Exercise 39 Page 543

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 Since we want the points to lie on the graph, they must satisfy this equation. Let's substitute (0,32) into the formula and simplify.

y=ab^x

SubstituteII

x= 0, y= 32

32=ab^0

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Solve for a

ExponentZero

a^0=1

32=a(1)

IdPropMult

Identity Property of Multiplication

32=a

RearrangeEqn

Rearrange equation

a=32

Now we can partially write our equation. y= ab^x ⇒ y= 32b^x Next, let's substitute the second given point, (3, 4), into our partial equation and solve for b.

y=32b^x

SubstituteII

x= 3, y= 4

4=32b^3

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Solve for b

DivEqn

.LHS /32.=.RHS /32.

4/32=b^3

ReduceFrac

a/b=.a /4./.b /4.

1/8=b^3

RadicalEqn

sqrt(LHS)=sqrt(RHS)

sqrt(1/8) = sqrt(b^3)

RootPowToNumber

sqrt(a^3)=a

sqrt(1/8) = b

RootQuot

sqrt(a/b)=sqrt(a)/sqrt(b)

sqrt(1)/sqrt(8) = b

CalcRoot

Calculate root

1/2 = b

RearrangeEqn

Rearrange equation

b=1/2

Finally, we can write the full equation of the exponential function. y=32b^x ⇒ y=32(1/2)^x