Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Graph each inequality. $$y>2(x+3)^{2}-1$$

Step-by-Step Solution

Answer

Shade the region above the parabola, using a dashed line for the parabola.

Step 1: Identify the Base Graph

First, recognize that the inequality is related to the quadratic function \( y = 2(x + 3)^2 - 1 \). This is a parabola opening upward.

Step 2: Determine the Vertex

The vertex form of the quadratic equation \( y = a(x-h)^2 + k \) is given by the expression \( y = 2(x+3)^2 - 1 \). Here, the vertex \((h, k)\) is \((-3, -1)\).

Step 3: Identify the Direction and Shape

Since the coefficient of the \((x + 3)^2\) term is positive (\(2\)), the parabola opens upwards and is narrower than the standard parabola \( y = x^2 \).

Step 4: Sketch the Parabola

Plot the vertex at \((-3, -1)\) on the coordinate plane. Sketch the parabola, ensuring it is centered at the vertex and opens upward.

Step 5: Identify the Inequality Condition

Since the inequality is \( y > 2(x + 3)^2 - 1 \), this means all points above the parabola satisfy the inequality.

Step 6: Shading the Correct Region

Shade the region above the parabola to represent the solutions to the inequality. Since the inequality is strict (\(>\)), use a dashed line for the parabola to indicate that points on the parabola are not included.

Key Concepts

Parabola

A parabola is a U-shaped curve that can open either upwards or downwards. It is defined by a quadratic equation like \( y = ax^2 + bx + c \). In the exercise, the base equation we examined is a parabola of the form \( y = 2(x + 3)^2 - 1 \). Parabolas are symmetric, and the highest or lowest point on a parabola is called the vertex. The direction in which a parabola opens depends on the coefficient of the squared term (\(a\)). If \(a > 0\), the parabola opens upwards, forming a U shape. If \(a < 0\), it opens downwards, resembling an upside-down U. Parabolas appear naturally in various phenomena, such as projectile motion, and are crucial in physics and engineering.

Vertex Form

The vertex form of a quadratic equation presents the equation in a way that clearly shows the vertex of the parabola. The vertex form is expressed as \( y = a(x - h)^2 + k \). The values \(h\) and \(k\) represent the coordinates of the vertex. For the inequality in our exercise, the vertex form \( y = 2(x + 3)^2 - 1 \) reveals the vertex at \((-3, -1)\). Compared to the standard form \( y = ax^2 + bx + c \), the vertex form provides an immediate visual clue about where the parabola is located on the coordinate plane. This form makes it easier to graph the parabola by directly plotting the vertex and understanding its transformations, such as horizontal and vertical shifts.

Graphing Inequalities

Graphing inequalities involves several additional steps compared to graphing equalities. With inequalities, you're not only interested in the line or curve itself but also in shading the region that satisfies the inequality. The inequality \( y > 2(x + 3)^2 - 1 \) tells us that the solution set includes points above the parabola. Here are the steps to graph such an inequality:

• Plot the parabola from the quadratic expression.
• Since the inequality is strict (\(>\)), use a dashed line for the parabola, indicating that points on the parabola are not included.
• Shade the region above this dashed line to represent all the points that satisfy the inequality.

Achieving accurate shading involves correctly identifying the half-plane that represents the set of solutions, which can typically be verified by testing a point not on the line.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can graph equations and inequalities. It consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants. The coordinate plane allows us to visually represent equations and inequalities, making abstract math problems tangible. When you plot the vertex at \((-3, -1)\) on the coordinate plane, you're establishing a reference point for the parabola's graph. Understanding its layout helps in graphing the given inequality and verifying the shape and direction of the parabola. Each point on the plane corresponds to an (x, y) coordinate, making it straightforward to sketch the graph, adjust curves, and interpret regions defined by inequalities. Expertise with the coordinate plane is essential for solving a multitude of mathematical problems.