Question

Sketch the graph of the inequality. $$y>-4 x^{2}-8 x-4$$

Short answer

The solution to the inequality \(y > -4x^{2} -8x -4 \) is represented by the shaded region above the downward-opening parabola whose vertex is at the point (-2,-4). The line forming the boundary (the parabola itself) is not included in the solution set.

Step-by-step solution

Identify the vertex of the parabola

The equation of the parabola is in the form \(y = ax^{2} + bx + c \) where a = -4, b = -8, and c = -4. The vertex of this parabola is given by \((-b/(2a), f(-b/(2a))\). Substituting the given values, we find the x-value of the vertex to be \(4/2=-2\) with a corresponding y-value of -4.

Determine the direction of opening of the parabola

As the coefficient of \(x^{2}\) is a negative number, the parabola will open downwards. This means the graph of this parabola would look like an upside-down 'U'.

Draw the graph of the equation

The best way to draw this is to plot some points. Besides the vertex, we can pick x-coordinates (for instance, -3 and -1), compute the respective y's and plot these on the graph. Graph the line \(y = -4x^{2} -8x -4 \), which is a dashed, downward-opening parabola with the vertex at \(-2, -4\). The dashed line implies that the points on the line are not part of the solution.

Shade the area to indicate the solution set

The inequality is \(y > -4x^{2} -8x -4 \), not \(y \geq -4x^{2} -8x -4 \). Therefore, we shade the area that does not include the dashed parabola line. Since our inequality sign is >, we shade the region above the parabola, to indicate that the solutions lie in this region.