Algebra Examples
Find the Exponential Function (2,25)
Step 1
To find an exponential function, , containing the point, set in the function to the value of the point, and set to the value of the point.
Step 2
Rewrite the equation as .
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Pull terms out from under the radical, assuming positive real numbers.
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
Step 3
Substitute each value for back into the function to find each possible exponential function.
Instructions: Use this step-by-step Exponential Function Calculator, to find the function that describe the exponential function that passes through two given points in the plane XY. You need to provide the points \((t_1, y_1)\) and \((t_2, y_2)\), and this calculator will estimate the appropriate exponential function and will provide its graph. The idea of this calculator is to estimate the parameters \(A_0\) and \(k\) for the function \(f(t)\) defined as: \[f(t) = A_0 e^{kt}\] so that this function passes through the given points \((t_1, y_1)\) and \((t_2, y_2)\). Technically, in order to find the parameters you need to solve the following system of equations: \[y_1 = A_0 e^{k t_1}\] \[y_2 = A_0 e^{k t_2}\]
Solving this system for \(A_0\) and \(k\) will lead to a unique solution, provided that \(t_1 = \not t_2\). Indeed, by dividing both sides of the equations: \[\displaystyle \frac{y_1}{y_2} = \frac{e^{k t_1}}{e^{k t_2}}\] \[\displaystyle \Rightarrow \, \frac{y_1}{y_2} = e^{k (t_1-t_2)}\] \[\displaystyle \Rightarrow \, \ln\left(\frac{y_1}{y_2}\right) = k (t_1-t_2)\] \[\displaystyle \Rightarrow \, k = \frac{1}{t_1-t_2} \ln\left(\frac{y_1}{y_2}\right)\] In order to solve for
\(A_0\) we notice from the first equation that: \[A_0 = y_1 e^{-k t_1} = y_1 \frac{y_2}{y_1 e^{k t_2}} =\frac{y_2}{e^{k t_2}} \] It is not always growth. Indeed, if the parameter \(k\) is positive, then we have exponential growth, but if the parameter \(k\) is negative, then we have exponential decay. The parameter \(k\) will be zero only if \(y_1 = y_2\) (the two points have the same height). For specific
exponential behaviors you can check our exponential growth calculator and the exponential decay calculator , which use specific parameters for that kinds of exponential behavior. Exponential Function Calculator from Two
Points
But, how do you find an exponential function from points?
How do you calculate exponential growth?
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1 Expert Answer
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
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An exponential function is in the general form
#y=a(b)^x#
We know the points #(-1,8)# and #(1,2)#, so the following are true:
#8=a(b^-1)=a/b#
#2=a(b^1)=ab#
Multiply both sides of the first equation by #b# to find that
#8b=a#
Plug this into the second equation and solve for #b#:
#2=(8b)b#
#2=8b^2#
#b^2=1/4#
#b=+-1/2#
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#:
#2=a(1/2)#
#a=4#
Giving us the equation: #color(green)(y=4(1/2)^x#
If#b=-1/2#:
#2=a(-1/2)#
#a=-4#
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
#color(green)(y=4(1/2)^x#