Problem 18 - College Algebra

Question

In Exercises 1–26, graph each inequality. $$(x+2)^{2}+(y-1)^{2}<16$$

Step-by-Step Solution

Answer

The graph of the inequality \((x+2)^{2}+(y-1)^{2}<16\) is a circle with a dashed border, centered at (-2,1) with radius 4 and the inside of this circle is shaded.

Step 1: Identify the Circle Parameters

The given inequality is \((x+2)^{2}+(y-1)^{2}<16\). From this, one can recognize the form of a circle's equation \((x-h)^{2}+(y-k)^{2} = r^{2}\), where (h,k) are the center of the circle and r is the circle's radius. Comparing the inequality and equation, one can identify the center of the circle as (-2,1) and the radius as 4.

Step 2: Graphing the Circle

Graph the circle with center (-2,1) and radius 4. To do this, simply plot the center and draw the circle around it, with a circle's point at a distance of 4 units from the center in all directions. However, because of the '<' sign, the circle border is not part of the solution, so it should be represented with a dashed line.

Step 3: Representing the Inequality Solution

Based on the direction of the inequality '<', the solution will be inside of the circle. To visually represent this on the graph, one can shade it.

Key Concepts

Circle Equations

When faced with problems involving circles in the coordinate plane, understanding circle equations is essential. The general equation of a circle is given by \[(x-h)^2 + (y-k)^2 = r^2\]where

• \( h \)is the x-coordinate of the center,
• \( k \)is the y-coordinate of the center, and
• \( r \)is the radius of the circle.

The circle's equation beautifully encapsulates the relationship between any point \((x, y)\)on the circle and its center \((h,k)\). By shifting \( h \) and \( k \), the circle moves in the plane without altering its shape. Changing \( r \)modulates its size. Understanding these parameters provides the foundation for transforming the equation into a specific graph.

Inequality Solutions

In the context of graphing inequalities, it's important to interpret how the inequality sign affects the graph and solution set. When you encounter an inequality such as \((x+2)^2 + (y-1)^2 < 16\),this signifies that instead of a definite boundary,symbolized by a circle in the equation of a circle form,the solution includes a region. Specifically:

• The '<' sign indicates that the solution comprises points inside the circle.
• The border, or circle itself, does not belong to the solution set.

To reflect this in the graph, use a dashed line for the circle's border, highlighting that points on this line are excluded. The inside of the circle, however, which the inequality describes, will consist of all points closer to the center than the radius allows.

Circle Parameters

Understanding circle parameters, such as the center and radius, is crucial when working with circle equations. In an expression like \((x+2)^{2}+(y-1)^{2} = r^{2}\),we identify:

• The center of the circle as the opposite of the numbers paired with x and y, which are \(h = -2\) and \(k = 1\),respectively.
• The radius as the square root of the number on the right side, which is \(r = 4\).

Recognizing these parameters in an equation comes from spotting the pattern of difference squared, producing a precise representation of the circle's attributes on the coordinate plane. Proper indentification simplifies plotting and interpreting the region related to inequality.

Graphing Techniques

Graphing circles and inequalities requires careful plotting and shading techniques to accurately represent solutions. With an equation like \((x+2)^2+(y-1)^2<16\),a systematic approach is:

• First plot the center of the circle, (-2,1).
• Use the radius to draw a reference circle with a radius of 4distances from the center.
• Since the inequality excludes the boundary, draw the circle with a dashed line.
• Next, shade the interior of the circle to indicate that this area meets the inequality's criteria, showing all potential solutions.

Breaking the graphing task into these individual steps not only improves accuracy but also offers a clear visual understanding of the inequality's meaning. With practice, these graphing techniques become intuitive and a valuable component of working with inequalities.