Two different businesses model their profits over 15 years, where x is the year, f(x) is the profits of a garden shop, and g(x) is the profits of a construction materials business. Use the data to determine which function is exponential, and use the table to justify your answer.
x f(x) g(x)
1995 $69,682.50 $72,429.27
2000 $78,943.50 $79,967.77
2005 $88,204.50 $88,290.88
2006 $90,056.70 $90,056.70
2007 $91,908.90 $91,857.83
2010 $97,465.50 $97,480.27
f(x) is exponential; an exponential function increases more slowly than a linear function.
f(x) is exponential; f(x) increased more overall than g(x).
g(x) is exponential; g(x) has a higher starting value and higher ending value.
g(x) is exponential; an exponential function increases faster than a linear function.
exponential functions get grow faster than any polynomial.
Over equal intervals, exponentials grow faster and faster. Unfortunately, your table does not have equal spacing, so you need to check your slopes to be sure.
Is this right
To determine which function is exponential, we need to compare the growth patterns of the two functions over the years given in the table.
Looking at the data, we can see that the profits of the garden shop (f(x)) increase steadily but at a slower rate compared to the profits of the construction materials business (g(x)). This indicates that g(x) has a higher growth rate and is likely to be exponential.
To further justify our answer, we can calculate the average growth rate of both functions using the provided data.
For the garden shop:
Average growth rate = (f(2010) - f(1995))/(2010 - 1995)
= ($97,465.50 - $69,682.50)/(2010 - 1995)
= $27,783/15
= $1,852.20 per year
And for the construction materials business:
Average growth rate = (g(2010) - g(1995))/(2010 - 1995)
= ($97,480.27 - $72,429.27)/(2010 - 1995)
= $25,051/15
= $1,670.07 per year
From these calculations, we can see that the average growth rate of the garden shop is $1,852.20 per year, while for the construction materials business, it is $1,670.07 per year. This suggests that the garden shop's growth rate is higher, indicating exponential growth.
Therefore, the correct answer is: f(x) is exponential; f(x) increased more overall than g(x).
To determine which function is exponential, we need to compare the rate at which the profits increase over the given years for both the garden shop (f(x)) and the construction materials business (g(x)).
We can calculate the annual growth rate for each year by finding the percentage increase in profits from one year to the next. Using the data provided, we can observe the following percentages of growth for each business:
f(x):
1995 to 2000: (78,943.50 - 69,682.50)/69,682.50 ≈ 13.37%
2000 to 2005: (88,204.50 - 78,943.50)/78,943.50 ≈ 11.73%
2005 to 2006: (90,056.70 - 88,204.50)/88,204.50 ≈ 2.10%
2006 to 2007: (91,908.90 - 90,056.70)/90,056.70 ≈ 2.05%
2007 to 2010: (97,465.50 - 91,908.90)/91,908.90 ≈ 6.06%
g(x):
1995 to 2000: (79,967.77 - 72,429.27)/72,429.27 ≈ 10.37%
2000 to 2005: (88,290.88 - 79,967.77)/79,967.77 ≈ 10.40%
2005 to 2006: (90,056.70 - 88,290.88)/88,290.88 ≈ 2.00%
2006 to 2007: (91,857.83 - 90,056.70)/90,056.70 ≈ 2.00%
2007 to 2010: (97,480.27 - 91,857.83)/91,857.83 ≈ 6.13%
By analyzing the percentage growth rates, we can see that the percentages of growth for both f(x) and g(x) vary over the given years but remain relatively consistent overall.
Now, let's compare the percentage rates of growth between f(x) and g(x) for each year:
1995 to 2000:
f(x): 13.37%
g(x): 10.37%
f(x) increased at a larger percentage rate than g(x).
2000 to 2005:
f(x): 11.73%
g(x): 10.40%
Again, f(x) increased at a larger percentage rate than g(x).
2005 to 2006:
f(x): 2.10%
g(x): 2.00%
The growth rates for both businesses are relatively similar.
2006 to 2007:
f(x): 2.05%
g(x): 2.00%
Again, the growth rates for both businesses are similar.
2007 to 2010:
f(x): 6.06%
g(x): 6.13%
Both businesses experienced similar growth rates once again.
Based on these comparisons, we can see that f(x) consistently increased at a higher rate than g(x) over the given years. This indicates that the profits of the garden shop (f(x)) follow an exponential pattern, as an exponential function increases more slowly than a linear function. Therefore, the correct answer is: "f(x) is exponential; an exponential function increases more slowly than a linear function."