Question

Sketch the graph of the function. $$y=-3 x^{2}-x-4$$

Short answer

The graph of the function \(y=-3x^2 - x - 4\) is a downward opening parabola with its vertex at (1/6, -4.1667) and the axis of symmetry at \(x=1/6\).

Step-by-step solution

Determine the vertex of the parabola

The vertex of a parabola given by the equation \(y=ax^2+bx+c\) is at \((-b/2a, y)\), where \(y\) is the value of the function at \(x=-b/2a\). For the function \(y=-3x^2 - x - 4\), we have \(a=-3\), \(b=-1\), and \(c=-4\). Therefore, the x-coordinate of the vertex itself is \(-(-1)/(2*-3) = 1/6\). To find the y-coordinate, we substitute \(x=1/6\) back into the equation to get \(y = -3*(1/6)^2 - (1/6) - 4 = -4.1667\).

Identify the axis of symmetry

The axis of symmetry of a parabola is a vertical line that passes through the vertex. The equation of the line is \(x=h\), where \(h\) is the x-coordinate of the vertex. So for this parabola, the axis of symmetry is \(x=1/6\).

Determine the direction of the parabola

The direction of the parabola depends on the sign of the leading coefficient, which is -3 in this case. Since the leading coefficient is negative, the parabola opens downwards.

Sketch the graph of the function

Having all this information, we can sketch a graph of the function. First, draw the axis of symmetry \(x=1/6\). Then plot the vertex at (1/6, -4.1667). Since the parabola opens downwards, draw the graph accordingly. Remember, the graph is symmetric with respect to the axis of symmetry.