Problem 32 - Algebra 2

Question

Use the center and the radius to graph each circle. $$ (x+3)^{2}+(y-9)^{2}=49 $$

Step-by-Step Solution

Answer

The graph of the circle has a center at (-3, 9) with a radius of 7 units.

Step 1: Find the center

The center (h,k) of the circle can be found by identifying the constants in the equation. In this case, the center is at (-3, 9).

Step 2: Identify the radius

The radius of the circle, r, can be found by taking the square root of the constant on the right side of the equals sign in the equation. Since the equation contains \(r^2 = 49\), the radius is found to be \(r = \sqrt{49} = 7\) units.

Step 3: Draw the circle

Now that we have the center and the radius, the circle can be plotted on a graph. Start by drawing a point at the center (-3,9). Then, use a compass, or the graph's scales, to draw the circle at 7 units away from this point in all directions.

Key Concepts

Center of a Circle

When you encounter an equation of a circle in standard form \[(x-h)^2 + (y-k)^2 = r^2,\]the variables \(h\) and \(k\) represent the coordinates of the circle's center. In our example, the equation \[(x+3)^2 + (y-9)^2 = 49,\]we identify the center by comparing with the standard form.
Since the equation has \( (x+3)^2\), this implies \(x - (-3)\) was squared, making the \(x\)-coordinate of the center \(-3\). Similarly, from \((y-9)^2\), we deduce the \(y\)-coordinate is 9. Therefore, the center of the circle is at (-3, 9).
Understanding the center helps in planning how the circle is positioned in a graph, making it easier to visualize how far and in what direction the circle stretches from its center.

Radius of a Circle

The radius is another essential component in circle equations, determining how large or small the circle appears on a graph. In circle equations like \[(x-h)^2 + (y-k)^2 = r^2,\]the term \(r^2\) represents the radius squared.
In our specific equation, \((x+3)^2 + (y-9)^2 = 49\), we see that \(r^2 = 49\). To find the actual radius \(r\), you need to take the square root of \(49\), giving us \(r = 7\).
This means that from the circle's center, the edge extends 7 units out in all directions. Knowing the radius is vital for accurately drawing the circle, as it ensures the circle's size is consistent and proportional when graphed.

Graphing Circles

Graphing circles effectively combines your understanding of the circle's center and radius. Start by plotting the center of your circle, like the point (-3, 9), on the graph. This fixed point acts as an anchor.
With the radius in hand, use it to draw the circumference around the center.

• First, if you're using graph paper, count 7 units from the center in all cardinal directions: left, right, up, and down.
• Next, you can connect these points smoothly to form a circular shape.
• Consider using tools like a compass for precision or the graph's scales to measure 7 units accurately from the center in all directions.

Graphing circles may require a bit of practice, but once you've mastered locating your center and calculating the radius, drawing the circle becomes much more straightforward.