To answer this question, we have that the parent function is x². Then, to finally have the graph above, this parent function needs the next transformations:
1. Translation of 3 units to the right and this is represented by:
[tex]f(x)=x^2\Rightarrow f(x-3)=(x-3)^2[/tex]
And we can see this graphically as follows:
2. We need to reflect the function in the x-axis as follows:
[tex]-f(x)=-(x-3)^2[/tex]
3. Now, we need to translate the function two units downward as follows:
[tex]-f(x-3)-2\Rightarrow g(x)=-(x-3)^2-2[/tex]
In summary, therefore, the equation for the function graphed above is:
[tex]g(x)=-(x-3)^2-2[/tex]
