Graph f(x)=x(x-2)^3(x+2)^2

Problem

ƒ(x)=x*(x−2)3*(x+2)2

Solution

1. Identify the zeros and their multiplicities by setting each factor to zero.

x=0* (multiplicity 1)

x−2=0⇒x=2* (multiplicity 3)

x+2=0⇒x=−2* (multiplicity 2)

2. Determine the behavior at each x-intercept based on the multiplicity.

At *x=0*, the graph crosses the x-axis linearly.

At *x=2*, the graph crosses the x-axis with a flattening effect (inflection point).

At *x=−2*, the graph touches the x-axis and turns around (local extremum).

3. Determine the end behavior by finding the leading term of the polynomial.

Degree=1+3+2=6

Leading coefficient=1

As *x→∞,ƒ(x)→∞

As *x→−∞,ƒ(x)→∞

4. Find the y-intercept by evaluating the function at x=0

ƒ(0)=0*(0−2)3*(0+2)2=0

5. Sketch the graph by starting from the upper left, touching the x-axis at x=−2 crossing at x=0 crossing with a flat shape at x=2 and heading toward positive infinity.

Final Answer

ƒ(x)=x*(x−2)3*(x+2)2

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