We want to determine whether the data show an exponential relationship. Then we will write a function that models the data. Let's do those things one at a time.

Determining the Type of the Model

If the ratios of consecutive $y -$ values are equal, then the data can be modeled by an exponential function. If the difference of consecutive $y -$ values is constant, then the data can be modeled by a linear function. Consider the given table.

$x$	$- 3$	$- 1$	$1$	$3$	$5$
$y$	$2$	$7$	$24$	$68$	$194$

Let's calculate the difference between consecutive

y -

values.

7 - 2 68 - 24 = 5, 24 - 7 = 17, = 44, 194 - 68 = 126

We can see that the differences are not constant, so the data cannot be modeled by a linear function. Let's determine the ratios of the consecutive

y -

values.

\frac{2}{7} \frac{2 4}{6 8} \approx 0.286, \frac{7}{2 4} \approx 0.292, \approx 0.353, \frac{6 8}{1 9 4} \approx 0.351

Each ratio is around

0.3,

so the data can be modeled by an exponential function.

y = a b^{x}

Writing the Model

To find the values of

a

and

b,

we will use two of the ordered pairs given in the table. For simplicity, we will use

(- 1, 7)

and

(1, 24) .

Let's start by substituting

- 1

for

x

and

7

for

y .

y = a b^{x}

SubstituteII

$x = - 1$ , $y = 7$

7 = a b^{- 1}

▼

Solve for

a

MultEqn

$LHS \cdot b = RHS \cdot b$

7 b = a

RearrangeEqn

Rearrange equation

a = 7 b

Now that we know that

a = 7 b,

we can partially write the equation.

y = 7 b (b)^{x}

To find the value of

b,

we will substitute

1

for

x

and

24

for

y

in the above equation.

y = 7 b (b)^{x}

SubstituteII

$x = 1$ , $y = 24$

24 = 7 b (b)^{1}

▼

Solve for

b

ExponentOne

$a^{1} = a$

24 = 7 b (b)

ProdToPowTwoFac

$a \cdot a = a^{2}$

24 = 7 b^{2}

DivEqn

$LHS / 7 = RHS / 7$

\frac{2 4}{7} = b^{2}

SqrtEqn

$LHS = RHS$

\frac{2 4}{7} = b

UseCalc

Use a calculator

1.85 \approx b

RearrangeEqn

Rearrange equation

b \approx 1.85

Now that we know that

b \approx 1.85,

we can calculate the coefficient

a = 7 b .

a = 7 b

Substitute

$b = 1.85$

a = 7 (1.85)

Multiply

a = 12.95

Now that we know that

a = 12.95,

we can write the full equation that models the data in the given table.

y = 12.95 (1.85)^{x}

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