Exercise 1 Page 294

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.

Let's calculate the difference between consecutive y-values. 2-1&= 1, 4-2= 2, 8-4&=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. 2/1&= 2, 4/2= 2, [0.8em] 8/4&= 2, 16/8= 2 Each ratio is equal to 2, so the data can be modeled by an exponential function. y=ab^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.

Now that we know that a=1, we can partially write the equation. y=1b^x ⇔ y=b^x To find the value of b, we will substitute 1 for x and 2 for y in the above equation.

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.