For x > 1 order the following functions from steepest to shallowest.
a. y=2(10)^x
b. y=2(3)^x
c. y=2(5)^x
d. y=1/2(10)^x
e. y=1/2(3)^x
f. y=1/2(5)^x
I think the order would be b,d,e,a,f,g but I am not very good at this, could someone help me with what the order would be?
the larger the base, the steeper the growth. Now, around x=0, the steepness also depends on the leading coefficient, but once you get to larger x values, any kind of 10^x will be steeper than any possible 3^x
So the order will be 10^x > 5^x > 3^x
after that, order by decreasing order of leading coefficients. That makes the correct order
a, d, c, f, b, e
Using some calculus, you can find just where faster functions overtake the slower ones, but that appears to be beyond the scope of this question.
try them for x = 2 and x = 3
a. 200, 2000
b. 18, 54
c. 50, 250
d. 50, 500
e. 4.5 , 13.5
f. 12.5 , 62.5
To compare the steepness of these exponential functions, we can examine their exponents and determine how quickly they grow as x increases. Here's how we can order them from steepest to shallowest:
1. First, let's compare the bases of the exponential functions. In this case, we have 10, 3, and 5 as the bases.
- Since 10 is the largest base, any exponential function with base 10 will grow faster than those with smaller bases. So, between options a and d, we can conclude that option a will be steeper than option d.
- Similarly, between options b and e, since 3 is smaller than 5, we can conclude that option b will be steeper than option e.
2. Now, within the remaining options a, b, e, d, and f, we need to compare the coefficients (multipliers) of the base.
- Comparing options a and b, we see that they have the same base (10), but option a has a coefficient of 2 while option b has a coefficient of 1. Since a higher coefficient leads to a steeper function, we can conclude that option a is steeper than option b.
- Similarly, comparing options d and e, both with base 3, we find that option d has a coefficient of 1/2 while option e has a coefficient of 1/2. Since they have the same coefficients, the steepness will be the same.
- Finally, comparing options a and f, both with base 5, we see that option f has a coefficient of 1/2 while option a has a coefficient of 2. Therefore, option a will be steeper than option f.
So, based on the above comparison, we can order these exponential functions from steepest to shallowest as follows:
a. y=2(10)^x
b. y=2(3)^x
e. y=1/2(3)^x
d. y=1/2(10)^x
f. y=1/2(5)^x
Option c, "y=2(5)^x," was not included in the ordering because the base (5) is smaller than the other bases and, therefore, leads to a slower growth rate.
Remember that this ordering is based on a relative comparison of the steepness of the functions and may not account for specific numerical values of y for different values of x.