Problem 17 - Intermediate Algebra

Question

Graph each inequality. See Examples 1 and \(2 .\) $$ y<(x-2)^{2}+1 $$

Step-by-Step Solution

Answer

Shade below the parabola \( y = (x-2)^2 + 1 \) drawn with a dashed line; exclude the parabola itself.

Step 1: Identify the Basic Parabola

The given inequality is based on a parabola. The basic parabola for the equation would be \( y = (x-2)^2 + 1 \). This is a standard upward-facing parabola with a vertex at the point \((2, 1)\).

Step 2: Determine the Vertex

From the expression \((x-2)^2 + 1\), we identify the vertex of the parabola as \((2, 1)\). This is because the \(x\) term indicates a horizontal shift to the right by 2 units, and the \(+1\) indicates a vertical shift up by 1 unit.

Step 3: Sketch the Parabola

Draw the graph of the parabola starting at the vertex. For \( y = (x-2)^2 + 1 \), points such as \( (1, 2) \) and \( (3, 2) \) lie on the parabola, since both satisfy the equation as their height is calculated by inputting the respective \(x\)-values into the equation.

Step 4: Determine the Inequality Region

Since the inequality is \( y < (x-2)^2 + 1 \), we are interested in the region below the parabola. This means shading the area beneath the curve. This shaded region represents all points \((x, y)\) that satisfy the inequality.

Step 5: Draw a Dashed Line for the Parabola

Because the inequality symbol is \( < \) and not \( \leq \), the line representing \( y = (x-2)^2 + 1 \) itself is not included in the solution set. Therefore, draw the parabola as a dashed curve to show that the points on the curve are not part of the solution.

Key Concepts

Graphing Parabolas

When graphing parabolas, it is important to understand the basic features and structure of a parabola. A parabola is a symmetrical curve that either opens upward or downward, and it can be described by a quadratic equation like \(y = ax^2 + bx + c\). The vertex form of a quadratic equation, which is \(y = a(x-h)^2 + k\), is especially useful for graphing because it easily identifies the vertex of the parabola.

The vertex is the highest or lowest point on a parabola, depending on whether it opens upwards or downwards. By examining the vertex form, \(y = (x-2)^2 + 1\), we can see the vertex here is located at \((2, 1)\). This indicates the parabola has been shifted horizontally to the right by 2 and vertically upward by 1 from the origin.

• Vertex: The point \((h, k)\) in \(y = a(x-h)^2 + k\)
• Opening direction: Upwards if \(a > 0\) and downwards if \(a < 0\)

When graphing, identify the vertex and plot key points equally spaced on either side. Since parabolas are symmetric, once you find these key points, you can easily sketch the curve.

Inequality Regions

Understanding inequality regions is crucial when dealing with inequalities involving parabolas. These inequalities define areas on the graph where all solutions lie. For an inequality like \(y < (x-2)^2 + 1\), the region of interest is all points below the curve of the parabola.

To find this region, first graph the corresponding equation \(y = (x-2)^2 + 1\) as if it were an equality. However, instead of a solid curve, you should use a dashed curve. This is because the inequality sign is a strict \(<\), meaning the boundary line itself is not included in the solution.

• Shaded Region: Indicates all points satisfying the inequality
• Dashed Line: Used for \(<\) and \(>\) symbol to show the boundary is not included

The inequality region must be shaded to clearly reflect the area where the inequality holds true to ensure clarity for interpretation.

Quadratic Equations

Quadratic equations are fundamental in understanding parabolas and their graphs, given their ability to express the classic \(y = ax^2 + bx + c\) structure. These equations form the basis for curves we deal with in problems involving parabolas and, consequently, inequality regions.

In standard practice, one can use the quadratic formula to find the roots of the equation if needed, though in graphing cases involving inequalities, the vertex form is particularly practical. A quadratic equation will usually have a unique vertex represented in its vertex form, \(y = a(x-h)^2 + k\), providing quick and clear plot points for graph interpretation.

• Factors: \(ax^2 + bx + c\) gives the parabola its specific shape and orientation
• Roots: Calculate using the quadratic formula when solving for specific intersection points

By using these principles, quadratic equations allow us to understand the algebraic and geometric characteristics of parabolas. This understanding of quadratics is vital in solving real-world problems involving parabolic paths and regions.