Question

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ x^{2}+y^{2}=4 $$

Short answer

The graph is a circle centered at the origin with a radius of 2, intercepting x-axis at (2, 0) and (-2, 0), and y-axis at (0, 2) and (0, -2).

Step-by-step solution

Identify the Shape of the Graph

The equation is of the form \( x^2 + y^2 = r^2 \), which represents a circle. Here, \( r^2 = 4 \), so \( r = 2 \). This means the circle is centered at the origin (0,0) and has a radius of 2.

Check for Symmetries

The equation is symmetric with respect to the x-axis, y-axis, and the origin because it has the same terms in \( x^2 \) and \( y^2 \). As both variables are squared, the graph is symmetric about both axes and through the origin.

Find the X-intercepts

To find the x-intercepts, set \( y = 0 \) in the equation: \[ x^2 + 0^2 = 4 \] Simplifying gives \( x^2 = 4 \) so \( x = \pm 2 \). Therefore, the x-intercepts are (2, 0) and (-2, 0).

Find the Y-intercepts

To find the y-intercepts, set \( x = 0 \) in the equation: \[ 0^2 + y^2 = 4 \] Simplifying gives \( y^2 = 4 \) so \( y = \pm 2 \). Therefore, the y-intercepts are (0, 2) and (0, -2).

Plot the Graph

Draw the Cartesian plane. Mark the intercepts (2,0), (-2,0), (0,2), and (0,-2). Since you are plotting a circle centered at the origin with a radius of 2, draw a circle passing through these points, maintaining equidistance from the origin.

Key concepts

Understanding X-Intercepts

To determine the points where a graph intersects the x-axis, known as x-intercepts, we can set the value of the y-coordinate to zero in the equation of the circle. For the equation of a circle in standard form, such as \[ x^2 + y^2 = r^2 \], this simplifies to identifying the real values of \( x \) that satisfy the equation when \( y = 0 \).
For our specific problem with the equation \( x^2 + y^2 = 4 \), set \( y = 0 \), resulting in \( x^2 = 4 \). To solve for \( x \), take the square root of both sides, giving you \( x = \pm 2 \). Thus, the x-intercepts are (2, 0) and (-2, 0).

• The x-intercepts help us understand where the circle crosses the x-axis.
• Since our circle is centered at the origin, the x-intercept calculations are consistent and straightforward.

Exploring Y-Intercepts

Y-intercepts occur when the graph of an equation crosses the y-axis. To find the y-intercepts of a circle's graph, set the x-coordinate to zero in the equation. For the circle described by \( x^2 + y^2 = r^2 \), we resolve the equation with \( x = 0 \).
In the case of the equation \( x^2 + y^2 = 4 \), substitute \( x = 0 \) to obtain \( y^2 = 4 \). Solving for \( y \) involves taking the square root of both sides, which yields \( y = \pm 2 \). The circle graph thus intersects the y-axis at (0, 2) and (0, -2).

• Finding y-intercepts involves similar methods as x-intercepts, emphasizing symmetry in mathematical calculations.
• The y-intercepts are important for sketching an accurate representation of the circle on a graph.

Symmetry in Graphs

Symmetry plays a crucial role in understanding the shape and nature of graphs, especially circles. For our equation \( x^2 + y^2 = 4 \), symmetry is easily observed because both \( x \) and \( y \) are squared, and the form involves similar terms. This specific format ensures symmetry about:

• The x-axis: If you replace \( y \) with \( -y \), the equation remains unchanged.
• The y-axis: Replacing \( x \) with \( -x \) does not alter the equation.
• The origin: Substituting \( x \) with \( -x \) and \( y \) with \( -y \) keeps the equation intact.

These symmetrical properties verify that our circle is perfectly balanced. The graph appears the same on both sides of each axis and through the center, enabling easier plotting and predicting where x- and y-intercepts will be.

Circle Equations Simplified

A circle's equation often appears in standard form as \( x^2 + y^2 = r^2 \), where \( r \) represents the radius. The beauty of this form is its simplicity and ease in identifying key features:

• **Center:** For the equation \( x^2 + y^2 = 4 \), the center is at (0,0). This is typical when no specific terms shift or translate the circle from the origin.
• **Radius:** Calculating the radius involves solving \( r^2 = 4 \), yielding a radius \( r = 2 \).
• **Uniform Shape:** The equation confirms that the curve is a circle, a perfectly round shape.

Understanding circle equations enhances graph plotting skills, aiding in determining intercepts and analyzing symmetrical properties with ease. Knowing these basics allows for quicker identification of the graph's critical points and general form.