Finding the Model

We want to tell whether the table of values represents a linear, exponential, or quadratic function. To do so, we will analyze how the consecutive terms are related to each other.

$x$	$- 1$	$0$	$1$	$2$	$3$
$y$	$1$	$3$	$9$	$27$	$81$

Let's begin with calculating the first differences.

The first differences are not all equal. Therefore, the table of values does not represent a linear function. Let's find the second differences and compare them.

The second differences are not all equal. Therefore, the table of values does not represent a quadratic function. Let's find the ratios of the

y -

values and compare them.

The ratios of successive

y -

values are equal. Therefore, the table of values can be modeled by an exponential function.

Finding the Equation

Let's recall the general form of this type of function.

y = a b^{x}

We will use two ordered pairs given in the table to find the values of

a

and

b .

For simplicity, let's use

(0, 3)

and

(1, 9) .

We will start by substituting

0

and

3

for

x

and

y,

respectively.

y = a b^{x}

SubstituteII

$x = 0$ , $y = 3$

3 = a b^{0}

▼

Solve for

a

ExponentZero

$a^{0} = 1$

3 = a (1)

IdPropMult

Identity Property of Multiplication

3 = a

RearrangeEqn

Rearrange equation

a = 3

We can write a partial equation of the function represented by the table.

y = 3 b^{x}

To find the value of

b

we will substitute

1

for

x

and

9

for

y

into our partial equation.

y = 3 b^{x}

SubstituteII

$x = 1$ , $y = 9$

9 = 3 b^{1}

▼

Solve for

b

ExponentOne

$a^{1} = a$

9 = 3 b

DivEqn

$LHS / 3 = RHS / 3$

3 = b

RearrangeEqn

Rearrange equation

b = 3

Now we can write the equation of the function represented by the table.

y = 3 \times 3^{x}

Exercise

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