What is the end behavior of the function
f(x)=-5x^(3)+2x^(2)-4x+6
down to the left, up to the right
up to the left, down to the right
up to the left, up to the right
down to the left, down to the right
The end behavior of a function refers to the behavior of the function as x approaches positive or negative infinity. To determine the end behavior of the given function f(x) = -5x^3 + 2x^2 - 4x + 6, we look at the leading term, which is -5x^3.
As x approaches negative infinity, -5x^3 approaches negative infinity as well. Therefore, the function goes down to the left.
As x approaches positive infinity, -5x^3 approaches negative infinity as well since the coefficient is negative. Therefore, the function goes down to the right.
Thus, the end behavior of the function is down to the left, down to the right.
To determine the end behavior of a function, we need to look at the leading term, which is the term with the highest exponent. In this case, the leading term is -5x^3.
When the leading term has an odd exponent like this, the end behavior will be opposite as we go to the left and to the right.
So, the end behavior of the function f(x) = -5x^3 + 2x^2 - 4x + 6 is:
- Down to the left (as x approaches negative infinity)
- Up to the right (as x approaches positive infinity)
Therefore, the correct answer is down to the left, up to the right.
To determine the end behavior of a function, we examine the behavior of the function as x approaches positive infinity and as x approaches negative infinity.
For the given function f(x) = -5x^3 + 2x^2 - 4x + 6, we can analyze the leading term and its coefficient (-5x^3) to determine how the function behaves as x goes towards positive and negative infinity.
The leading term of the function is -5x^3. Since the exponent is odd (3), as x approaches positive infinity, the leading term will also approach negative infinity. This means that the function will be heading down to the left as x increases.
Similarly, as x approaches negative infinity, the leading term will still approach negative infinity. So, again, the function will be heading down to the left as x goes towards negative infinity.
Therefore, the end behavior of the function f(x)=-5x^(3)+2x^(2)-4x+6 is down to the left, down to the right.
Find the zeroes of the polynomial equation . 2x^3+6x^2-x-3=0
To find the zeroes of the polynomial equation 2x^3 + 6x^2 - x - 3 = 0, we can use various methods such as factoring, the rational root theorem, or synthetic division.
One approach is to use synthetic division to test possible rational roots. Possible rational roots can be determined using the rational root theorem, which states that the rational roots of a polynomial equation with integer coefficients are of the form p/q, where p is a factor of the constant term (-3) and q is a factor of the leading coefficient (2).
The factors of -3 are ±1 and ±3, and the factors of 2 are ±1 and ±2. So, the possible rational roots are ±1, ±3, ±1/2, and ±3/2.
By synthetic division, we can test these possible roots to find any that satisfy the equation:
Testing x = 1:
1 | 2 6 -1 -3
| 2 8 7
___________
2 8 7 4
The remainder is 4, so x = 1 is not a root.
Testing x = -1:
-1 | 2 6 -1 -3
| -2 -4 5
___________
2 4 -5 2
The remainder is 2, so x = -1 is not a root.
Testing x = 3:
3 | 2 6 -1 -3
| 6 36 105
___________
2 12 35 102
The remainder is 102, so x = 3 is not a root.
Testing x = -3:
-3 | 2 6 -1 -3
| -6 0 3
___________
2 0 -1 0
The remainder is 0, so x = -3 is a root.
So far, we have found one root, x = -3.
To find the remaining roots, we can divide the polynomial equation by (x + 3) using long division or synthetic division.
By dividing 2x^3 + 6x^2 - x - 3 by (x + 3), we have:
2x^2 - 1
Now, we can solve the equation 2x^2 - 1 = 0 for x:
2x^2 - 1 = 0
2x^2 = 1
x^2 = 1/2
x = ±√(1/2)
x = ±(1/√2)
x = ±(1/√2) * (√2/√2)
x = ±√(2)/2
Therefore, the zeroes of the polynomial equation 2x^3 + 6x^2 - x - 3 = 0 are -3, √(2)/2, and -√(2)/2.
Describe the end of the function
f(x)=-2x^4-x^3+3
A. f(x)-> ∞ as x -> -∞ ; f(x) -> ∞ as x -> ∞
B. f(x) -> -∞ as x -> -∞ ; f(x) -> -∞ as x -> ∞
C. f(x) -> ∞ as x -> -∞ ; f(x) -> -∞ as x -> ∞
D. f(x) -> -∞ as x -> -∞ ; f(x) -> ∞ as x -> ∞
To determine the end behavior of the function f(x) = -2x^4 - x^3 + 3, we need to examine the leading term, which is -2x^4.
As x approaches negative infinity, -2x^4 approaches positive infinity since the coefficient is negative. Therefore, f(x) goes to positive infinity as x goes to negative infinity.
As x approaches positive infinity, -2x^4 also approaches positive infinity. Therefore, f(x) goes to positive infinity as x goes to positive infinity.
Therefore, the correct answer is A. f(x) -> ∞ as x -> -∞ ; f(x) -> ∞ as x -> ∞