Graph f(x)=-2x^3(x-3)^2(x+5)

Problem

ƒ(x)=−2*x3*(x−3)2*(x+5)

Solution

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

x=0* (multiplicity 3)

x=3* (multiplicity 2)

x=−5* (multiplicity 1)

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

At *x=0*, the graph crosses the x-axis with an inflection point (cubic behavior).

At *x=3*, the graph touches the x-axis and turns around (tangent behavior).

At *x=−5*, the graph crosses the x-axis linearly.

3. Determine the degree of the polynomial by summing the multiplicities of the factors.

Degree=3+2+1=6

4. Determine the end behavior using the leading coefficient and the degree.

Leading coefficient=−2

Since the degree is even and the leading coefficient is negative, *ƒ(x)→−∞* as *x→±∞.

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

ƒ(0)=−2*(0)3*(0−3)2*(0+5)=0

6. Sketch the graph by starting from the bottom left, crossing at x=−5 flattening and crossing at x=0 touching and turning at x=3 and heading back down to negative infinity.

Final Answer

ƒ(x)=−2*x3*(x−3)2*(x+5)

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