Problem 20 - College Algebra

Question

In Exercises 1–26, graph each inequality. $$y < x^{2}-9$$

Step-by-Step Solution

Answer

The solution to the inequality \(y < x^{2} - 9\) is the shaded region beneath the parabola of the graph of the function \( y = x^{2} - 9 \), excluding the points on the actual parabola line.

Step 1: Graph the Equation

Start by graphing the function \(y = x^{2} - 9\). This is a parabola opening upwards, shifted down 9 units from the origin. Sketch this parabola and denote it with a dashed line, as the inequality is 'less than', not 'less than or equal to', meaning the points on the line \(y = x^{2} - 9\) are not included in the solution.

Step 2: Determine the solution region

Next, to identify where to shade for the inequality \(y < x^{2} - 9\), select a test point not on the parabola. The most convenient point is the origin (0,0). Substitute these coordinates into the inequality, if it makes the inequality true, shade the region that contains the origin. If it makes the inequality false, shade the region that does not contain the origin.

Step 3: Shade the region

Substitute (0,0) into the inequality: \(0 < 0^{2} - 9\), this simplifies to \(0 < -9\), which is false. Thus, shade the region that does not contain the origin, which is the area below the parabola.

Key Concepts

Parabola

Understanding the shape and properties of a parabola is crucial when graphing quadratic inequalities. A parabola is a U-shaped curve that can open upwards or downwards, depending on the sign of the quadratic term in the equation. In the inequality \(y < x^{2} - 9\), the parabola opens upward because the coefficient of \(x^{2}\) is positive.

When graphing, we often begin by sketching the corresponding parabola of the equation \(y = x^{2} - 9\). This parabola has its vertex, the lowest point, at (0, -9). The axis of symmetry, a vertical line that divides the parabola into two mirror-image halves, is the y-axis in this case. Visualizing this symmetry can help when drawing the parabola.

Remember, in the context of inequalities, the parabola is typically represented by a dashed line to indicate that points lying exactly on the parabola are not part of the solution set for a '<' or '>' inequality.

Quadratic Functions

Quadratic functions form the backbone of parabolic equations and are typically expressed in the standard form \(y = ax^{2} + bx + c\), where 'a', 'b', and 'c' are constants, and \(a eq 0\). The sign of 'a' determines the direction of the parabola—positive for upwards and negative for downwards.

To locate the vertex of a quadratic function, one can use the formula \(h = -\frac{b}{2a}\) for the x-coordinate of the vertex, and then plug this value into the original quadratic equation to find the y-coordinate. The vertex represents the highest or lowest point on the graph, depending on the parabola's orientation. For the function \(y = x^{2} - 9\), the vertex is straightforward as there is no 'bx' term, making it easy to identify the vertex at (0, -9).

Graphing the function's parabola is instrumental in visualizing solutions to quadratic inequalities, as it sets the boundary for where to test and shade the region of solutions.

Inequality Shading

Inequality shading is a visual method used to represent the solution set of an inequality on a graph. Once the boundary—our parabola in the case of a quadratic inequality—is drawn, we use a test point to determine which side of the boundary contains the solutions to the inequality.

For the inequality \(y < x^{2} - 9\), after graphing the parabola, you choose a point not on the graph, frequently (0,0) if it isn't on the parabola, and substitute it into the inequality to see if the statement remains true. If the test point satisfies the inequality, the region containing that point is shaded. Conversely, if the test point does not satisfy the inequality, the other side of the boundary is shaded.

It is important to use a simple and obvious test point because it reduces the likelihood of calculation errors that can lead to incorrect shading. Shading correctly is critical as it represents the entire set of points that satisfy the inequality, hence ensuring that your solutions are accurately depicted on the graph.