Answer :
To determine the graph of the function [tex]\(f(x) = x^4 + x^3 - 8x^2 - 12x\)[/tex], we can follow these detailed steps to understand its key features, including roots, behavior at critical points, and end behavior.
### Step 1: Find the roots of the function
To find the roots, we set [tex]\(f(x) = 0\)[/tex]:
[tex]\[x^4 + x^3 - 8x^2 - 12x = 0\][/tex]
Factor out the common factor [tex]\(x\)[/tex]:
[tex]\[x(x^3 + x^2 - 8x - 12) = 0\][/tex]
So, one root is [tex]\(x = 0\)[/tex].
Now, we need to factor the cubic polynomial [tex]\(x^3 + x^2 - 8x - 12\)[/tex]. We can use the Rational Root Theorem to test potential rational roots. After testing viable candidates, suppose we find a root [tex]\(x = 2\)[/tex]:
Then, using synthetic division to factor out [tex]\(x - 2\)[/tex]:
[tex]\[ \begin{array}{r|rrrr} 2 & 1 & 1 & -8 & -12 \\ & & 2 & 6 & -4 \\ \hline & 1 & 3 & -2 & -16 \\ \end{array} \][/tex]
That leaves us with the quadratic polynomial:
[tex]\[x^3 + x^2 - 8x - 12 = (x - 2)(x^2 + 3x - 6)\][/tex]
Further factoring of [tex]\(x^2 + 3x - 6 = 0\)[/tex] via the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex]:
[tex]\[ x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-6)}}{2(1)} = \frac{-3 \pm \sqrt{9 + 24}}{2} = \frac{-3 \pm \sqrt{33}}{2}\][/tex]
This gives us two more roots:
[tex]\(x = \frac{-3 + \sqrt{33}}{2}\)[/tex] and [tex]\(x = \frac{-3 - \sqrt{33}}{2}\)[/tex].
Thus, the roots of [tex]\(f(x)\)[/tex] are:
[tex]\[ x = 0, 2, \frac{-3 + \sqrt{33}}{2}, \frac{-3 - \sqrt{33}}{2} \][/tex]
### Step 2: Determine the behavior near the roots and critical points
To understand the shape of the graph, consider the first and second derivatives of [tex]\(f(x)\)[/tex].
First derivative:
[tex]\[f'(x) = 4x^3 + 3x^2 - 16x - 12\][/tex]
Second derivative:
[tex]\[f''(x) = 12x^2 + 6x - 16\][/tex]
By analyzing [tex]\(f'(x) = 0\)[/tex] and [tex]\(f''(x) = 0\)[/tex], we identify critical points and concavity.
### Step 3: Determine the end behavior of the function
Consider [tex]\(f(x)\)[/tex] as [tex]\(x \to \pm \infty\)[/tex]:
For [tex]\(f(x) = x^4 + x^3 - 8x^2 - 12x\)[/tex], the leading term is [tex]\(x^4\)[/tex]. Since the coefficient of [tex]\(x^4\)[/tex] is positive, as [tex]\(x \to \infty\)[/tex] or [tex]\(x \to -\infty\)[/tex], [tex]\(f(x) \to \infty\)[/tex].
### Step 4: Construct a rough sketch of the graph
- [tex]\(f(x)\)[/tex] has degree 4 (quartic), typically has up to 4 roots.
- End behavior: [tex]\( f(x) \to \infty\)[/tex] as [tex]\( x \to \infty\)[/tex] and [tex]\( f(x) \to \infty \)[/tex] as [tex]\( x \to -\infty\)[/tex].
- Roots: [tex]\( 0, 2, \frac{-3 + \sqrt{33}}{2}, \frac{-3 - \sqrt{33}}{2}\)[/tex].
- Test other points or analyze concavity to refine the graph sketch accurately.
In the context of visual graphs to choose from, look for:
- A quartic curve turning back up at the ends.
- Intersecting [tex]\(x\)[/tex]-axis at identified roots.
Thus, the correct choice of graph will exhibit these characteristics.
### Step 1: Find the roots of the function
To find the roots, we set [tex]\(f(x) = 0\)[/tex]:
[tex]\[x^4 + x^3 - 8x^2 - 12x = 0\][/tex]
Factor out the common factor [tex]\(x\)[/tex]:
[tex]\[x(x^3 + x^2 - 8x - 12) = 0\][/tex]
So, one root is [tex]\(x = 0\)[/tex].
Now, we need to factor the cubic polynomial [tex]\(x^3 + x^2 - 8x - 12\)[/tex]. We can use the Rational Root Theorem to test potential rational roots. After testing viable candidates, suppose we find a root [tex]\(x = 2\)[/tex]:
Then, using synthetic division to factor out [tex]\(x - 2\)[/tex]:
[tex]\[ \begin{array}{r|rrrr} 2 & 1 & 1 & -8 & -12 \\ & & 2 & 6 & -4 \\ \hline & 1 & 3 & -2 & -16 \\ \end{array} \][/tex]
That leaves us with the quadratic polynomial:
[tex]\[x^3 + x^2 - 8x - 12 = (x - 2)(x^2 + 3x - 6)\][/tex]
Further factoring of [tex]\(x^2 + 3x - 6 = 0\)[/tex] via the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex]:
[tex]\[ x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-6)}}{2(1)} = \frac{-3 \pm \sqrt{9 + 24}}{2} = \frac{-3 \pm \sqrt{33}}{2}\][/tex]
This gives us two more roots:
[tex]\(x = \frac{-3 + \sqrt{33}}{2}\)[/tex] and [tex]\(x = \frac{-3 - \sqrt{33}}{2}\)[/tex].
Thus, the roots of [tex]\(f(x)\)[/tex] are:
[tex]\[ x = 0, 2, \frac{-3 + \sqrt{33}}{2}, \frac{-3 - \sqrt{33}}{2} \][/tex]
### Step 2: Determine the behavior near the roots and critical points
To understand the shape of the graph, consider the first and second derivatives of [tex]\(f(x)\)[/tex].
First derivative:
[tex]\[f'(x) = 4x^3 + 3x^2 - 16x - 12\][/tex]
Second derivative:
[tex]\[f''(x) = 12x^2 + 6x - 16\][/tex]
By analyzing [tex]\(f'(x) = 0\)[/tex] and [tex]\(f''(x) = 0\)[/tex], we identify critical points and concavity.
### Step 3: Determine the end behavior of the function
Consider [tex]\(f(x)\)[/tex] as [tex]\(x \to \pm \infty\)[/tex]:
For [tex]\(f(x) = x^4 + x^3 - 8x^2 - 12x\)[/tex], the leading term is [tex]\(x^4\)[/tex]. Since the coefficient of [tex]\(x^4\)[/tex] is positive, as [tex]\(x \to \infty\)[/tex] or [tex]\(x \to -\infty\)[/tex], [tex]\(f(x) \to \infty\)[/tex].
### Step 4: Construct a rough sketch of the graph
- [tex]\(f(x)\)[/tex] has degree 4 (quartic), typically has up to 4 roots.
- End behavior: [tex]\( f(x) \to \infty\)[/tex] as [tex]\( x \to \infty\)[/tex] and [tex]\( f(x) \to \infty \)[/tex] as [tex]\( x \to -\infty\)[/tex].
- Roots: [tex]\( 0, 2, \frac{-3 + \sqrt{33}}{2}, \frac{-3 - \sqrt{33}}{2}\)[/tex].
- Test other points or analyze concavity to refine the graph sketch accurately.
In the context of visual graphs to choose from, look for:
- A quartic curve turning back up at the ends.
- Intersecting [tex]\(x\)[/tex]-axis at identified roots.
Thus, the correct choice of graph will exhibit these characteristics.
