A function $f$ is defined by $f\left(x\right)=3{\left(x-1\right)}^2-18, x\in R$.

Find the equation of the axis of symmetry of the graph of $f$.

A function $f$ is defined by $f\left(x\right)=3{\left(x-1\right)}^2-18, x\in R$.

State the range of $f$.

A function $f$ is defined by $f\left(x\right)=3{\left(x-1\right)}^2-18, x\in R$.

The graph of $f$ is translated through the vector $\left(\begin{array}{c}2\\ -1\end{array}\right)$ to form a curve representing a new function $g\left(x\right)$. Find $g\left(x\right)$ in the form of $p{x}^2+qx+r$, where $p, q$ and $r$ are constants.

Solve the equation $8x^2+6x-5=0$ by factorization.

Determine the range of values of $k$ for which $8x^2+6x-5=k$ has no real solutions.

Consider the function $f\left(x\right)=x^2-10x+27, x\in R$

Show that function $f$ can be expressed in the form $f\left(x\right)=a{\left(x-h\right)}^2+k$, where $a, h$ and $k$ are constants.

Consider the function $f\left(x\right)=x^2-10x+27, x\in R$

Hence write down the coordinates of the vertex of the graph of $y=f\left(x\right)$.

Consider the function $f\left(x\right)=x^2-10x+27, x\in R$

Hence write down the equation of the line of symmetry of the graph of $y=f\left(x\right)$.

The quadratic curve $y=x^2+bx+c$ intersects the $x$-axis at $\left(10, 0\right)$ and has an equation of a line of symmetry $x=\frac{5}{2}$.

Find the values of $b$ and $c$.