Problem 18 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the inequality

We are given the inequality \(y > (x-2)^2 + 1\). This inequality describes the region where the y-values are greater than the values of the quadratic function \((x-2)^2 + 1\).

Step 2: Rewrite the quadratic function

The function \((x-2)^2 + 1\) represents a parabola opening upwards with its vertex at the point \((2, 1)\). It can be expressed in the standard vertex form \((x-h)^2 + k\), where \(h = 2\) and \(k = 1\).

Step 3: Determine the boundary (a parabola)

The equality \(y = (x-2)^2 + 1\) can be graphed as a parabola. This is the boundary of the region for the inequality. Graph this parabola to understand the region being referred to.

Step 4: Shading the region above the parabola

Since our inequality is \(y > (x-2)^2 + 1\), we shade the region above the parabola to represent all the points where the y-values are greater than the values on the parabola. Use a dotted line for the parabola to indicate that points on the parabola itself are not included.

Key Concepts

Quadratic Functions Overview

Quadratic functions are a type of polynomial function that describe curves known as parabolas. They are originally expressed in the form

• \( f(x) = ax^2 + bx + c \)

where \(a\), \(b\), and \(c\) are constants. The leading coefficient \(a\) determines the direction of the parabola's opening. If \(a\) is positive, the parabola opens upwards and resembles a smiling face. Conversely, if \(a\) is negative, it opens downwards like a frowning face.
The quadratic function plays a crucial role in various real-world applications. They can model phenomena like the trajectory of a thrown ball or the area of a garden.
Beyond real applications, understanding quadratic functions is essential for solving inequalities involving parabolas. Such inequalities help determine the regions in a coordinate plane that satisfy certain conditions, often shown by shading on a graph.

Basics of Parabola Graphing

Graphing a parabola involves representing the curve of a quadratic function on a coordinate plane. Every quadratic equation plots as a parabola.

• Start by identifying the vertex of the parabola, which is the highest or lowest point, depending on the opening direction.
• Understand that the vertex influences the positioning of the parabola on the plane.
• Determine the axis of symmetry, a vertical line that runs through the vertex, dividing the parabola into two identical halves.

For example, in the expression \( y = (x-2)^2 + 1 \), we see that the parabola vertex is at \( (2, 1) \), and it opens upwards due to the positive coefficient of the squared term.
Using this information, you can graph the basic shape of the parabola, enabling you to further analyze and solve related inequalities. The parabola's symmetry and shape help in understanding which regions of the graph satisfy certain inequalities.

Exploring the Vertex Form

The vertex form of a quadratic function is a significant format that helps identify and graph parabolas efficiently. It is typically written as

• \( y = a(x-h)^2 + k \)

where \( (h, k) \) is the vertex of the parabola. This format is advantageous because it instantly reveals the parabola's vertex and the direction of its opening.
For instance, the vertex form \( y = (x-2)^2 + 1 \) indicates a parabola with the vertex at \( (2, 1) \), opening upwards due to the positive coefficient.
Utilizing the vertex form makes graph transformations more manageable, such as translating the graph to different positions or adjusting its width. Understanding and using the vertex form is fundamental when solving graphing inequalities, as it guides in the proper shading of regions related to the given inequality expressions.