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Big Ideas Math Algebra 2, 2014

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Big Ideas Math Algebra 2, 2014 View details

Mathematical Practices

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Exercise 4 Page 294

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.

Linear or Exponential? Exponential.
Model: y=3^x
Explanation: See solution.
Value of y when x=10: 59049

Practice makes perfect

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	3	9	27	81

Let's calculate the difference between consecutive y-values. 3-1= 2, 9-3= 6, [1em] 27-9=18, 81-27=54 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. 1/3, 3/9= 1/3, 9/27= 1/3, 27/81= 1/3 Each ratio is equal to 13, 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 (1,3). Let's start by substituting 0 for x and 1 for y.

y=ab^x

x= 0, y= 1

1=ab^0

▼

Solve for a

a^0=1

1=a(1)

Identity Property of Multiplication

1=a

Rearrange equation

a=1

Now that we know that a=1, we can partially write the equation. y=1 * b^x ⇔ y=b^x To find the value of b, we will substitute 1 for x and 3 for y in the above equation.

y=b^x

x= 1, y= 3

3=b^1

a^1=a

3=b

Rearrange equation

b=3

Now that we know that b=3, we can write the full equation that models the data in the given table. y=3^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=3^x

x= 10

y=3^(10)

Calculate power

y=59049

Subchapter links

Monitoring Progress p.294