Which statements correctly describes the graph of the function f(x)=6x^2-x^5-8+3x^3

Question

•The degree of f is odd

•f(x) approaches ♾ as x approaches -♾

•f has a zero at x=1

•The leading coefficient of f is positive

•f(x) approaches -♾ as x approaches ♾

•f has a zero at x=2

Answer 1

•The degree of f is odd TRUE

The highest exponent on the variable is 5, which is odd, so the function is odd. This means that the function values tend towards opposite signs of ♾ as x goes to opposite signs of ♾.

•f(x) approaches ♾ as x approaches -♾ TRUE

An odd function which has a negative leading coefficient (the coefficient on the highest power of the variable, -1 in this case, on x⁵) approaches +♾ as x approaches -♾, and the function approaches -♾ as x approaches +♾.

•f has a zero at x=1 TRUE

If f has a zero at this value of x, the function will evaluate to be zero when x is 1. Simply substitute, and evaluate:

f(1) = 6(1)² - (1)⁵ - 8 + 3(1)³ = 6 - 1 - 8 + 3 = 0

•The leading coefficient of f is positive FALSE

We already determined the leading coefficient (-1), and it is not positive.

•f(x) approaches -♾ as x approaches ♾ TRUE

See the explanation above, in the second option.

•f has a zero at x=2 FALSE

See the explanation above, in the third option. At x=2, the function evaluates to:

f(2) = 6(2)² - (2)⁵ - 8 + 3(2)³ = 6*4 - 32 - 8 + 3*8 = 24 - 32 - 8 + 24 = 48 - 40 = 8 ≠ 0