To convert the equation of a parabola from its vertex form to its standard form, let's start with the given vertex form:
[tex]\[ y = (x - 3)^2 + 36 \][/tex]
The next step is to expand the squared term:
[tex]\[ (x - 3)^2 \][/tex]
We can use the formula for expanding a binomial squared [tex]\((a - b)^2\)[/tex]:
[tex]\[ (x - 3)^2 = x^2 - 6x + 9 \][/tex]
Now, substitute this expanded form back into the original equation:
[tex]\[ y = x^2 - 6x + 9 + 36 \][/tex]
Next, we simplify the equation by combining like terms:
[tex]\[ y = x^2 - 6x + 45 \][/tex]
Hence, the standard form of the equation is:
[tex]\[ y = x^2 - 6x + 45 \][/tex]
From the given options, the correct answer is:
D. [tex]\( y = x^2 - 6x + 45 \)[/tex]
