First, let's determine the vertex of the parabola, we can do it by using the expression that gives us the coordinate x and y of the vertex, for the x-coordinate we have
[tex]x_V=-\frac{b}{2a}[/tex]
And the y-coordinate
[tex]y_V=-\frac{\Delta}{4a}[/tex]
And the vertex is
[tex]V=(x_V,y_V)[/tex]
Applying the formula
[tex]\begin{gathered} x_V=\frac{-12}{2\cdot(-2)}=3 \\ \\ y_V=\frac{144-4\cdot(-2)\cdot(-19)}{4\cdot(-2)}=\frac{-8}{8}=-1 \end{gathered}[/tex]
The vertex is
[tex]V=(3,-1)[/tex]
Let's plot two points to the right of the vertex, which means x > 3, I'll pick x = 4 and x = 5
[tex]\begin{gathered} f(x)=-2x^2+12x-19 \\ \\ f(4)=-2\cdot4^2+12\cdot4-19=-3 \\ \\ f(5)=-2\cdot5^2+12\cdot5-19=-9 \end{gathered}[/tex]
Then two points at the right of the vertex are
[tex]\begin{gathered} (4,-3) \\ \\ (5.-9) \end{gathered}[/tex]
And at the left, I'll pick 0 and 1
[tex]\begin{gathered} f(0)=-19 \\ \\ f(1)=-9 \end{gathered}[/tex]
Therefore the points are
[tex]\begin{gathered} (0,-19) \\ \\ (1,-9) \end{gathered}[/tex]
Final answer:
[tex]\begin{gathered} \text{ vertex:} \\ V=(3,-1) \\ \\ \text{ right points:} \\ (4,-3) \\ \\ (5,-9) \\ \\ \text{ left points:} \\ (0,-19) \\ \\ (1,-9) \end{gathered}[/tex]
