Answer
Vertex: $(-4, -1/2)$
Opens upward
x-intercepts: -5, -3
y-intercept: 0
Work Step by Step
$f(x)=1/2*x^2+4x+15/2$
$y=1/2(x^2+8x+15)$
$y+1/2 = 1/2(x^2+8x+15+1)$
$y+1/2 = 1/2 (x^2+8x+16)$
$y+1/2 = 1/2(x+4)^2$
$y+1/2-1/2 = 1/2(x+4)^2-1/2$
$y= 1/2(x+4)^2-1/2$
Vertex: $(-4, -1/2)$
Opens upward
$x=0$
$f(x)=1/2*x^2+4x+15/2$
$f(0)=1/2*0^2+4*0+15/2$
$f(0)=1/2*0+0+15/2$
$f(0)=0+15/2$
$f(0)=15/2$
$y=0$
$y+1/2 = 1/2(x+4)^2$
$0+1/2 = 1/2(x+4)^2$
$1/2 = 1/2(x+4)^2$
$1/2*2 =2* 1/2(x+4)^2$
$1 = (x+4)^2$
$\sqrt 1 = \sqrt {(x+4)^2}$
$±1 = x+4$
$-1=x+4$
$-1-4=x+4-4$
$-5=x$
$1=x+4$
$1-4=x+4-4$
$-3=x$
x-intercepts: -5, -3
y-intercept: 0
