Answer
See below:
Work Step by Step
(a)
General equation of a parabola on \[y-\text{axis}\]is \[y=a{{x}^{2}}+bx+c\].
Now, for the parabola \[{{x}^{2}}+8x+7\], coefficient of \[{{x}^{2}}\]is 1.
As, \[a>0\], thus the parabola opens upwards.
(b)
Compare the given equation of parabola\[y={{x}^{2}}+8x+7\]with the general equation and obtain the values of \[a,b\]and\[c\]as,
\[\begin{align}
& a=1 \\
& b=8 \\
& c=7
\end{align}\]
Now,\[x\]coordinate of the vertex is obtained as,
\[\begin{align}
& x=-\frac{b}{2a} \\
& =-\frac{8}{2\times 1} \\
& =-\frac{8}{2} \\
& =-4
\end{align}\]
Now, substitute above \[x\]coordinate in the equation of the parabola to get the\[y\]coordinate of the vertex as,
\[\begin{align}
& y={{\left( -4 \right)}^{2}}+8\left( -4 \right)+7 \\
& y=16-32+7 \\
& y=-9
\end{align}\]
Thus, vertex is\[\left( -4,-9 \right)\].
(c)
Substitute\[y=0\]in the parabola equation and solve for\[x\]as,
\[\begin{align}
& {{x}^{2}}+8x+7=0 \\
& {{x}^{2}}+x+7x+7=0 \\
& x\left( x+1 \right)+7\left( x+1 \right)=0 \\
& \left( x+1 \right)\left( x+7 \right)=0
\end{align}\]
Thus, the given parabola cuts the \[x-\text{axis}\]at two points, that is, \[x=-1\]and \[x=-7\].
(d)
Substitute\[x=0\]in the parabola equation and solve for\[y\]as,
\[\begin{align}
& y={{x}^{2}}+8x+7 \\
& y=0+0+7 \\
& y=7
\end{align}\]
Thus, the given parabola cuts the\[y\]axis at\[y=7\].
