SOLUTION: Hello all I would like your help one more time. I've got most of what my teacher asked us to do but I am stuck on the following the task is to: Evaluate the functions for the va

Algebra ->  Functions -> SOLUTION: Hello all I would like your help one more time. I've got most of what my teacher asked us to do but I am stuck on the following the task is to: Evaluate the functions for the va      Log On


   

Question 150910: Hello all I would like your help one more time. I've got most of what my teacher asked us to do but I am stuck on the following the task is to:
Evaluate the functions for the values of x given as 1, 2, 4, 8, and 16. Describe the differences in the rate at which each function changes with increasing values of x. f(x) = x2 - 3x + 2
f(x) = 2x3 + 7x2 - x - 1
f(x) = 10x
f(x) = ln x (I don't know if this is (one n x) or the letter (l)n x, whatever that means)
Any help would be GREATLY appreciated!!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 1





Let's find the function value when x=1:


y=x%5E2-3x%2B2 Start with the given equation.


y=%281%29%5E2-3%281%29%2B2 Plug in x=1.


y=1%281%29-3%281%29%2B2 Square 1 to get 1.


y=1-3%281%29%2B2 Multiply 1 and 1 to get 1.


y=1-3%2B2 Multiply -3 and 1 to get -3.


y=0 Combine like terms.


So if x=1, then y=0.

-------------

Let's find the function value when x=2:


y=x%5E2-3x%2B2 Start with the given equation.


y=%282%29%5E2-3%282%29%2B2 Plug in x=2.


y=1%284%29-3%282%29%2B2 Square 2 to get 4.


y=4-3%282%29%2B2 Multiply 1 and 4 to get 4.


y=4-6%2B2 Multiply -3 and 2 to get -6.


y=0 Combine like terms.


So if x=2, then y=0.

-------------

Let's find the function value when x=4:


y=x%5E2-3x%2B2 Start with the given equation.


y=%284%29%5E2-3%284%29%2B2 Plug in x=4.


y=1%2816%29-3%284%29%2B2 Square 4 to get 16.


y=16-3%284%29%2B2 Multiply 1 and 16 to get 16.


y=16-12%2B2 Multiply -3 and 4 to get -12.


y=6 Combine like terms.


So if x=4, then y=6.

-------------

Let's find the function value when x=8:


y=x%5E2-3x%2B2 Start with the given equation.


y=%288%29%5E2-3%288%29%2B2 Plug in x=8.


y=1%2864%29-3%288%29%2B2 Square 8 to get 64.


y=64-3%288%29%2B2 Multiply 1 and 64 to get 64.


y=64-24%2B2 Multiply -3 and 8 to get -24.


y=42 Combine like terms.


So if x=8, then y=42.

-------------

Let's find the function value when x=16:


y=x%5E2-3x%2B2 Start with the given equation.


y=%2816%29%5E2-3%2816%29%2B2 Plug in x=16.


y=1%28256%29-3%2816%29%2B2 Square 16 to get 256.


y=256-3%2816%29%2B2 Multiply 1 and 256 to get 256.


y=256-48%2B2 Multiply -3 and 16 to get -48.


y=210 Combine like terms.


So if x=16, then y=210.

-------------

Now let's make a table of the values we just found.





Jump to Top of Page

Table of Values:





xy
10


20


46


842


16210






# 2




Let's find the function value when x=1:


y=2x%5E3%2B7x%5E2-x-1 Start with the given equation.


y=2%281%29%5E3%2B7%281%29%5E2-1-1 Plug in x=1.


y=2%281%29%2B7%281%29%5E2-1-1 Cube 1 to get 1.


y=2%281%29%2B7%281%29-1-1 Square 1 to get 1.


y=2%2B7%281%29-1-1 Multiply 2 and 1 to get 2.


y=2%2B7-1-1 Multiply 7 and 1 to get 7.


y=7 Combine like terms.


So if x=1, then y=7.

-------------

Let's find the function value when x=2:


y=2x%5E3%2B7x%5E2-x-1 Start with the given equation.


y=2%282%29%5E3%2B7%282%29%5E2-2-1 Plug in x=2.


y=2%288%29%2B7%282%29%5E2-2-1 Cube 2 to get 8.


y=2%288%29%2B7%284%29-2-1 Square 2 to get 4.


y=16%2B7%284%29-2-1 Multiply 2 and 8 to get 16.


y=16%2B28-2-1 Multiply 7 and 4 to get 28.


y=41 Combine like terms.


So if x=2, then y=41.

-------------

Let's find the function value when x=4:


y=2x%5E3%2B7x%5E2-x-1 Start with the given equation.


y=2%284%29%5E3%2B7%284%29%5E2-4-1 Plug in x=4.


y=2%2864%29%2B7%284%29%5E2-4-1 Cube 4 to get 64.


y=2%2864%29%2B7%2816%29-4-1 Square 4 to get 16.


y=128%2B7%2816%29-4-1 Multiply 2 and 64 to get 128.


y=128%2B112-4-1 Multiply 7 and 16 to get 112.


y=235 Combine like terms.


So if x=4, then y=235.

-------------

Let's find the function value when x=8:


y=2x%5E3%2B7x%5E2-x-1 Start with the given equation.


y=2%288%29%5E3%2B7%288%29%5E2-8-1 Plug in x=8.


y=2%28512%29%2B7%288%29%5E2-8-1 Cube 8 to get 512.


y=2%28512%29%2B7%2864%29-8-1 Square 8 to get 64.


y=1024%2B7%2864%29-8-1 Multiply 2 and 512 to get 1024.


y=1024%2B448-8-1 Multiply 7 and 64 to get 448.


y=1463 Combine like terms.


So if x=8, then y=1463.

-------------

Let's find the function value when x=16:


y=2x%5E3%2B7x%5E2-x-1 Start with the given equation.


y=2%2816%29%5E3%2B7%2816%29%5E2-16-1 Plug in x=16.


y=2%284096%29%2B7%2816%29%5E2-16-1 Cube 16 to get 4096.


y=2%284096%29%2B7%28256%29-16-1 Square 16 to get 256.


y=8192%2B7%28256%29-16-1 Multiply 2 and 4096 to get 8192.


y=8192%2B1792-16-1 Multiply 7 and 256 to get 1792.


y=9967 Combine like terms.


So if x=16, then y=9967.

-------------

Now let's make a table of the values we just found.



Table of Values:





xy
17


241


4235


81463


169967






# 3




Let's find the function value when x=1:


y=10x Start with the given equation.


y=10%281%29 Plug in x=1.


y=10 Multiply 10 and 1 to get 10.


y=10 Combine like terms.


So if x=1, then y=10.

-------------

Let's find the function value when x=2:


y=10x Start with the given equation.


y=10%282%29 Plug in x=2.


y=20 Multiply 10 and 2 to get 20.


y=20 Combine like terms.


So if x=2, then y=20.

-------------

Let's find the function value when x=4:


y=10x Start with the given equation.


y=10%284%29 Plug in x=4.


y=40 Multiply 10 and 4 to get 40.


y=40 Combine like terms.


So if x=4, then y=40.

-------------

Let's find the function value when x=8:


y=10x Start with the given equation.


y=10%288%29 Plug in x=8.


y=80 Multiply 10 and 8 to get 80.


y=80 Combine like terms.


So if x=8, then y=80.

-------------

Let's find the function value when x=16:


y=10x Start with the given equation.


y=10%2816%29 Plug in x=16.


y=160 Multiply 10 and 16 to get 160.


y=160 Combine like terms.


So if x=16, then y=160.

-------------

Now let's make a table of the values we just found.


Table of Values:





xy
110


220


440


880


16160







# 4

Note: ln(x) also looks like LN(x) (to pronounce it, simply read off the letters "L" "N")

This is the natural log of x. So it is a logarithmic function.

Let's find the y value when x=1


y=ln%28x%29 Start with the given equation.


y=ln%281%29 Plug in x=1.


y=0 Take the natural log of 1 to get 0


So when x=1, y=0.


----------------------------


Let's find the y value when x=2


y=ln%28x%29 Start with the given equation.


y=ln%282%29 Plug in x=2.


y=0.693 Take the natural log of 2 to get 0.693


So when x=2, y=0.693.



----------------------------


Let's find the y value when x=4


y=ln%28x%29 Start with the given equation.


y=ln%284%29 Plug in x=4.


y=1.386 Take the natural log of 4 to get 1.386


So when x=4, y=1.386.

-------------------------------

Let's find the y value when x=8


y=ln%28x%29 Start with the given equation.


y=ln%288%29 Plug in x=8.


y=2.079 Take the natural log of 8 to get 2.079


So when x=8, y=2.079.


-------------------------------------


Let's find the y value when x=16


y=ln%28x%29 Start with the given equation.


y=ln%2816%29 Plug in x=16.


y=2.773 Take the natural log of 16 to get 2.773


So when x=16, y=2.773.


Now let's make a table of the values we just found.



Table of Values:





xy
10


20.693


41.386


82.079


162.773





===========================================================

So to compare the rates, simply compare the tables:

Table for y=x%5E2-3x%2B2 




xy
10


20


46


842


16210



Table for y=2x%5E3%2B7x%5E2-x-1 





xy
17


241


4235


81463


169967


Table for y=10x 





xy
110


220


440


880


16160


Table for y=ln%28x%29 





xy
10


20.693


41.386


82.079


162.773



From the tables, the functions can be sorted from slowest growing to fastest growing like this:

y=ln%28x%29, y=10x, y=x%5E2-3x%2B2, and y=2x%5E3%2B7x%5E2-x-1



So y=ln%28x%29 is the slowest growing function and y=2x%5E3%2B7x%5E2-x-1 is the fastest growing function.



If all of that is confusing, simply graph all of the functions to get

Graph of y=x%5E2-3x%2B2 (red), y=2x%5E3%2B7x%5E2-x-1 (green), y=10x(blue), and y=ln%28x%29 (purple)

From the graph, we can see that the order from slowest growing to fastest growing is

y=ln%28x%29, y=10x, y=x%5E2-3x%2B2, and y=2x%5E3%2B7x%5E2-x-1