Question

(08.07 MC)

A polynomial function is shown below:

f(x) = x3 − 3x2 − 4x + 12

Which graph best represents the function? (5 points)

Answer 1

Graph A (first graph from top to bottom)

Explanation:

Given
`f(x)=x^3-3x^2-4x+12`, since the degree of the polynomial is 3, the function must be odd and will resemble the shown shape in the graphs. The degree of 3 indicates that there are 3 zeroes, whether distinct or non-distinct. Therefore, the graph must intersect the x-axis at these three points.

Factoring the polynomial:

`f(x)=x^3-3x^2-4x+12,\\f(x)=(x+2)(x-2)(x-3),\\\begin{cases}x+2=0, x=-2\\x-2=0, x=2\\x-3=, x=3\end{cases}`

Thus, the three zeroes of this function are
`x=-2, x=2, x=3` and the graph must intersection the x-axis at these points. The y-intercept of any graph occurs when
`x=0`. Thus, the y-coordinate of the y-intercept is equal to
`y=0^3-3(0^2)-4(0)+12,\\y=12` and the y-intercept is (0, 12).

The graph that corresponds with this is graph A.