Problem 23 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

Graph \( y = (x-1)^2 + 2 \) and shade the region above it.

Step 1: Understanding the Inequality

The given inequality is \( y > (x-1)^2 + 2 \). This inequality represents a region in the coordinate plane, specifically the area above the parabola \( y = (x-1)^2 + 2 \). This parabola opens upwards with its vertex at \((1, 2)\).

Step 2: Graph the Parabola

Begin by graphing the function \( y = (x-1)^2 + 2 \). This is a parabola that has been shifted 1 unit to the right and 2 units up from the standard \( y = x^2 \) parabola. Mark the vertex at the point \((1, 2)\) and draw the parabola, ensuring it opens upwards. Use a dashed line to indicate that points on the parabola are not included in the solution (as it is a strict inequality).

Step 3: Shade the Solution Region

The inequality \( y > (x-1)^2 + 2 \) indicates that the solution region is above the parabola. Shade the entire region above the parabola to represent the set of points where \( y \) is greater than \( (x-1)^2 + 2 \).

Step 4: Verify with a Test Point

Choose a test point not on the parabola (e.g., \((0,3)\)). Substitute \((0, 3)\) into the inequality: \( 3 > (0-1)^2 + 2 \), which simplifies to \( 3 > 1 + 2 \) or \( 3 > 3 \). This is false, therefore do not include the region where this test point is. This confirms we should shade above the curve.

Key Concepts

Parabolas

Parabolas are unique and fascinating curves that appear in many mathematical contexts. A parabola is the graph of a quadratic function, and it has a distinct U-shape. Often, they are symmetrical about a vertical line called the axis of symmetry.
Parabolas can open upwards or downwards, and on a coordinate plane, they are represented by the equation \( y = ax^2 + bx + c \). The value of \( a \) determines the direction in which the parabola opens: if \( a > 0 \), it opens upwards, and if \( a < 0 \), it opens downwards.
The point where the parabola appears to "turn around" is known as the vertex. This is a crucial feature and acts as the minimum or maximum point of the curve, depending on its orientation.
Parabolas have many practical uses in fields such as physics, engineering, and economics, often describing trajectories, signal paths, and profit models.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which points are graphed based on their coordinates. These coordinates are typically given as ordered pairs \( (x, y) \).
In this plane, there are two axes: the horizontal axis, known as the x-axis, and the vertical axis, referred to as the y-axis. The point where these two axes intersect is called the origin, denoted by \( (0, 0) \).
Each point on this plane can be located precisely using its x and y coordinates, which tell how far the point is from the origin horizontally and vertically, respectively.

• The x-coordinate specifies the point's position left or right of the origin.
• The y-coordinate specifies the position above or below the origin.

Graphing inequalities on the coordinate plane involves shading regions that satisfy the inequality. These are usually found either above, below, or to the sides of lines or curves.

Vertex Form

The vertex form of a quadratic equation provides a simple way to quickly identify the vertex of a parabola. It is given as \( y = a(x-h)^2 + k \).
In this form, \( (h, k) \) is directly the vertex of the parabola. The vertex form makes it easy to graph the parabola and understand how it has been transformed from the basic parabola \( y = x^2 \).

• \( h \) represents a horizontal shift from the origin.
• \( k \) represents a vertical shift.
• The coefficient \( a \) affects the width and direction of the parabola.

A positive \( a \) makes the parabola open upwards, while a negative \( a \) opens it downwards. This form is particularly useful for graphing because it shows transformations clearly, including translations and reflections without needing to manipulate the standard form equation.