Evaluate each of the functions below at x = 1, 2, 4, 8, and 16. Plot the graph of each function. Classify each as linear, quadratic, polynomial, exponential, or logarithmic, and explain the reasons for your classifications. Compare how quickly each function increases, based on the evaluations and graphs, and rank the functions from fastest to slowest growing.
f(x) = x3 - 3x2 - 2x + 1
f(x) = ex
f(x) = 3x - 2
f(x) = log x
f(x) = x2 - 5x + 6
I saw your earlier posting of this same question.
I don't know what you expect us to do here.
Just plug in the given values of x for each function using your calculator for the more complicated ones, and create a set of points for each equation.
I would graph each function on the same graph using different colours for each one, that way you can see which one increases more quickly than another.
btw, the choice of x=16, and even x=8, is a poor choice, since the function values are very large and would be unreasonable choices for graphing.
e.g. if x=16 , for your first equation f(16) = 3297, rather silly for graphing.
So if you saw my posting earlier why wouldn't you answer it, I just didn't plug it in because I wanted to for my health. I needed help that what I though you were here for not sarcism. I can't help the problem the professor gives me to work with, you may think they are silly. But I am still required to anwser the question. But thank you for the information that you gave me along with your sarcism.
To evaluate each of the functions at x = 1, 2, 4, 8, and 16, we can substitute the given values into the functions and calculate the corresponding y-values.
1. f(x) = x^3 - 3x^2 - 2x + 1:
f(1) = 1^3 - 3(1)^2 - 2(1) + 1 = -3.
f(2) = 2^3 - 3(2)^2 - 2(2) + 1 = -7.
f(4) = 4^3 - 3(4)^2 - 2(4) + 1 = 9.
f(8) = 8^3 - 3(8)^2 - 2(8) + 1 = 329.
f(16) = 16^3 - 3(16)^2 - 2(16) + 1 = 3545.
2. f(x) = e^x (where e is the base of the natural logarithm, approximately 2.71828...):
f(1) = e^1 ≈ 2.71828.
f(2) = e^2 ≈ 7.38906.
f(4) = e^4 ≈ 54.59815.
f(8) = e^8 ≈ 2980.95799.
f(16) = e^16 ≈ 8886110.52051.
3. f(x) = 3x - 2:
f(1) = 3(1) - 2 = 1.
f(2) = 3(2) - 2 = 4.
f(4) = 3(4) - 2 = 10.
f(8) = 3(8) - 2 = 22.
f(16) = 3(16) - 2 = 46.
4. f(x) = log(x) (base 10 logarithm):
f(1) = log(1) = 0.
f(2) = log(2) ≈ 0.30103.
f(4) = log(4) ≈ 0.60206.
f(8) = log(8) ≈ 0.90309.
f(16) = log(16) ≈ 1.20412.
5. f(x) = x^2 - 5x + 6:
f(1) = 1^2 - 5(1) + 6 = 2.
f(2) = 2^2 - 5(2) + 6 = 0.
f(4) = 4^2 - 5(4) + 6 = 6.
f(8) = 8^2 - 5(8) + 6 = 46.
f(16) = 16^2 - 5(16) + 6 = 222.
Now, let's classify each function based on their graphs and formulas:
1. f(x) = x^3 - 3x^2 - 2x + 1:
This is a polynomial function of degree 3 because the highest power of x in the function is 3.
2. f(x) = e^x:
This is an exponential function because it has the base e raised to the power of x.
3. f(x) = 3x - 2:
This is a linear function because it has a constant rate of change, with the degree of x being 1.
4. f(x) = log(x):
This is a logarithmic function because it represents the logarithm of x to the base 10.
5. f(x) = x^2 - 5x + 6:
This is a polynomial function of degree 2 because the highest power of x in the function is 2.
To compare how quickly each function increases, we can look at the values obtained by evaluating the functions at the given x-values:
For the evaluations:
- The exponential function f(x) = e^x grows the fastest, with the y-values increasing rapidly as x increases.
- The polynomial function f(x) = x^3 - 3x^2 - 2x + 1 grows moderately fast.
- The linear function f(x) = 3x - 2 increases at a constant rate.
- The logarithmic function f(x) = log(x) grows slowly, with the y-values increasing at a decreasing rate.
- The polynomial function f(x) = x^2 - 5x + 6 grows the slowest.
Based on the evaluations and the behavior of their graphs, we can rank the functions in terms of their growth speed from fastest to slowest:
1. Exponential function (f(x) = e^x)
2. Polynomial function (f(x) = x^3 - 3x^2 - 2x + 1)
3. Linear function (f(x) = 3x - 2)
4. Logarithmic function (f(x) = log(x))
5. Polynomial function (f(x) = x^2 - 5x + 6)