Problem 29 - Algebra 1

Question

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

Step-by-Step Solution

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 1: 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.

Step 2: 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'.

Step 3: 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.

Step 4: 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.

Key Concepts

Parabola

Parabolas are a common geometric shape in mathematics, especially when dealing with quadratic functions. A parabola is essentially the set of all points equidistant from a single point (called the focus) and a line (called the directrix). When you're dealing with quadratic equations, the parabola's orientation—whether it opens upwards or downwards—depends on the coefficient of the squared term. In the function \(y = ax^{2} + bx + c\), the sign of \(a\):

• If \(a > 0\), the parabola opens upwards, resembling a 'U'.
• If \(a < 0\), it opens downwards, resembling an inverted 'U'.

Parabolas have unique properties that make them valuable in various fields, such as physics and engineering. Understanding the basic structure and behavior of parabolas is fundamental for graphing quadratic inequalities.

Quadratic Inequalities

When graphing quadratic inequalities like \(y > -4x^{2} -8x -4\), the goal is to find the set of points that satisfies the inequality. Unlike quadratic equations, which involve finding particular values or sets of values for \(x\), inequalities involve shading regions on a graph. This shading represents all possible solutions.Here are key steps when handling quadratic inequalities:

• Identify the related quadratic equation by setting the inequality to equality (e.g., if \(y > -4x^{2} -8x -4\), consider \(y = -4x^{2} -8x -4\)).
• Determine the vertex and the direction it opens to quickly sketch the parabola.
• Use a dashed or solid line for the parabola: dashed for ">" or "<", solid for "≥" or "≤".
• Test points to determine which side of the parabola satisfies the inequality and shade that region.

The inequality \(y > -4x^{2} -8x -4\) specifically means shading the area above the parabola because we are interested in the values of \(y\) that exceed those on the parabola itself.

Vertex of Parabola

The vertex of a parabola is a critical point that can represent either a maximum or a minimum point, depending on the parabola's orientation. It is often the first step in graphing a quadratic function or inequality.To find the vertex of a parabola given by the equation \(y = ax^2 + bx + c\):

• Use the formula \(x = \frac{-b}{2a}\) to find the x-coordinate of the vertex.
• Substitute this x-value back into the function to find the corresponding y-coordinate.

For the equation \(y = -4x^2 -8x -4\), the vertex calculation involves:

• Finding the x-coordinate: \(\frac{-(-8)}{2 \times -4} = -1\).
• The y-coordinate by substituting \(x = -1\) back into the equation: \(y = -4(-1)^2 - 8(-1) - 4 = -4 + 8 - 4 = 0\).

Thus, the vertex is at \((-1, 0)\). Knowing the vertex helps in sketching the overall shape and position of the parabola on the graph, which is especially important when working with inequalities.