Question 306779
You can factor to find the zeros.
{{{x^2 - 5x - 24 = 0}}}
{{{(x-8)(x+3)=0}}}
Zeros at x=-3 and x=8.
The vertex is then at {{{x=(8-3)/2=5/2}}}.
We can switch to vertex mode by completing the square.
{{{(x^2-5x+25/4)-24-25/4=0}}}
{{{(x-5/2)^2-96/4-25/4=0}}}
{{{(x-5/2)^2-121/4=0}}}
So the vertex is (5/2,-121/4).
Now you can hand sketch the parabola with all of these points.
{{{ drawing( 300, 300, -10, 10, -40, 10,grid(1),circle(-3,0,.3),circle(8,0,.3),circle(5/2,-121/4,.3),green(line(5/2,-50,5/2,50)), graph( 300, 300, -10, 10, -40, 10, x^2 - 5x - 24)) }}}