Question

List the intercepts and test for symmetry the graph of $$ (x+12)^{2}+y^{2}=16 $$

Short answer

x-intercepts: (-8,0) and (-16,0). No y-intercepts. Symmetric with respect to the x-axis only.

Step-by-step solution

Identifying x-intercepts

To find the x-intercepts, set \( y = 0 \) in the equation and solve for \( x \). So, the equation becomes: \( (x+12)^2 + 0^2 = 16 \). This simplifies to \( (x+12)^2 = 16 \). Taking the square root of both sides, \( x+12 = 4 \) or \( x+12 = -4 \). So, \( x = -8 \) or \( x = -16 \). The x-intercepts are \((-8, 0) \) and \((-16, 0) \).

Identifying y-intercepts

To find the y-intercepts, set \( x = 0 \) in the equation and solve for \( y \). So, the equation becomes: \( (0+12)^2 + y^2 = 16 \). This simplifies to \( 144 + y^2 = 16 \). Subtract 144 from both sides to get: \( y^2 = -128 \). Since the square root of a negative number is not real, there are no y-intercepts.

Testing for symmetry with respect to the y-axis

To test for symmetry with respect to the y-axis, replace \( x \) with \( -x \) in the equation and see if we get an equivalent equation.Starting with the original equation, \( (x+12)^2 + y^2 = 16 \). Replacing \( x \) with \( -x \) gives: \( (-x+12)^2 + y^2 = 16 \) or \( (12-x)^2 + y^2 = 16 \). This is not equivalent to the original equation, so there is no symmetry with respect to the y-axis.

Testing for symmetry with respect to the x-axis

To test for symmetry with respect to the x-axis, replace \( y \) with \( -y \) in the equation and see if we get an equivalent equation.Starting with the original equation, \( (x+12)^2 + y^2 = 16 \). Replacing \( y \) with \( -y \) gives: \( (x+12)^2 + (-y)^2 = 16 \). This equation simplifies back to \( (x+12)^2 + y^2 = 16 \), which is the original equation. Therefore, there is symmetry with respect to the x-axis.

Testing for symmetry with respect to the origin

To test for symmetry with respect to the origin, replace both \( x \) with \( -x \) and \( y \) with \( -y \) in the equation and see if we get an equivalent equation.Starting with the original equation, \( (x+12)^2 + y^2 = 16 \). Replacing \( x \) with \( -x \) and \( y \) with \( -y \) gives: \( (-x+12)^2 + (-y)^2 = 16 \) or \( (12-x)^2 + y^2 = 16 \). This is not equivalent to the original equation, so there is no symmetry with respect to the origin.

Key concepts

x-intercepts

X-intercepts are points where the graph of an equation crosses the x-axis. To find them, we set the y-value to zero and solve for x. For the given equation \[ (x + 12)^2 + y^2 = 16 \], we set y to zero, turning the equation into: \[ (x + 12)^2 = 16 \]. Taking the square root of both sides, we get two possible solutions: \[ x + 12 = 4 \] or \[ x + 12 = -4 \]. This simplifies to \[ x = -8 \] and \[ x = -16 \]. Therefore, the x-intercepts are \[ (-8, 0) \] and \[ (-16, 0) \]. Identifying intercepts helps in graphing and understanding the behavior of the function.

y-intercepts

Y-intercepts are points where the graph crosses the y-axis. To find them, we set the x-value to zero and solve for y. Using the equation from before: \[ ( x + 12 )^2 + y^2 = 16 \], substituting \[ x = 0 \] gives: \[ (0 + 12)^2 + y^2 = 16 \]. This simplifies to \[ 144 + y^2 = 16 \]. Subtract 144 from both sides, and we get \[ y^2 = -128 \]. Since the square root of a negative number is not real, this equation has no real y-intercepts. This means the graph does not cross the y-axis. This tells us more about the graph's position and behavior.

Symmetry in Graphing

Understanding symmetry in graphing makes plotting easier and reveals important properties of the function. There are three main types of symmetry we test:

• Y-Axis Symmetry: Replace x with -x in the equation and see if the equation remains the same. For our equation, \[ (x + 12)^2 + y^2 = 16 \], replacing x with -x gives \[ (12 - x)^2 + y^2 = 16 \], which is not equivalent. Thus, it does not have y-axis symmetry.
• X-Axis Symmetry: Replace y with -y in the equation. For our equation, \[ (x + 12)^2 + y^2 = 16 \], replacing y with -y gives \[ (x + 12)^2 + (-y)^2 = 16 \], which simplifies back to the original. Thus, it has x-axis symmetry.
• Origin Symmetry: Replace both x with -x and y with -y. For our equation, \[ (x + 12)^2 + y^2 = 16 \], replacing both variables gives \[ (12 - x)^2 + y^2 = 16 \], which is not the original. Thus, it does not have origin symmetry.

Recognizing these symmetries helps to understand the function's shape and behavior.

Conic Sections

Conic sections are shapes created by slicing a double cone. These shapes include circles, ellipses, parabolas, and hyperbolas. Our original equation, \[ (x + 12)^2 + y^2 = 16 \], represents a circle. A circle's general form is \[ (x - h)^2 + (y - k)^2 = r^2 \], where (h, k) is the center, and r is the radius. Here, we can rewrite the equation as \[ (x + 12)^2 + y^2 = 4^2 \]. The center is \[ (-12, 0) \], and the radius is \[ 4 \]. Circles are fundamental in geometry and have unique properties. Understanding the center and radius of a circle helps in graphing and analyzing its behavior.