Problem 16 - Precalculus with Limits

Question

In Exercises 5-18, sketch the graph of the inequality. $$(x-1)^{2}+(y-4)^{2}>9$$

Step-by-Step Solution

Answer

The graph would show a dashed circle with center at (1,4) and radius 3, with the region outside the circle shaded to denote the solution set of the inequality.

Step 1: Identify the Center and the Radius

From the formula \((x-a)^{2} + (y-b)^{2} = r^{2}\), where (a, b) is the midpoint of the circle and r is the radius, determine the circle's center and radius. Given \((x-1)^{2}+(y-4)^{2} > 9\), it can be seen that the center of the circle is (1,4) and the radius \(\sqrt{9} = 3\).

Step 2: Sketch the Circle

Sketch the circle with the center at (1,4) with a radius of 3 on graph paper. Since our inequality does not include the points on the circle, the circle should be sketched as a dashed line.

Step 3: Shade the Appropriate Region

As the inequality is \((x-1)^{2}+(y-4)^{2} > 9\), the graph should include all points 'greater than' the radius. Therefore, the region outside the circle should be shaded to denote that all these points satisfy our inequality.

Key Concepts

Understanding the Center of a Circle

The center of a circle is an essential concept in graphing equations, especially when dealing with inequalities. When you see an equation like \((x-a)^2 + (y-b)^2 = r^2\), the center of the circle is the point \(a, b\). It is effectively the midpoint of the circle, and it determines where the circle is positioned on the graph.

• For the inequality \((x-1)^2 + (y-4)^2 > 9\), the center is located at \(1, 4\).
• This means the circle is centered at this exact point, which serves as the focal point for measuring the radius.

By grasping the concept of the center, you can accurately place the circle on a graph. It serves as the foundation for identifying how and where the circle exists within the coordinate plane.

Decoding the Radius of a Circle

The radius is another vital element when graphing circles. It represents the distance from the center of the circle to any point along its boundary. In mathematical terms, this is represented by \(r\), and it influences the size of the circle.

• For the given inequality \((x-1)^2 + (y-4)^2 > 9\), the radius is found by taking the square root of 9, which means \(r = 3\).
• This defines a circle with all boundary points exactly 3 units from the center (1, 4).

The radius is crucial in defining the extent of the circle, dictating how far it stretches across both the x and y directions from its center. Understanding this helps you sketch the circle accurately on the graph.

Mastering Shading Regions in Inequalities

When graphing inequalities that involve a circle, it's important to understand the regions that need shading. Shading plays a key role in representing which areas satisfy the given inequality.

• For the inequality \((x-1)^2 + (y-4)^2 > 9\), the focus is on points that are outside the circle.
• To represent the inequality, draw a dashed circle, indicating that points on the circle itself are not included.
• Then, shade the region outside the circle to show that it contains points satisfying \((x-1)^2 + (y-4)^2\) greater than 9.

Shading the correct region ensures clarity in your graph, making it easy to visually understand which points meet the inequality's conditions. Mastery of shading techniques is essential in correctly interpreting and presenting graphical inequalities.