which statements correctly describe the graph of the function (f(x)=x^{3}-2x^{2}-19x + 20)? select three…

which statements correctly describe the graph of the function (f(x)=x^{3}-2x^{2}-19x + 20)? select three options.\nas the x - values increase, the y - values always increase.\nas x approaches negative infinity, y approaches negative infinity.\nthe domain of the function is all real numbers.\nthe range of the function is (ygeq20).\nthe graph has a positive y - intercept.

which statements correctly describe the graph of the function (f(x)=x^{3}-2x^{2}-19x + 20)? select three options.\nas the x - values increase, the y - values always increase.\nas x approaches negative infinity, y approaches negative infinity.\nthe domain of the function is all real numbers.\nthe range of the function is (ygeq20).\nthe graph has a positive y - intercept.

Answer

Explanation:

Step1: Analyze end - behavior

For a cubic function (y = ax^{3}+bx^{2}+cx + d) (here (a = 1), (b=-2), (c = - 19), (d = 20)), when (a>0), as (x\to-\infty), (y\to-\infty) and as (x\to+\infty), (y\to+\infty). So, as (x) approaches negative infinity, (y) approaches negative infinity.

Step2: Determine domain

The domain of a polynomial function (f(x)=x^{3}-2x^{2}-19x + 20) is all real numbers since we can substitute any real - number value for (x) into the function.

Step3: Find y - intercept

To find the (y) - intercept, set (x = 0). Then (f(0)=0^{3}-2\times0^{2}-19\times0 + 20=20>0), so the graph has a positive (y) - intercept.

Step4: Analyze increasing/decreasing

The derivative of (f(x)) is (f^\prime(x)=3x^{2}-4x - 19). The function is not always increasing as (x) increases because (f^\prime(x)) is a quadratic function and changes sign.

Step5: Analyze range

Since it is a cubic function with a positive leading coefficient, the range is all real numbers, not (y\geq20).

Answer:

As (x) approaches negative infinity, (y) approaches negative infinity; The domain of the function is all real numbers; The graph has a positive (y) - intercept.