Big Ideas Math Algebra 2, 2014
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Big Ideas Math Algebra 2, 2014 View details
7. Modeling with Exponential and Logarithmic Functions
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Exercise 26 Page 347

Begin by creating a table of data pairs (x,ln y) using the given information. Then, make the scatter plot of the resulting ordered pairs.

Possible Model: y=6.34(1.42)^x

Practice makes perfect

We will first create a table of data pairs (x,ln y) using the given table.

x y ln y
1 9 2.20
2 14 2.64
3 19 2.94
4 25 3.22
5 37 3.61
6 53 3.97
7 71 4.26

Next, we will plot the transformed points.

The points lie close to a line, so an exponential model should be a good fit for the original data. To write an exponential model y=ab^x, let's use the points ( 1, 2.20) and ( 5, 3.61) because they appear to be on the line. With this, we will write an equation in point-slope form. ln y- ln y_1= m(x- x_1) In this form, m is the slope and ( x_1, ln y_1) is a reference point that is on the line. Using the Slope Formula, we can substitute the chosen points and find the slope.
m = y_2-y_1/x_2-x_1
m = 3.61- 2.20/5- 1
m=1.41/4
m=0.3525
Now that we found the slope, we can write the equation in point-slope form. Let the point ( 1, 2.20) be our reference point. ln y-2.20=0.3525(x-1) Next, we will isolate y to have the model in the form of y=ab^x.
ln y-2.20=0.3525(x-1)
â–Ľ
Solve for y
ln y-2.20=0.3525x-0.3525
ln y=0.3525x+1.8475

e^(LHS)=e^(RHS)

e^(ln y)=e^(0.3525x+1.8475)

a = e^(ln(a))

y=e^(0.3525x+1.8475)
y=e^(0.3525x)* e^(1.8475)
y=(e^(0.3525))^x * e^(1.8475)
y=(1.42261...)^x* 6.34393...
y=6.34393...(1.42261...)^x
y=6.34(1.42)^x
As a result, one possible model is y=6.34(1.42)^x depending on the choice of the points.