First let's find the vertex of the equation, using the formula:
[tex]x_v=-\frac{b}{2a}[/tex]
In the equation y = -x^2 - 3, we have that a = -1, b = 0 and c = -3, so:
[tex]x_v=-\frac{0}{-2}=0[/tex]
And the y-coordinate of the vertex is:
[tex]y=-(0)^2-3=0-3=-3[/tex]
So the vertex is (0, -3).
The axis of symmetry is a vertical line that passes through the x-coordinate of the vertex.
So the axis of symmetry is x = 0.
The domain of the function is all values of x that the function can assume.
In the function y = -x^2 - 3, x can assume any value, so the domain is all real numbers.
The range of the function is all values of y that the function can assume.
The expression x^2 can only result in positive numbers, so since there is a minus sign in front of it, the maximum value of y is when we have x = 0:
[tex]y=-0^2-3=-3[/tex]
For any other values of x, the y-value will decrease towards minus infinity. So the range is:
[tex]\mleft\lbrace y\mright|y\le-3\}[/tex]
The parent function is y = x^2, so the first transformation is multiplying the function by -1, that is, the first transformation is a reflection over the x-axis.
Then, we add -3 to the function, and this represents a translation of 3 units down.
Filling the table with some ordered pairs and drawing the equation, we have:
