Questions: Graph f(x) = x^3 - 3x^2 - 4x + 12. Label each intercept.

Solution Steps

To find the y-intercept, set $x = 0$ in the function $f(x) = x^3 - 3x^2 - 4x + 12$.

$$f(0) = 0^3 - 3(0)^2 - 4(0) + 12 = 12$$

So, the y-intercept is $(0, 12)$.

To find the x-intercepts, set $f(x) = 0$ and solve for $x$.

$$x^3 - 3x^2 - 4x + 12 = 0$$

We can use the Rational Root Theorem to test possible rational roots. Testing $x = 2$:

$$2^3 - 3(2)^2 - 4(2) + 12 = 8 - 12 - 8 + 12 = 0$$

So, $x = 2$ is a root. We can factor $x - 2$ out of the polynomial using synthetic division or polynomial division.

Using synthetic division with $x = 2$:

$$\begin{array}{r|rrrr} 2 & 1 & -3 & -4 & 12 \\ & & 2 & -2 & -12 \\ \hline & 1 & -1 & -6 & 0 \\ \end{array}$$

The quotient is $x^2 - x - 6$. We can factor this quadratic:

$$x^2 - x - 6 = (x - 3)(x + 2)$$

So, the polynomial factors as:

$$(x - 2)(x - 3)(x + 2) = 0$$

The x-intercepts are $x = 2$, $x = 3$, and $x = -2$.

Plot the intercepts on the graph:

• y-intercept: $(0, 12)$
• x-intercepts: $(2, 0)$, $(3, 0)$, and $(-2, 0)$

Final Answer