Exercise 121 Page 715

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 a function. y=ab^x Since we want the points to lie on the graph, they must both satisfy this equation. By substituting these points into the formula, we get two equations. |c|c| [-0.8em] Point & y=ab^x [0.3em] [-0.8em] ( 1, 6) & 6=ab^1 [0.3em] [-0.8em] ( 5, 30.375) & 30.375=ab^5 [0.3em] If we combine these equations we get a system of equations we can solve for a and b.

6=ab^1 & (I) 30.375=ab^5 & (II)

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(I): Solve for a

ExponentOne

(I): a^1=a

6=ab 30.375=ab^5

RearrangeEqn

(I): Rearrange equation

ab=6 30.375=ab^5

DivEqn

(I): .LHS /b.=.RHS /b.

a= 6b 30.375=ab^5

Substitute

(II): a= 6/b

a= 6b 30.375=( 6b)b^5

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(II): Solve for b

MoveRightFacToNum

(II): a/c* b = a* b/c

a= 6b 30.375= 6b^5b

SimpQuot

(II): Simplify quotient

a= 6b 30.375=6b^4

RearrangeEqn

(II): Rearrange equation

a= 6b 6b^4=30.375

DivEqn

(II): .LHS /6.=.RHS /6.

a= 6b b^4=5.0625

RadicalEqn

(II): sqrt(LHS)=sqrt(RHS)

a= 6b b=±1.5

(II): b > 0

a= 6b b=1.5

Having solved for b, we can substitute this into the first equation to find a.

a= 6b b=1.5

Substitute

(I): b= 1.5

a= 6 1.5 b=1.5

CalcQuot

(I): Calculate quotient

a=4 b=1.5

Finally, we can write the full equation of the exponential function. y=4(1.5)^x