The y intercept is easy to find, because it’s the value of y when x=0, so the y intercept=3/2.
We can write y=x²/8+11x/8+3/2 as y=⅛(x²+11x)+3/2, y=⅛(x²+11x+121/4-121/4)+3/2.
So y=⅛((x+11/2)²-121/4)+3/2,
y=⅛(x+11/2)²-121/32+3/2,
y=⅛(x+11/2)²-73/32. When y=0 we get the x intercepts, so ⅛(x+11/2)²=73/32, (x+11/2)²=73/4.
Take square roots of each side: x+11/2=±√73/2 and the x intercepts are -11/2+√73/2=-1.228 and -11/2-√73/2=-9.772 approx. The vertex lies on the axis of symmetry and is found by finding where y has its minimum (smallest) value. We know y=⅛(x+11/2)²-73/32, so y has a minimum value of -73/32 when x=-11/2. The axis of symmetry is a vertical line parallel to the y axis, so x has the same value everywhere on the line, regardless of what y is. The equation of the line of symmetry must therefore be x=-11/2. This line reflects one half of the parabola on to the other.
