Answer
See graph and explanations.
Work Step by Step
Step 1. Rewrite the function as $f(x)=-2x^2+6x-3=-2(x^2-3x+\frac{9}{4})-3+\frac{9}{2}=-2(x-\frac{3}{2})^2+\frac{3}{2}$
Step 2. We can find the vertex as $(\frac{3}{2},\frac{3}{2})$
Step 3. The x-intercepts can be found when $f(x)=0$ which gives $x=\frac{3}{2}\pm\frac{\sqrt 3}{2}$
Step 4. The y-intercepts can be found as $f(0)=-3$
Step 5. The axis of symmetry is $x=\frac{3}{2}$
Step 6. The domain is $(-\infty,\infty)$,
Step 7. The range is $(-\infty, \frac{3}{2}]$
Step 8. The function is increasing over $(-\infty, \frac{3}{2})$
Step 9. The function is decreasing. over $(\frac{3}{2},\infty)$
Step 10. Use test points as necessary to graph the function as shown in the figure.
