1. which of the following is a fourth degree polynomial function? select all that apply.
a. f(x)= 4x^3 - x^2 + 2x - 7
b. f(x)= 5-x^4
c. f(x)= 1 / 2x^4 + x^2 -5
d. f(x)= 3x^4 + 2x^3 -4x +1
2. which function below has the end behavior f(x) approaches neg. infinity as x approaches infinity and f(x) approaches infinity as x approaches neg. infinity?
a. f(x)= 3x^5 - 2x^2 + 5x -1
b. f(x)= 5x^2 - 7 +6
c. f(x)= -x^3 -4x^2 +x
d. f(x)= -3x^4 + 2x^3 +6x - 10
3. the graph of a polynomial function has 3 turning points. what do you know about the degree and the extrema of this polynomial?
a. the polynomial is degree 3 and has 3 extrema
b. the polynomial is at least degree 3 and has 4 extrema
c. the polynomial is degree 4 and has 4 extrema
d. the polynomial is at least degree 4 and has 3 extrema
4. what is the maximum for the graphed function on the interval [-1,3]?
(idk if the graph is important but it starts out increasing from the left, theres an arrow at the end so i guess that means its infinite, crosses the x axis at (-1,0), there's a turning point at (1,3.5) and it starts decreasing, another turning point at (3,1.5) it starts increasing again, another turning point at (5,4) then decreases and keeps going past the graph.)
a. 1.5
b. 2
c. 3.5
d. 4
my answers for 1 are A and D, my answer for 2 is A, my answer for 3 is C, and my answer for 4 is C. are these right? i'm really stumped.
#1. is B and D -- 4 is highest exponent
#2. C (A is just the opposite)
#3. D
#4. C is correct
Okay, they're right for the practice, I've got the qc below:
1) C. The function is a polynomial. The degree is 4 and the leading coefficient is -3
2) B. f (x) -> -∞ as x -> -∞; f (x) -> -∞ as x -> ∞
(It's negative, negative, negative, positive)
3) C. 5
4) A. (-2, -2) is a local minimum
C. (4, 4) is the global maximum
thank you sm dude
oobleck is correct
1. Well, let's take a closer look at the options:
a. f(x)= 4x^3 - x^2 + 2x - 7: This is a third degree polynomial, not fourth degree.
b. f(x)= 5-x^4: Now we're talking! This is indeed a fourth degree polynomial.
c. f(x)= 1 / 2x^4 + x^2 -5: Sorry, but this one is not a polynomial at all. It involves division, so it's not a polynomial function.
d. f(x)= 3x^4 + 2x^3 -4x +1: Well, look at that! Another fourth degree polynomial.
So, the correct answers are B and D.
2. Let's find which function satisfies the given end behavior:
a. f(x)= 3x^5 - 2x^2 + 5x -1: Sorry, but as x approaches infinity, this one approaches infinity, not negative infinity.
b. f(x)= 5x^2 - 7 +6: Nope, this one approaches positive infinity as x approaches positive or negative infinity.
c. f(x)= -x^3 -4x^2 +x: Sorry, but as x approaches infinity, this one approaches negative infinity. However, as x approaches negative infinity, it approaches positive infinity. So, close but no cigar.
d. f(x)= -3x^4 + 2x^3 +6x - 10: Ding, ding, ding! This is the one that has the desired end behavior.
So, the correct answer is D.
3. Turning points, huh? That tells us something interesting. Let's see:
a. the polynomial is degree 3 and has 3 extrema: Sorry, but if a polynomial has 3 turning points, that means it must be at least degree 4.
b. the polynomial is at least degree 3 and has 4 extrema: Nope, not quite. The number of extrema is always one less than the degree of the polynomial.
c. the polynomial is degree 4 and has 4 extrema: Well, well, well! This is the correct answer we're looking for.
d. the polynomial is at least degree 4 and has 3 extrema: Sorry, but we need at least 4 turning points for a polynomial to be at least degree 4.
So, the correct answer is C.
4. Ah, the maximum! Let's analyze the graph and find out:
Based on the information given, we know that the graph has a maximum at (1,3.5). Since the interval we're interested in is [-1,3], the maximum occurs at x = 1.
So, the correct answer is 3.5.
Hope that helps clarify things for you!
To determine the degree of a polynomial function, we look at the highest power of the variable in the function.
1. Let's analyze each option:
a. f(x) = 4x^3 - x^2 + 2x - 7: This is a third-degree polynomial function.
b. f(x) = 5 - x^4: This is a fourth-degree polynomial function.
c. f(x) = 1 / (2x^4 + x^2 - 5): This is not a polynomial function because the exponent on the variable is -1.
d. f(x) = 3x^4 + 2x^3 - 4x + 1: This is a fourth-degree polynomial function.
Thus, options B and D are fourth-degree polynomial functions.
2. To determine the end behavior of a polynomial function, we examine the leading term, which is the term with the highest power of the variable.
a. f(x) = 3x^5 - 2x^2 + 5x - 1: The leading term is 3x^5, which means that as x approaches infinity, f(x) approaches positive infinity. However, as x approaches negative infinity, f(x) approaches negative infinity. So, this does not match the given end behavior.
b. f(x) = 5x^2 - 7x + 6: The leading term is 5x^2, which means that as x approaches infinity, f(x) approaches positive infinity. Similarly, as x approaches negative infinity, f(x) also approaches positive infinity. Therefore, this function satisfies the given end behavior.
c. f(x) = -x^3 - 4x^2 + x: The leading term is -x^3, which means that as x approaches infinity, f(x) approaches negative infinity. As x approaches negative infinity, f(x) approaches positive infinity. Hence, this function does not match the specified end behavior.
d. f(x) = -3x^4 + 2x^3 + 6x - 10: The leading term is -3x^4, which means that as x approaches infinity, f(x) approaches negative infinity. As x approaches negative infinity, f(x) also approaches negative infinity. Thus, this function does not satisfy the given end behavior.
From the above analysis, option B (f(x) = 5x^2 - 7x + 6) is the correct choice.
3. Regarding the number of turning points of a polynomial function, it is equal to the degree of the polynomial minus one. Therefore, a polynomial function with three turning points must have a degree of at least 4.
Hence, the correct answer is option D (the polynomial is at least degree 4 and has 3 extrema).
4. To find the maximum value of the function on the given interval, we need to examine the turning points within that interval. In this case, the graphed function has a turning point at (3, 1.5). Since the value of f(x) is 1.5 at x = 3, it represents the maximum value on the interval [-1,3].
Therefore, the correct answer is option C (3.5).
In summary, the correct answers are:
1. B and D
2. B
3. D
4. C