15. Which statement describes the graph and orientation of the parametric equations x = 9 cos and y = 9sin t' ?
The graph is a circle centered at the origin with a radius of 9, and the orientation is counterclockwise.
Well, let me put on my clown hat and bring out the colorful balloons to describe this graph for you! Imagine a circus ring with a juggling performer. Now, the graph of the parametric equations x = 9cos(t) and y = 9sin(t) forms a beautiful circular shape, just like a hula hoop!
As for the orientation, it's like watching a circus performer twirl those hula hoops on their arms. The orientation is counterclockwise, just like the juggling act that defies gravity! So, in summary, the graph is a circus-worthy circle, and it spins counterclockwise like a skilled performer in the big top!
The parametric equations given are x = 9cos(t) and y = 9sin(t), where t is the parameter.
The equation x = 9cos(t) represents the x-coordinate of a point on the x-y plane, and y = 9sin(t) represents the y-coordinate of the same point.
The graph of these equations creates a curve in the x-y plane. Since both x and y are multiplied by 9, the curve is scaled by a factor of 9 compared to the standard cos(t) and sin(t) curves.
The graph of the parametric equations represents a circle with a radius of 9 units, centered at the origin (0,0) of the coordinate system. As the parameter t increases, the point moves counterclockwise around the circle.
Hence, the graph is a circle and the orientation is counterclockwise.
To answer this question, we need to understand the given parametric equations and their relationship to the graph.
The parametric equations given are:
x = 9 cos(t)
y = 9 sin(t)
These equations represent a parametric curve, where the x and y coordinates of a point on the curve are determined by the parameter t. The values of t control the position of the point on the curve.
In this case, the equations describe a circular path with a radius of 9 units. The parameter t represents the angle at which a point lies on the circle.
The cosine function (9 cos(t)) determines the x-coordinate of each point on the circle, while the sine function (9 sin(t)) determines the y-coordinate. By varying t from 0 to 2π (a complete revolution in radians), we can trace out the entire circle.
Since both the x and y coordinates are multiplied by 9, the circle is scaled up by a factor of 9.
To determine the orientation of the graph, we must look at the trigonometric functions used. The cosine function (cos t) is an even function, which means it is symmetric about the y-axis. So, the x-values will be positive or negative depending on the sign of cos(t).
Similarly, the sine function (sin t) is an odd function, which means it is symmetric about the origin. So, the y-values will be positive or negative depending on the sign of sin(t).
Based on these properties, we can conclude that the graph of the parametric equations x = 9 cos(t), y = 9 sin(t) is a circle centered at the origin with a radius of 9 units. The orientation of the graph is counterclockwise, starting from the positive x-axis.