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

Transcript text: 3. Graph $f(x)=x^{3}-3 x^{2}-4 x+12$. Label each intercept.

Solution

Step 1: Identify the y-intercept

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) $.

Step 2: Find the x-intercepts

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 $.

Step 3: Graph the function and label intercepts

Plot the intercepts on the graph:

The intercepts of the function $ f(x) = x^3 - 3x^2 - 4x + 12 $ are:

Graph the function with these intercepts labeled.

Was this solution helpful?

Unhelpful

Helpful