Question

How do you graph `y=-3x^2-4x`?

Answer 1

You'll get a parabola.

Explanation:

This function is a Quadratic (maximum degree of `x` is`=2`) so it will give you a PARABOLA.
Observing the coefficient of `x^2` you see that is `-3` which is `<0` so yours will be a downward parabola.
Now let us try to find interesting points of your parabola that will help us to plot it:

1] y-intercept:
set `x=0`; you get: `y=0`

2] x-intercept:
set `y=0`; you get `-3x^2-4x=0` solving it you find:
`-x(3x+4)=0`
so you have two solutions:
`x_1=0`
`x_2=-4/3`
so the x- intercepts will be:
`x=0, y=0`
`x=-4/3, y=0`

3] Vertex: this point represents the highest point reached by your parabola. the function can be written as `y=ax^2+bx+c`
where in your case:
`a=-3`
`b=-4`
`c=0`
The `x` coordinate of the vertex can then be found considering that `color(red)(x_v=-b/(2a))=4/(-6)=-4/6`;
The `y` coordinate of the vertex can then be found considering that `color(red)(y_v=-(Delta)/(4a))=-(b^2-4ac)/(4a)=(-16)/(-12)=4/3`

Graphically:
graph{-3x^2-4x [-11.25, 11.25, -5.625, 5.625]}