,000 in an account earning a nominal interest rate of 10% per year compounded continuously. How much was in the account at the end of one year? Try ItA person invests $100,000 at a nominal 12% interest per year compounded continuously. What will be the value of the investment in 30 years? Example: Calculating Continuous DecayRadon-222 decays at a continuous rate of 17.3% per day. How much will 100 mg of Radon-222 decay to in 3 days? Try ItUsing the data in the previous example, how much radon-222 will remain after one year? Contribute!Did you have an idea for improving this content? We’d love your input. Improve this pageLearn More |