A circle measures 360 degrees. If this circle were marked with numbers like a clock, every number would represent 30 degrees farther from 0 and closer to 360 degrees. If an angle marker typically found at 11:00 were to rotate to the place normally marked for 4:00, what is the degree measure of the angle formed moving clockwise?
options are 120 degrees, 240 degrees, 150 degrees, and 330 degrees
From 11:00 to 4:00, we are moving 5 "spaces" clockwise on the circle. Since each "space" represents an increase of 30 degrees, the angle formed by moving clockwise is $5 \cdot 30 = \boxed{150}$ degrees.
the function f(x) = x^2 - 1 is reflected over the y-axis. choose the equation that correctly identifies g(x).
g(x) -x^2 - 2
g(x)= x^2 - 1
g(x)= -x^2 + 1
g(x)= x^2 + 1
Reflecting a function over the y-axis involves changing the signs of the x-coordinates. Since the original function is $f(x) = x^2 - 1$, reflecting it over the y-axis would result in the x-coordinates becoming negative. Therefore, the correct equation for the reflected function $g(x)$ is $\boxed{g(x) = -x^2 - 1}$.
the point P (-1,2) is rotated to become P' (2,1). describe the rotation by degree and direction
To describe the rotation, we need to find the angle and direction of rotation.
First, let's plot the points P (-1,2) and P' (2,1) on a coordinate plane.
Now, let's find the angle of rotation. We can use the slope of the line connecting the points P and P' to find the angle of rotation. The slope of this line is calculated as:
slope = (change in y-coordinates)/(change in x-coordinates) = (1-2)/(2-(-1)) = -1/3
We can also find this slope by calculating the tangent of the angle of rotation:
tan(angle of rotation) = (1/3)
Using an inverse tangent (or arctan) function, we find that the angle of rotation is approximately -18.43 degrees.
From the coordinates of P and P', we can see that P' is to the right and slightly down from P. Therefore, the rotation is in a clockwise direction.
In conclusion, the rotation is approximately -18.43 degrees in a clockwise direction.
the point P (-1,2) is rotated to become P' (2,1). describe the rotation by degree and direction
180 rotation
-90 rotation
-270 rotation
90 rotation
To describe the rotation, we need to find the angle and direction of rotation.
The transformation that takes the point P (-1,2) to P' (2,1) is a rotation of 180 degrees about the origin (0,0).
To find the angle of rotation, we can calculate the slope of the line connecting the origin and P' (2,1).
slope = (change in y-coordinates)/(change in x-coordinates) = (1-0)/(2-0) = 1/2
We can also find this slope by calculating the tangent of the angle of rotation:
tan(angle of rotation) = (1/2)
Using an inverse tangent (or arctan) function, we find that the angle of rotation is approximately 26.57 degrees.
Since the angle of rotation is 180 degrees, the rotation itself is a half-turn. The direction of the rotation can be described as "clockwise" or "counterclockwise", but since a half-turn can be considered as either direction, we can say the direction is ambiguous.
In conclusion, the rotation is a 180 degree rotation (half-turn) about the origin and the direction is ambiguous.
the point Q(-5,-9) is rotated -270 degrees about the origin. select the location of Q'.
Q'(9,-5)
Q'(-5,9)
Q'(9,5)
Q'(5,9)
To find the location of Q' after a rotation of -270 degrees about the origin, we can use the following formula for rotating a point (x, y) counterclockwise about the origin:
Q' = (x*cos(-270) - y*sin(-270), x*sin(-270) + y*cos(-270))
Since (-270 degrees) is a counterclockwise rotation, we will use positive trigonometric functions (cosine and sine).
Using the coordinates of Q(-5,-9), the formula becomes:
Q' = (-5*cos(-270) - (-9)*sin(-270), -5*sin(-270) + (-9)*cos(-270))
Simplifying the expression, we get:
Q' = (-5*0 - (-9)*(-1), -5*(-1) + (-9)*0)
= (0 + 9, 5 + 0)
= (9, 5)
Therefore, Q' is located at (9,5). The correct option is Q'(9,5).