We want to write an appropriate model for the given data. Then we will use the model to find $y$ when $x = 10 .$ Let's do those things one at a time.

Writing 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.

$x$	$0$	$1$	$2$	$3$	$4$
$y$	$1$	$2$	$4$	$8$	$16$

Let's calculate the difference between consecutive

y -

values.

2 - 1 8 - 4 = 1, 4 - 2 = 2, = 4, 16 - 8 = 8

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}{1} \frac{8}{4} = 2, \frac{4}{2} = 2, = 2, \frac{1 6}{8} = 2

Each ratio is equal to

2,

so the data can be modeled by an exponential function.

y = a b^{x}

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

(0, 1)

and

(1, 2) .

Let's start by substituting

0

for

x

and

1

for

y .

y = a b^{x}

SubstituteII

$x = 0$ , $y = 1$

1 = a b^{0}

▼

Solve for

a

ExponentZero

$a^{0} = 1$

1 = a (1)

IdPropMult

Identity Property of Multiplication

1 = a

RearrangeEqn

Rearrange equation

a = 1

Now that we know that

a = 1,

we can partially write the equation.

y = 1 b^{x} \Leftrightarrow y = b^{x}

To find the value of

b,

we will substitute

1

for

x

and

2

for

y

in the above equation.

y = b^{x}

SubstituteII

$x = 1$ , $y = 2$

2 = b^{1}

ExponentOne

$a^{1} = a$

2 = b

RearrangeEqn

Rearrange equation

b = 2

Now that we know that

b = 2,

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

y = 2^{x}

Finding $y$ when $x = 10$

Finally, we will find the value of

y

when

x = 10 .

To do so, we will substitute

10

for

x

in the obtained formula.

y = 2^{x}

Substitute

$x = 10$

y = 2^{10}

CalcPow

Calculate power

y = 1024

Writing the Model

Finding y when x=10

Finding $y$ when $x = 10$

Exercise