Question

Write the equation of each circle in standard form. Graph. center at (0,0)\(;\) radius \(=7\)

Short answer

The equation of the circle in standard form is \( x^2 + y^2 = 49 \) and it can be graphed with a center at (0,0) and a radius of 7 units.

Step-by-step solution

Write the standard form equation of a circle

The standard form of a circle's equation with a center at \(h, k\) and radius \(r\) is given by \( (x - h)^2 + (y - k)^2 = r^2 \). Since the center is at (0,0), that means \(h = 0\) and \(k = 0\).

Substitute the center and radius into the equation

Replace \(h, k\) with \(0, 0\) and \(r\) with \(7\) to get the specific equation for our circle: \( (x - 0)^2 + (y - 0)^2 = 7^2 \).

Simplify the equation

Simplify the equation by squaring \(7\) to get \(49\) and removing the zero subtractions to get the final standard form equation of the circle: \( x^2 + y^2 = 49 \).

Graph the circle

To graph the circle, plot the center at (0,0) and use a compass or a circular drawing tool to draw a circle with a radius of \(7\) units around the center.

Key concepts

Circle Graphing

Graphing a circle on a coordinate plane can be a surprisingly straightforward process if approached methodically. The graph of a circle centers on identifying its core elements – the center and radius. With the coordinates of the center \( (h,k) \) and the radius \( r \) known, a visual representation can be drawn.

In our example, the center of the circle is at the origin, or \( (0,0) \), which is where we begin our sketch. From the center, measure out the length of the radius in all directions. In this exercise, with a radius of \( 7 \), this means marking points that are \( 7 \) units away from the origin in the horizontal and vertical directions, as well as at diagonal angles. Then, using a compass or drawing tool, a smooth curve connecting these points is drawn to complete the circle. This basic approach enables students to accurately visualize and sketch the given circle, improving their understanding of circle graphing.

Algebraic Equations

An algebraic equation represents a mathematical statement where two expressions are set equal to each other. These equations can describe a myriad of shapes and relationships, with circle equations being a particular instance.

The general standard form of the equation for a circle in algebra is \( (x - h)^2 + (y - k)^2 = r^2 \), as seen in the exercise. Here, \( h \) and \( k \) denote the x and y coordinates of the circle’s center, while \( r \) stands for its radius. By plugging in the specific values for \( h \) and \( k \) (in this case, zero), and the radius \( r \), we transform this general formula into an equation that concretely describes the circle in question.
When students master this algebraic form, they fortify their foundation for solving a variety of problems involving circles, and extend this knowledge to understanding more complex geometric shapes.

Radius of a Circle

The radius of a circle is a line segment that extends from the center of the circle to any point on its edge. It is a fundamental concept as it determines the size of the circle and appears in many geometrical formulas.

In the provided example, the radius is given as \( 7 \) units. This radius is not only central to graphing the circle, as previously described, but also to computing the circle's area and circumference. The standard form equation incorporates the radius squared, \( r^2 \) – in our case, \( 49 \). This power of two relationship reflects the nature of a circle's area as well – the area being \( \pi r^2 \), which once again underscores the pivotal role of the radius in various circle-related calculations. Understanding the role of the radius as a bridge between linear dimensions and area can significantly enhance a student's grasp on the spatial aspects of geometry.