Problem 30 - Algebra 2

Question

Use the center and the radius to graph each circle. $$ (x-1)^{2}+(y+3)^{2}=16 $$

Step-by-Step Solution

Answer

The center of the circle is (1,-3), and it has a radius of 4. The circle is plotted on the graph accordingly.

Step 1: Identify the center and the radius

From the equation \((x-1)^2+(y+3)^2=16\), the value of (h,k) can be identified where h=1 and k=-3. This is because the formula is \((x-h)^2+(y-k)^2 = r^2\). The value of r(radius) can be inferred from the equation which is √16=4.

Step 2: Plotting the center

The center of the circle is at the point (1,-3). This point is plotted on the graph.

Step 3: Draw the circle using the radius

From the center (1,-3), count outwards 4 units in all directions (up, down, right and left) due to radius being 4. Plot these points. Draw the circle through these points to complete the graph.

Key Concepts

Center-Radius Form

A widely used method to represent circles in math is the center-radius form. It is a way of expressing the equation of a circle to easily identify its center and radius. It typically looks like \[(x - h)^2 + (y - k)^2 = r^2\] where:

• \( h \) and \( k \) are the coordinates of the center of the circle.
• \( r \) represents the radius.

To read it, you just need to match these components with the equation you're dealing with. In our exercise, the equation \[(x-1)^2+(y+3)^2=16\]can be compared with the general form. Here, you can see:- The center is at \((h, k) = (1, -3)\),- The radius \( r \) is \( \sqrt{16} = 4 \). Understanding the center-radius form allows you to quickly graph the circle or make modifications.

Equation of a Circle

An equation of a circle defines all the points that make up the circle's boundary. The general formula, specifically when using the center-radius form, \[(x - h)^2 + (y - k)^2 = r^2\]reveals crucial information about the circle:

• The left-hand side, \((x - h)^2 + (y - k)^2\), measures how far any point \((x, y)\) is from the center \((h, k)\).
• When this distance equals \( r^2 \), the point lies on the circle.

For example, consider \[(x - 1)^2 + (y + 3)^2 = 16\]This tells us:

• The circle has a center at \((1, -3)\).
• It has a radius of \(4\) {since \(\sqrt{16} = 4\)}.

This mathematical framework allows us to visualize and map circles consistently.

Plotting Coordinates

Plotting coordinates involves locating specific points on a graph, which is critical in graphing circles. For the given equation \[(x-1)^2+(y+3)^2=16\]we start with the center \((1, -3)\) by marking it on the graph. This point acts as the reference for drawing the circle. Once the center is set:

• From the center, count \(4\) units (the radius) upwards, downwards, rightwards, and leftwards to find boundary points.
• These points are key: they ensure the circle remains equidistant from the center in all directions.

Finally, connect these boundary points with a smooth curve to complete the circle. This method underscores how coordinate plotting helps visualize mathematical relationships like circles effectively.