Problem 16 - A Graphical Approach to College Algebra

Question

Graph each inequality. Do not use a calculator. $$y>2(x+3)^{2}-1$$

Step-by-Step Solution

Answer

Shade above the parabola with vertex at (-3, -1) and opening upwards, using a dashed line.

Step 1: Understand the inequality and related equation

The given inequality is expressed as \( y > 2(x+3)^2 - 1 \). To graph it, first consider the related equation \( y = 2(x+3)^2 - 1 \). This is the equation of a parabola.

Step 2: Determine the vertex of the parabola

The standard form of a parabola is \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex. For our equation \( y = 2(x+3)^2 - 1 \), we have \( h = -3 \) and \( k = -1 \). So, the vertex of the parabola is \((-3, -1)\).

Step 3: Identify the direction of the parabola

The coefficient \( a \) in the equation \( y = 2(x+3)^2 - 1 \) is \( 2 \), which is positive. Therefore, the parabola opens upwards.

Step 4: Plot the vertex and additional points

Plot the vertex at \((-3, -1)\). To find additional points, choose \( x \)-values on either side of \(-3\). Calculate \( y \) for those \( x \)-values. For instance, if \( x = -2, \ y = 2(-2+3)^2 - 1 = 1 \) and if \( x = -4, \ y = 2(-4+3)^2 - 1 = 1 \). Plot these points: \((-2, 1)\) and \((-4, 1)\).

Step 5: Sketch the parabola

Draw a smooth curve through the plotted points and vertex. Ensure the curve opens upwards and reflects symmetry about the line \( x = -3 \). This curve represents the boundary for the inequality.

Step 6: Shade the solution region

Since the original inequality is \( y > 2(x+3)^2 - 1 \), shade the region above the parabola, as it includes points where \( y \) is greater than the parabola's y-values. Use a dashed line for the parabola to indicate that points on the line are not included in the solution.

Key Concepts

Parabola

A parabola is a symmetric, U-shaped curve used in mathematics to represent quadratic functions. The basic equation for a parabola can be written as \( y = ax^2 + bx + c \). A key property of the parabola is its symmetry, which refers to the fact that it is identical on both sides of its vertical axis. This unique geometric shape is important in many branches of mathematics, especially in algebra, where it helps to understand quadratic relationships.

• The shape of a parabola depends on the coefficient \( a \) in its equation. If \( a \) is positive, the parabola opens upwards, and if \( a \) is negative, it opens downwards.
• Every parabola has an axis of symmetry, which is a vertical line that divides it into two equal halves.
• The points on the parabola satisfy the quadratic equation, meaning when you plug in any x-value, you can solve for the corresponding y-value.

Vertex Form

The vertex form of a parabola provides an efficient way to identify its most significant features, such as the vertex and direction it opens. The form is written as \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. This point is either the highest or lowest point on the curve, depending on the direction the parabola opens.

• The vertex \((h, k)\) allows for easy graphing because it is a central reference point from which the parabola expands outward.
• Keen observers can quickly assess whether a parabola opens upwards or downwards by looking at the sign of \( a \).
• In the case of \( y = 2(x+3)^2 - 1 \), \( h = -3 \) and \( k = -1 \), so the vertex is at \((-3, -1)\).
• This form clearly shows the translation of the parabola along the x and y axes from its origin.

Solution Region

The solution region in inequalities involving parabolas refers to the set of points that satisfy the inequality. In the problem \( y > 2(x+3)^2 - 1 \), the solution region is the area where the y-values of points are greater than those on the parabola \( y = 2(x+3)^2 - 1 \).

• Graphically, this is represented by shading the part of the graph above the parabola since it signifies all y-values greater than the boundary line.
• To indicate that points on the parabola itself do not satisfy the inequality, the parabola is often drawn with a dashed line.
• Understanding the solution region helps determine which input values (x-values) will produce results within the desired range (y-values).

Upward Opening Parabola

An upward opening parabola is characterized by its U-shape and symmetry about its vertical axis. When graphing the quadratic function, it's important to recognize the mathematical properties that define its direction.

• The coefficient \( a \) in the equation \( y = a(x-h)^2 + k \) is key. If \( a \) is positive, the parabola opens upwards, resembling a smile. In the equation \( y = 2(x+3)^2 - 1 \), since \( a = 2 \), the parabola opens upwards.
• This direction impacts the solutions to the inequality. For \( y > 2(x+3)^2 - 1 \), we focus on regions above the curve where y-values exceed those of points on the parabola.
• Understanding whether a parabola opens upwards or downwards influences how we interpret the vertex and the equation's graph.