Answer: The vertex form of the given function is [tex]y=5\left(x+\dfrac{3}{2}\right)^2-\dfrac{53}{4}[/tex] and its axis of symmetry is [tex]x=-\dfrac{3}{2}.[/tex]
Step-by-step explanation: The given quadratic function is
[tex]y=5x^2+15x-2~~~~~~~~~~~~~~(i)[/tex]
We are to rewrite the above function in vertex form and to determine its axis of symmetry.
We have from equation (i),
[tex]y=5x^2+15x-2\\\\\Rightarrow y=5(x^2+3x)-2\\\\\Rightarrow y=5\left(x^2+2\times x\times \dfrac{3}{2}+\dfrac{9}{4}\right)-\dfrac{45}{4}-2\\\\\Rightarrow y=5\left(x+\dfrac{3}{2}\right)^2-\dfrac{45+8}{4}\\\\\Rightarrow y=5\left(x+\dfrac{3}{2}\right)^2-\dfrac{53}{4}.[/tex]
So, the given function is a parabola with vertex at the point [tex]\left(-\dfrac{3}{2},-\dfrac{53}{4}\right).[/tex]
Therefore, the axis of symmetry is given by
[tex]x=-\dfrac{3}{2}.[/tex]
Thus, the vertex form of the given function is [tex]y=5\left(x+\dfrac{3}{2}\right)^2-\dfrac{53}{4}[/tex] and its axis of symmetry is [tex]x=-\dfrac{3}{2}.[/tex]
