Question

describe the graph of the function at its roots.

(f(x) = (x - 2)^3(x + 6)^2(x + 12))

at (x = 2), the graph crosses / does not intersect / touches the (x)-axis.
at (x = -6), the graph crosses / does not intersect / touches the (x)-axis.
at (x = -12), the graph crosses / does not intersect / touches the (x)-axis.

Explanation:

⚡ Using what you learned: Zeros and Multiplicity

Step 1: Identify roots and multiplicities

The given polynomial function is:
\[ f(x) = (x - 2)^3(x + 6)^2(x + 12) \]

Identify the root and multiplicity for each factor:

• For \( (x - 2)^3 \): root is \( x = 2 \) with multiplicity \( 3 \) (odd).
• For \( (x + 6)^2 \): root is \( x = -6 \) with multiplicity \( 2 \) (even).
• For \( (x + 12) \): root is \( x = -12 \) with multiplicity \( 1 \) (odd).

Step 2: Determine behavior at each root

• Odd multiplicity (multiplicity of \( 1 \) or \( 3 \)): the graph crosses the \( x \)-axis.
• Even multiplicity (multiplicity of \( 2 \)): the graph touches (and turns around at) the \( x \)-axis.

Applying these rules:

• At \( x = 2 \) (multiplicity \( 3 \)): the graph crosses the \( x \)-axis.
• At \( x = -6 \) (multiplicity \( 2 \)): the graph touches the \( x \)-axis.
• At \( x = -12 \) (multiplicity \( 1 \)): the graph crosses the \( x \)-axis.

Answer:

• At \( x = 2 \), the graph crosses the \( x \)-axis.
• At \( x = -6 \), the graph touches the \( x \)-axis.
• At \( x = -12 \), the graph crosses the \( x \)-axis.