In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. center (0, 4), radius 4

Verified step by step guidance

Identify the given information: the center of the circle is at the point (0, 4) and the radius is 4.

Recall the center-radius form of a circle's equation: \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius.

Substitute the center coordinates \((h, k) = (0, 4)\) and the radius \(r = 4\) into the formula: \( (x - 0)^2 + (y - 4)^2 = 4^2 \).

Simplify the equation: \( x^2 + (y - 4)^2 = 16 \).

To graph the circle, plot the center at (0, 4) on the coordinate plane, then draw a circle with radius 4 units extending in all directions from the center.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Equation of a Circle in Center-Radius Form

The center-radius form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. This form directly shows the circle's location and size, making it easier to graph and analyze.

Identifying the Center and Radius

To write the equation of a circle, you must know its center coordinates (h, k) and radius r. The center is the fixed point from which all points on the circle are equidistant, and the radius is that constant distance.

Graphing a Circle

Graphing a circle involves plotting its center and using the radius to mark points in all directions. Drawing a smooth curve through these points forms the circle, helping visualize its position and size on the coordinate plane.

Recommended video:

5:18

Circles in Standard Form

Related Practice

Textbook Question

Determine whether each relation defines a function. {(9,-2),(-3,5),(9,1)}

Textbook Question

For the points P and Q, find (a) the distance d(P, Q) and (b) the coordinates of the mid-point M of line segment PQ. P(-5,-6), Q(7,-1)

Textbook Question

Determine the intervals of the domain over which each function is continuous.

Textbook Question

Determine whether each relation defines a function. {(-3,1),(4,1),(-2,7)}

views

Textbook Question

Let ƒ(x)=x²+3 and g(x)=-2x+6. Find each of the following. (ƒg)(4)

Textbook Question

Write an equation for each line described. Give answers in standard form for Exercises 11–20 and in slope-intercept form (if possible) for Exercises 21–32. through (-8,4), undefined slope

views