Answer
$f(x)=-3(x+2)^2-5$
Work Step by Step
Completing the square:
$f(x)=-3x^2-12x-17$
$f(x)=-3(x^2+4x)-17$
$f(x)=-3(x^2+4x+(\frac{4}{2})^2)+3\cdot(\frac{4}{2})^2-17$
$f(x)=-3(x^2+4x+2^2)+3\cdot2^2-17$
$f(x)=-3(x+2)^2+3\cdot4-17$
$f(x)=-3(x+2)^2+12-17$
$f(x)=-3(x+2)^2-5$
Graphing the function:
1st step (black graph): The parent function: $y=x^2$
2nd step (green graph): Reflect over the x-axis. $y=-x^2$
3rd step (orange graph): Stretch by a factor of 3. $y=-3x^2$
4th step (red graph): Shift 5 units down. $y=-3x^2-5$
Final step (blue graph): Shift left by 2 units. $y=-3(x+2)^2-5$
