Question

Sketch the graph of the inequality. $$x^{2}+(y-2)^{2}<16$$

Short answer

The graph of the inequality \(x^2 + (y - 2)^2 < 16\) is a circle with center at (0, 2) and radius 4, with the interior of the circle shaded to indicate all points inside the circle fulfill the inequality condition.

Step-by-step solution

Identify the Center and Radius of the Circle

The given inequality is \(x^2 + (y - 2)^2 < 16\). This can be rewritten in a standard circle formula as \((x - 0)^2 + (y - 2)^2 < 4^2\). Herein, the circle's center is at (0, 2) and the radius is 4.

Sketch the Circle

First, draw a pair of perpendicular axes. Then plot the center point (0, 2). From this point, draw the circle by sketching lines at a distance equal to the radius (4 units) in all directions from the center point.

Shade the Interior of the Circle

As the inequality presents a less than symbol (<), it indicates all points inside the circle but not on the circle itself are part of the solution. So, you need to shade or fill in the area inside the circle.

Key concepts

Circle Graphing

When it comes to graphing circles, it's essential to understand what an equation of a circle represents in a coordinate system. The general equation of a circle centered at \( (h, k) \) with a radius of \( r \) is \( (x - h)^2 + (y - k)^2 = r^2 \). In this context, \( h \) and \( k \) signify the x and y coordinates of the circle's center, whereas \( r \) represents the circle's radius.

For instance, the inequality \( x^2 + (y - 2)^2 < 16 \) refers to all the points that lie strictly inside a circle with a center at \( (0, 2) \) and a radius of 4 units. Since the inequality does not include the boundary (indicated by the < symbol), you would graph this circle by drawing a dashed line to signify the boundary is not part of the solution. Remember, the boundary line of the circle is as important as the area it encompasses, particularly when dealing with inequalities.

After plotting the center, you would draw a circle around it, using the radius length to determine how far from the center the curve should extend in all directions. Tools like compasses can be used for precision, but for a rough sketch, this can be done freehand with practice and care. What's especially important is to visualize the radius as extending equally in all directions from the center, creating a perfect circle.

Inequalities in Algebra

In algebra, inequalities are statements about the relative size or order of two values. They are used to define ranges or intervals that contain all numbers that make the inequality true. Common inequality symbols include \( < \) (less than), \( > \) (greater than), \( \leq \) (less than or equal to), and \( \geq \) (greater than or equal to).

Graphically, inequalities can be represented on a coordinate system where solutions to an inequality like \( x^2 + (y - 2)^2 < 16 \) form a region. In this case, the region consists of all points inside the circle but not including the circle itself. This means the points that lie exactly on the circle do not satisfy the inequality and are not part of the solution set.

Inequalities that involve a less than sign, \( < \) or a less than or equal to sign, \( \leq \) indicate shading below or to the left of a line or inside a region like a circle, whereas inequalities that involve a greater than sign, \( > \) or a greater than or equal to sign, \( \geq \) direct shading above or to the right of a line or outside a region. Inequalities can also involve absolute values and quadratic forms, thus allowing a vast range of graphical solution sets.

Coordinate System

The coordinate system, typically referred to as the Cartesian coordinate system, is a two-dimensional plane consisting of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants and allow us to locate any point with a pair of numerical coordinates: \( (x, y) \).

The origin, represented by the point \( (0, 0) \), is the intersection of the x-axis and y-axis and acts as the reference point for the system. To graph any equation or inequality, we first identify points or regions in relation to the origin and then plot accordingly.

For circle graphing, the coordinate system allows us to plot the center \( (h, k) \) by moving \( h \) units along the x-axis and \( k \) units along or against the y-axis, depending on the sign. The distance from the origin to this center and from the center to any point on the circumference is crucial to draw an accurate circle. Inequalities in the coordinate system are visualized by shading areas that represent all possible solutions, providing a clear visual of what values satisfy the inequality. Understanding how to navigate and use the coordinate system effectively is the foundation for graphing algebraic equations and inequalities.