Determine the equations for the cubic functions given their x-intercepts and a point on each graph.

The graph touches the x-axis at $x=-1$, indicating a double root (multiplicity 2). It crosses the x-axis at $x=3$, indicating a single root.
The factored form is $y = a(x+1)^2(x-3)$.

Substitute $x=2$ and $y=-6$ into the equation:

Expand $(x+1)^2$:

Multiply by $(x-3)$:

Multiply by $\frac{2}{3}$:

Graph (2)

The graph crosses the x-axis at $x=-2$, indicating a single root. It touches the x-axis at $x=3$, indicating a double root.
The factored form is $y = a(x+2)(x-3)^2$.

Substitute $x=0$ and $y=-18$ into the equation:

Expand $(x-3)^2$:

Multiply by $(x+2)$:

Multiply by $-1$:

Graph (3)

The graph crosses the x-axis at $x=-4$, indicating a single root. It touches the x-axis at $x=0$, indicating a double root.
The factored form is $y = a(x+4)(x-0)^2$, which simplifies to $y = ax^2(x+4)$.

Substitute $x=-2$ and $y=4$ into the equation:

Graph (4)

The graph crosses the x-axis at $x=0$, indicating a single root. It touches the x-axis at $x=5$, indicating a double root.
The factored form is $y = a(x-0)(x-5)^2$, which simplifies to $y = ax(x-5)^2$.

Substitute $x=1$ and $y=-4$ into the equation:

Expand $(x-5)^2$:

Multiply by $x$:

Multiply by $-\frac{1}{4}$: