Answer
See graph and explanations.
Work Step by Step
Step 1. Given $f(x)=x^3-5x^2+3x+9$, we can find its end behavior as fall to the left and rise to the right.
Step 2. List possible rational zeros as $\pm1,\pm3,\pm9$
Step 3. Use synthetic division as shown, we have $f(x)=(x+1)(x^2-6x+9)=(x+1)(x-3)^2$
Step 4. The zeros are $x=-1$ (multiplicity 1, graph crosses the x-axis) and $x=3$ (multiplicity 2, graph touches the x-axis and turns around)
Step 5. The y-intercept can be found as $f(0)=9$
Step 6. Use test points as necessary, graph the function as shown in the figure.