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A 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 .

( 0 , 129 ) and ( 2 , 236 ) ; N ( t ) = 129 ( 1 .3526 ) t

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Writing an exponential model when the initial value is not known

Find an exponential function that passes through the points ( 2 , 6 ) and ( 2 , 1 ) .

Because we don’t have the initial value, we substitute both points into an equation of the form f ( x ) = a b x , and then solve the system for a and b .

  • Substituting ( 2 , 6 ) gives 6 = a b 2
  • Substituting ( 2 , 1 ) gives 1 = a b 2

Use the first equation to solve for a in terms of b :

..

Substitute a in the second equation, and solve for b :

..

Use the value of b in the first equation to solve for the value of a :

..

Thus, the equation is f ( x ) = 2.4492 ( 0.6389 ) x .

We can graph our model to check our work. Notice that the graph in [link] passes through the initial points given in the problem, ( 2 ,  6 ) and ( 2 ,  1 ) . The graph is an example of an exponential decay function.

Graph of the exponential function, f(x)=2.4492(0.6389)^x, with labeled points at (-2, 6) and (2, 1).
The graph of f ( x ) = 2.4492 ( 0.6389 ) x models exponential decay.
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Given the two points ( 1 , 3 ) and ( 2 , 4.5 ) , find the equation of the exponential function that passes through these two points.

f ( x ) = 2 ( 1.5 ) x

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Do 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.

Given the graph of an exponential function, write its equation.

  1. First, identify two points on the graph. Choose the y -intercept as one of the two points whenever possible. Try to choose points that are as far apart as possible to reduce round-off error.
  2. If one of the data points is the y- intercept ( 0 , a ) , then a is the initial value. Using a , substitute the second point into the equation f ( x ) = a ( b ) x , and solve for b .
  3. If neither of the data points have the form ( 0 , a ) , substitute both points into two equations with the form f ( x ) = a ( b ) x . Solve the resulting system of two equations in two unknowns to find a and b .
  4. Write the exponential function, f ( x ) = a ( b ) x .

Writing an exponential function given its graph

Find an equation for the exponential function graphed in [link] .

Graph of an increasing exponential function with notable points at (0, 3) and (2, 12).

We can choose the y -intercept of the graph, ( 0 , 3 ) , as our first point. This gives us the initial value, a = 3. Next, choose a point on the curve some distance away from ( 0 , 3 ) that has integer coordinates. One such point is ( 2 , 12 ) .

   y = a b x Write the general form of an exponential equation .    y = 3 b x Substitute the initial value 3 for  a . 12 = 3 b 2 Substitute in 12 for  y  and 2 for  x .    4 = b 2 Divide by 3 .    b = ± 2 Take the square root .

Because we restrict ourselves to positive values of b , we will use b = 2. Substitute a and b into the standard form to yield the equation f ( x ) = 3 ( 2 ) x .

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Practice Key Terms 4

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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