Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Graph each inequality. $$x^{2}+(y+3)^{2} \leq 16$$

Step-by-Step Solution

Answer

Graph a circle centered at (0, -3) with a radius of 4 and shade its interior.

Step 1: Identify the Inequality Type

The given inequality is \(x^2 + (y+3)^2 \leq 16\). This represents a circle or disc in the coordinate plane, as it resembles the standard form of a circle's equation \((x-h)^2 + (y-k)^2 = r^2\). In this inequality, the center is \((h,k) = (0,-3)\) and the radius \(r\) is the square root of 16, which is 4.

Step 2: Determine the Center and Radius

From the comparison with the circle's equation, we identify the center of the circle as \((0, -3)\) and the radius as 4. This interpretation comes from the expression \((y+3)^2\) which indicates the circle is shifted 3 units down along the y-axis from the origin.

Step 3: Sketch the Circle

Draw a circle in the coordinate plane centered at \((0, -3)\) with a radius of 4. This involves plotting the center at \(x = 0, y = -3\) and then using the radius to draw the circle extending 4 units in all directions (up, down, left, and right) from the center.

Step 4: Shade the Solution Area

Since the inequality is \(\leq\), we need to shade the region inside the circle to represent all the points \((x, y)\) that satisfy the inequality. Also, include the boundary of the circle itself in the shaded region since the inequality is non-strict (\(\leq\)).

Step 5: Verify Boundary and Interior Points

Choose a few test points inside and on the boundary, such as \((0,-3)\), \((4,-3)\), and \((0,1)\). Check the inequality \(x^2 + (y+3)^2 \leq 16\) for these points to ensure they satisfy it. For example, for point \((0,-3)\), the inequality becomes \(0 + 0 \leq 16\), which is true. Ensure all chosen test points satisfy the inequality.

Key Concepts

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Each point on the plane can be identified using an ordered pair of numbers, \(x, y\), which represent its coordinates. The position of points along these axes is crucial when graphing inequalities, as it helps to depict regions that satisfy specific conditions.

When graphing inequalities that involve a circle, we use the coordinate plane to precisely locate the center of the circle and to determine the extent of the region represented by it. In our exercise, this understanding helps us position the circle centered at \(0, -3\), and effectively sketch the inequality's solution region.

Circle Equation

In algebra, the equation of a circle is traditionally expressed in the form: \((x - h)^2 + (y - k)^2 = r^2\), where \(h, k\) is the center of the circle and \(r\) is its radius. This equation is crucial for graphing circles as it allows us to determine the location and size of the circle on the coordinate plane.

In this exercise, the given inequality \(x^2 + (y+3)^2 \leq 16\) indicates a circle with a center at \(0, -3\) and a radius of 4, based on the equation's form. The term \(y + 3\) shows the vertical shift from the origin. It’s essential to recognize that the inequality \leq\ means we are considering all points on and inside this circle.

Shading Regions

Shading regions is a critical part of graphing inequalities. The solution to an inequality like \(x^2 + (y+3)^2 \leq 16\) involves not just a single line or curve, but an area on the coordinate plane.

To find the solution, start by shading the entire circle area. This action represents all conceivable points where the inequality holds true i.e., \(x^2 + (y+3)^2 \leq 16\). Since the inequality includes \leq\, the border of the circle also counts as part of the solution area. By shading this area, we effectively visualize which x and y combinations fulfill the inequality's conditions.

Test Points

Using test points is a supportive method to verify whether certain areas on the coordinate plane satisfy an inequality. Select any point within the shaded region or on the boundary, and substitute its coordinates back into the inequality to verify the truth of the condition.

For instance, consider the point \(0, -3\) in this exercise. Substituting yields the expression \(0^2 + (y+3)^2 = 0 \leq 16\), proving it's correct. Similarly, test points like \(4, -3\) are crucial. Each point that satisfies the inequality confirms that the shaded region correctly represents the solution.

This step acts as a simple yet powerful tool to ensure that your graphical representation aligns with the mathematical definition provided by the inequality.