Exercise 96 Page 471

a We want to write an exponential function for the graph that has a

y -

intercept of

2

and passes through the given point. Let's consider the general form for this type of function.

y = a b^{x}

Since

2

is the

y -

intercept, the graph of the given function starts at the point

(0, 2) .

We want the points to lie on the graph, so they must satisfy the above equation. Let's substitute

(0, 2)

into this formula.

y = a b^{x}

SubstituteII

$x = 0$ , $y = 2$

2 = a b^{0}

▼

Solve for

a

ExponentZero

$a^{0} = 1$

2 = a (1)

IdPropMult

Identity Property of Multiplication

2 = a

RearrangeEqn

Rearrange equation

a = 2

Now we can partially write our equation.

y = a b^{x} \Rightarrow y = 2 b^{x}

Next we will substitute the second given point,

(3, 128),

into the above equation and solve for

b .

y = 2 b^{x}

SubstituteII

$x = 3$ , $y = 128$

128 = 2 b^{3}

▼

Solve for

b

DivEqn

$LHS / 2 = RHS / 2$

64 = b^{3}

RadicalEqn

$3 LHS = 3 RHS$

364 = b

CalcRoot

Calculate root

4 = b

RearrangeEqn

Rearrange equation

b = 4

Finally, we can write the full equation of the exponential function.

y = 2 b^{x} \Rightarrow y = 2 (4)^{x}

b 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 = a b^{x}

Since we want the points to lie on the graph, they must satisfy this equation. Let's substitute

(0, 4)

into the above formula.

y = a b^{x}

SubstituteII

$x = 0$ , $y = 4$

4 = a b^{0}

▼

Solve for

a

ExponentZero

$a^{0} = 1$

4 = a (1)

IdPropMult

Identity Property of Multiplication

4 = a

RearrangeEqn

Rearrange equation

a = 4

Now we can partially write our equation.

y = a b^{x} \Rightarrow y = 4 b^{x}

Next we will substitute the second given point,

(2, 1),

into the above equation and solve for

b .

y = 4 b^{x}

SubstituteII

$x = 2$ , $y = 1$

1 = 4 b^{2}

▼

Solve for

b

DivEqn

$LHS / 4 = RHS / 4$

\frac{1}{4} = b^{2}

SqrtEqn

$LHS = RHS$

\frac{1}{4} = b

SqrtQuot

$\frac{a}{b} = \frac{a}{b}$

\frac{1}{4} = b

CalcRoot

Calculate root

\frac{1}{2} = b

RearrangeEqn

Rearrange equation

b = \frac{1}{2}

Finally, we can write the full equation of the exponential function.

y = 4 b^{x} \Rightarrow y = 4 (\frac{1}{2})^{x}

Subchapter links

8.2.1 Curve Fitting p.471-472

100

101

102

8.2.2 Curved Best-Fit Models p.478-480

108

109

110

111

112

113

114

8.2.3 Solving a System of Exponential Functions Graphically p.483-484

119

120

121

122

123

124

125