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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 3 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? Linear.
Model: y=3x+1
Explanation: See solution.
Value of y when x=10: 31

Practice makes perfect

We want to write an appropriate model for the given data. Then we will use that 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	4	7	10	13

Let's calculate the ratios of the consecutive y-values. 1/4&= 0.25, 4/7 ≈ 0.571, [1em] 7/10&=0.7 , 10/13 ≈ 0.769 We can see that the ratios are not equal to each other, so the data cannot be modeled by an exponential function. Let's determine the differences between the consecutive y-values. 4-1&= 3, 7-4= 3, 10-7&= 3, 13-10= 3 The difference of consecutive y-values is constant, so the data can be modeled by a linear function. y=mx+b To find the values of m and b, we will use two of the ordered pairs given in the table. For simplicity, we will use (0,1) and (1,4). Let's start by substituting 0 for x and 1 for y.

y=mx+b

x= 0, y= 1

1=m( 0)+b

Zero Property of Multiplication

1=b

Rearrange equation

b=1

Now that we know that b=1, we can partially write the equation. y=mx+1 ⇔ y=mx+1 To find the value of m, we will substitute 1 for x and 4 for y in the above equation.

y=mx+1

x= 1, y= 4

4=m( 1)+1

Identity Property of Multiplication

4=m+1

LHS-1=RHS-1

3=m

Rearrange equation

m=3

Now that we know that m=3, we can write the full equation that models the data in the given table. y=3x+1

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=3x+1

x= 10

y=3( 10)+1

Multiply

y=30+1

Add terms

y=31

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Monitoring Progress p.294