1. **State the problem:** We need to sketch the graph of the cubic function $$y = (x-3)(x+1)(x+4)$$. 2. **Formula and important rules:** This is a cubic polynomial in factored form. The roots (x-intercepts) are the values of $x$ that make each factor zero: $x=3$, $x=-1$, and $x=-4$. 3. **Find the roots:** - Set each factor equal to zero: $$x-3=0 \Rightarrow x=3$$ $$x+1=0 \Rightarrow x=-1$$ $$x+4=0 \Rightarrow x=-4$$ 4. **Determine the end behavior:** - The leading term of the expanded polynomial will be $x^3$ (since multiplying $x \cdot x \cdot x$). - For large positive $x$, $y \to +\infty$. - For large negative $x$, $y \to -\infty$. 5. **Find the y-intercept:** - Set $x=0$: $$y = (0-3)(0+1)(0+4) = (-3)(1)(4) = -12$$ 6. **Sketch key points:** - Roots at $x=-4, -1, 3$ (where $y=0$). - Y-intercept at $(0, -12)$. 7. **Plot and connect:** - The graph crosses the x-axis at the roots. - Since all roots are simple (multiplicity 1), the graph crosses the x-axis at each root. - The graph goes from $-\infty$ at left, crosses at $-4$, goes up and down crossing at $-1$, then crosses at $3$ and goes to $+\infty$. **Final answer:** The graph of $$y = (x-3)(x+1)(x+4)$$ is a cubic curve with roots at $-4$, $-1$, and $3$, y-intercept at $-12$, and end behavior going down to the left and up to the right.