Chapter Closure
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The general equation of an exponential function is y=ab^x. The asymptote at y=20 suggests that the function is translated.
y=8(1/4)^x+20
We want to write an exponential function for the graph that passes through the given points. Let's consider the general form for this type of function.
y=ab^x
Such functions have horizontal asymptote at y=0. However, we are told that the function has a horizontal asymptote at y=20. This means that the given function is translated 20 units up.
y=ab^x+20
Next, we will substitute the second given point, (3,20.125).
We received two equations, so to find the values of a and b we need to solve the system of equations. ab=2 & (I) ab^3=0.125 & (II) The first equation says that the product of a and b is 2. Notice that the second equation contain the same expression multiplied by b^2. ab=2 ab^3=0.125 ⇒ ab=2 ab* b^2=0.125 This allows us to substitute the value of ab to the second equation and solve for b. Keep in mind that b is the base of the exponential function, so it has to be positive.
(II): ab= 2
(II): Write as a fraction
(II): .LHS /2.=.RHS /2.
(II): .a/b /c.= a/b* c
(II): sqrt(LHS)=sqrt(RHS)
(II): sqrt(a^2)=a
(II): sqrt(a/b)=sqrt(a)/sqrt(b)
Once we know the value of b, let's substitute it to the first equation to find a.
Finally, we can write the full equation of the exponential function. y= a b^x+20 ⇒ y= 8( 1/4)^x+20