Answer:
[tex]y-1/2=2(x+7/2)^2[/tex]
Step-by-step explanation:
Equation of the Quadratic Function
The vertex form of the quadratic function has the following equation:
[tex]y-k=a(x-h)^2[/tex]
Where (h, k) is the vertex of the parabola that results when plotting the function, and a is a coefficient different from zero.
The given function is:
[tex]y=(x+3)^2+(x+4)^2[/tex]
Expanding the squares:
[tex]y=x^2+6x+9+x^2+8x+16[/tex]
Simplifying:
[tex]y=2x^2+14x+25[/tex]
Factoring 2 from the first two terms:
[tex]y=2(x^2+7x)+25[/tex]
The expression in parentheses must be completed to form the square of a binomial:
[tex]y=2(x^2+7x+49/4-49/4)+25[/tex]
Isolating the square of the binomial:
[tex]y=2(x^2+7x+49/4)-49/2+25[/tex]
Operating and Factoring:
[tex]y=2(x+7/2)^2+1/2[/tex]
Rearranging:
[tex]\boxed{y-1/2=2(x+7/2)^2}[/tex]
Comparing with the vertex form:
a=2, h=-7/2, k=1/2
