Problem 46 - College Algebra Essentials

Question

give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. $$ (x+1)^{2}+(y-4)^{2}=25 $$

Step-by-Step Solution

Answer

The center of the circle is (-1, 4), and the radius is 5. The domain of the function on the graph is [-6, 4], and the range is [-1, 9].

Step 1: Identify the center of the circle

In the given equation \((x + 1)^2 + (y - 4)^2 = 25\), comparing it to the general form, the center of the circle \((h, k)\) can be found by changing the sign of the constants attached to \(x\) and \(y\). Therefore, the center is \((-1, 4)\).

Step 2: Determine the radius of the circle

The radius of the circle can be found from the equation by taking the square root of the constant term on the right side. In this case, \(r^2 = 25\), so the radius \(r\) is \(5\).

Step 3: Identify the domain and range by graphing the circle

A graph of this circle would reveal a circle centered at point (-1, 4) with a radius of 5 units. The domain of a circle (possible values of x) is the set of all x-values from the x-coordinate of the center minus the radius to the x-coordinate of the center plus the radius. Consequently, the domain is from \(h - r = -1 - 5 = -6\) to \(h + r = -1 + 5 = 4\), which is [-6, 4]. Similary, the range (possible values of y) is from \(k - r = 4 - 5 = -1\) to \(k + r = 4 + 5 = 9\), which is [-1, 9].

Key Concepts

Center of a Circle

The center of a circle in math is the point from which all points on the circle are equidistant. It's like the heart of your circle, making it essential to understand its location.
To find the center using the circle equation, \[ (x + 1)^2 + (y - 4)^2 = 25 \] we compare it to the standard form: \[ (x - h)^2 + (y - k)^2 = r^2 \] where

• \(h\) is the x-coordinate of the center
• \(k\) is the y-coordinate of the center

In our function,

• \(h = -1\)
• \(k = 4\)

These values indicate the center of your circle is at point \((-1, 4)\).
Remember to switch the signs of the numbers in the equation to find \((h, k)\). This can often trip up people, but with practice, it becomes straightforward!

Radius Calculation

The radius of a circle is the distance from the center to any point on its edge. It's like the 'reach' of the circle.
To determine the radius, look at the circle equation again: \[ (x + 1)^2 + (y - 4)^2 = 25 \] You notice the number 25 on the right side? This is actually \( r^2 \), where \( r \) is the radius.
To find the radius, take the square root of 25, leading us to:

• \(r = \sqrt{25} = 5\)

Voila! The radius of your circle is 5. It's crucial always to find the square root of the number to get the actual 'reach' from the center.

Domain and Range of a Circle

When graphing a circle, its domain and range describe where the circle fits on the xy-plane.
The **domain** indicates the complete set of x-values (horizontal span). For our circle centered at \((-1, 4)\) with a radius of 5, calculate:

• Lower x-bound: \(h - r = -1 - 5 = -6\)
• Upper x-bound: \(h + r = -1 + 5 = 4\)

Therefore, the domain is \([-6, 4]\).
The **range** specifies all y-values (vertical span) the circle covers:

• Lower y-bound: \(k - r = 4 - 5 = -1\)
• Upper y-bound: \(k + r = 4 + 5 = 9\)

So, the range becomes \([-1, 9]\).
Plotting these on a graph would show how the circle spans horizontally and vertically. Grasping domain and range ensures you can visualize circles correctly on coordinate planes!