Problem 12 - Precalculus Mathematics for Calculus

Question

Graph the inequality. $$-x^{2}+y \geq 10$$

Step-by-Step Solution

Answer

Shade the region above and on the parabola \(y = x^2 + 10\).

Step 1: Rewrite the inequality in standard form

To graph the inequality, start by rewriting the inequality from \(-x^{2} + y \geq 10\)to \(y \geq x^{2} + 10\).

Step 2: Graph the equation y = x² + 10

Graph the equation \(y = x^{2} + 10\) on a coordinate plane. This will form a parabola that opens upwards, with its vertex at (0, 10).

Step 3: Determine the boundary line type

Since the inequality is \(y \geq x^{2} + 10\), the boundary is \(y = x^{2} + 10\) and must be drawn as a solid line, because values on the line satisfy the inequality.

Step 4: Shade the appropriate region

The inequality \(y \geq x^{2} + 10\) suggests that we shade the region above the parabola (y > x² + 10) or on the parabola (y = x² + 10).

Step 5: Check your work

To ensure correctness, select a test point not on the line, such as (0, 11). Substitute it into the original inequality: \(-0^2 + 11 \geq 10\). This simplifies to 11 \(\geq\) 10, which is true. Thus, the shading is correct.

Key Concepts

Graphing Inequalities

Graphing inequalities involves representing mathematical expressions on a graph by showing the area of the coordinate plane that satisfies the inequality. When dealing with inequalities, like the one in the exercise

• -x² + y ≥ 10

, it is crucial to convert it into a form that is easier to work with, often by rearranging terms. Here, it is transformed into

• y ≥ x² + 10

, which is more straightforward for graphing. The concept of inequalities involves shading a specific region of the graph. You decide which region to shade based on the inequality symbol. For ≥ or ≤ inequalities, the shaded region includes the boundary line. In our example, since the inequality is

• y ≥ x² + 10,

the boundary line is solid, and the region above the line is shaded because these are the solutions that satisfy the inequality. Testing a point helps confirm which side of the line fulfills the inequality. By choosing a point such as (0, 11) and substituting it back into the inequality, you verify the correct shading by checking if the inequality holds true.

Parabolas

A parabola is a U-shaped curve that is a key concept in graphing quadratic equations. The standard form of a quadratic equation is

• y = ax² + bx + c

, and the graph of any quadratic equation forms a parabola. In our inequality,

• y = x² + 10

, the parabola opens upwards. The direction in which the parabola opens depends on the coefficient of x². If it's positive, the parabola opens upwards, like in our case. If negative, it opens downwards. The vertex of the parabola is its highest or lowest point, found at the axis of symmetry. Here, the vertex is at (0,10). This vertex is crucial because it helps determine the shape and position of the parabola on the graph. To graph a parabola, you plot the vertex and several additional points either side of the vertex to illustrate the curve. Ensure the plotted points satisfy

• y = x² + 10

, so you graph accurately. This visual representation aids in understanding the quadratic relationship in the inequality.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which you can plot points, lines, and curves to solve and visualize mathematical problems. It consists of a horizontal axis, the x-axis, and a vertical axis, the y-axis. Every point in the plane is represented by an ordered pair (x, y), showing its position relative to these axes. In graphing inequalities like

• y ≥ x² + 10

, the coordinate plane allows us to see which points are included in the solution set. To graph, first identify the vertex of the parabola (

• (0,10)

in this case), then determine the shape of the parabola by plotting additional points. Mark these points precisely on the plane. Also, decide the line type (solid or dashed) based on whether or not the points on the line itself satisfy the inequality. Utilizing the coordinate plane correctly is vital as it visually demonstrates where the inequality holds true, enhancing understanding of the relationships between algebraic solutions and their graphical counterparts.