Problem 11 - Algebra 2

Question

Graph each inequality. $$ y \geq x^{2}+3 x-18 $$

Step-by-Step Solution

Answer

Graph the parabola \( y = x^2 + 3x - 18 \), shade above it, and include the boundary line.

Step 1: Identify the Equation of the Parabola

The given inequality is \( y \geq x^2 + 3x - 18 \). To graph the inequality, start by identifying the equation of the parabola, which is \( y = x^2 + 3x - 18 \). This will be our boundary line.

Step 2: Find the Vertex of the Parabola

The equation is in the standard form \( y = ax^2 + bx + c \), where \( a = 1 \), \( b = 3 \), and \( c = -18 \). Use the formula \( x = -\frac{b}{2a} \) to find the x-coordinate of the vertex: \( x = -\frac{3}{2(1)} = -1.5 \). Substitute \( x = -1.5 \) into the equation to find \( y \): \( y = (-1.5)^2 + 3(-1.5) - 18 \). Calculate \( y = 2.25 - 4.5 - 18 = -20.25 \). The vertex is \( (-1.5, -20.25) \).

Step 3: Identify the Direction of the Parabola

Since the coefficient \( a = 1 \) is positive, the parabola opens upwards.

Step 4: Draw the Parabola

Plot the vertex at \( (-1.5, -20.25) \) and draw the parabola opening upwards. Use additional points to accurately plot the parabola, such as intercepts and symmetric points around the vertex.

Step 5: Determine the Boundary Condition

The inequality is \( y \geq x^2 + 3x - 18 \), which means the region includes the parabola itself. Therefore, the boundary line (the parabola) should be drawn solid to indicate that points on this line are included in the solution set.

Step 6: Shade the Region of the Graph

Since we have \( y \geq x^2 + 3x - 18 \), shade the region above the parabola where \( y \) is greater than or equal to the value of \( x^2 + 3x - 18 \). This represents the solution to the inequality.

Key Concepts

Graphing Inequalities

Graphing inequalities involves visually representing the range of solutions that satisfy a given inequality on a coordinate plane. To graph an inequality like \( y \geq x^2 + 3x - 18 \), the first step is to consider the related equation, \( y = x^2 + 3x-18 \). This equation represents the boundary of your graph, known as the parabola.

• Start by treating the inequality as an equation to find the parabola.
• Plot the graph, focusing on the solid or dashed lines (solid for \( \geq \) or \( \leq \), dashed for \( > \) or \( < \)).

Once the parabola is graphed, you shade the region that satisfies the inequality. This process helps you communicate which points on the plane meet the conditions set by the inequality.

Vertex of a Parabola

The vertex of a parabola is a key point that represents either the maximum or minimum value of the quadratic function, depending on the direction the parabola opens. For the inequality \( y \geq x^2 + 3x - 18 \), the vertex is particularly important as it is used to balance the parabola symmetrically. To find the vertex, you use the formula for the x-coordinate: \( x = -\frac{b}{2a} \). This calculation gives you the position along the x-axis.
For our quadratic equation \( y = x^2 + 3x - 18 \):

• \( a = 1 \) and \( b = 3 \)
• x-coordinate of vertex: \( -\frac{3}{2(1)} = -1.5 \)
• Substitute \( x = -1.5 \) into the equation to obtain the y-coordinate: \( y = 2.25 - 4.5 - 18 = -20.25 \)
• Vertex: \( (-1.5, -20.25) \)

This vertex helps you position and sketch the parabola correctly on the graph. Identifying the vertex ensures the parabola is accurately represented in terms of shape and placement.

Parabola Direction

The direction of a parabola determines whether it opens upwards or downwards, influencing which side of the boundary you'll shade. The direction is determined by the coefficient of the squared term in the quadratic equation. For \( y = x^2 + 3x - 18 \), the squared term is \( x^2 \), and its coefficient, \( a = 1 \), is positive.

• Positive \( a \) means the parabola opens upwards.
• Negative \( a \) would mean the parabola opens downwards.

Since this parabola opens upwards, the arms will extend towards positive infinity along the y-axis. Understanding the direction helps determine the appropriate inequality solution region to shade.

Shading Regions of Inequalities

Shading regions of inequalities is an essential task that visually demonstrates the solution set on a graph. When dealing with quadratic inequalities, like \( y \geq x^2 + 3x - 18 \), shading helps you to identify which side of the parabola contains solutions that satisfy the inequality.In this example, because the inequality is \( y \geq x^2 + 3x - 18 \):

• You need to shade above the parabola, since the y-values in this region will be greater than or equal to those on the parabola.
• The solid line is used for the parabola to show inclusion, meaning the points on the parabola are part of the solution set.

Proper shading and drawing help make the solution clear and help you determine if a given point satisfies the inequality. Consider testing a simple point, like the origin, to verify you've shaded the correct area.