To perform a 90° rotation clockwise around the origin, you take the coordinates of the point A(x, y) and transform them to A'(y, -x). Since 180° and 270° are both "steps" of 90°, we can do this in succession and achieve our goal.
1) (5, 2) 90° = (2, -5) [(y, -x)]
2) (5, 2) 180° = two 90° turns = (2, -5) rotated 90° = (-5, -2)
3) (5, 2) 270° = three 90° turns = (-5, -2) rotated 90° = (-2, 5)°
4) (-5, 2) 90° = (y, -x) = (2, 5)
5) (-5, 2) 180° = two 90° turns = (2, 5) rotated 90° = (5, -2)
6) (-5, 2) 270° = three 90° turns = (5, -2) rotated 90° = (-2, -5)
7) (-2, 5) 90° = (y, -x) = (5, 2)
8) (5, -2) 180° = (y, -x) with another 90° turn = (-2, -5) rotated 90° = (-5, 2)
Answer:
105 ways
Step-by-step explanation:
Because the order of selection does not matter, we know that we will be using a combination instead of a permutation.
₁₅C₂ = 15! / (2! * 13!)
= 15 * 14 * ... * 2 * 1 / 2 * 1 * 13 * 12 * ... * 2 * 1
= 15 * 14 / 2 * 1 (The 13! cancels out)
= 105
Answer:
When kicked, the height of the ball is 1 feet. The highest point for the ball's trajectory is 12 feet.
Step-by-step explanation:
We are given that the height is given by 
. The distance between the ball to the point where it was kicked is 0 right at the moment the ball was kicked. So, the height of the ball, when it was kicked is f(0) = -0.11*0 + 2.2*0 +1 = 1.
To determine the highest point, we will proceed as follows. Given a parabola of the form x^2+bx + c, we can complete the square by adding and substracting the factor b^2/4. So, we get that
.
In this scenario, the highest/lowest points is ![c-\frac{b^2}{4}[/tex} (It depends on the coefficient that multiplies x^2. If it is positive, then it is the lowest point, and it is the highest otherwise). Then, we can proceed as follows. [tex] f(x) = -0.11x^2+2.2x+1 = -0.11(x^2-20x)+1](https://tex.z-dn.net/?f=c-%5Cfrac%7Bb%5E2%7D%7B4%7D%5B%2Ftex%7D%20%28It%20depends%20on%20the%20coefficient%20that%20multiplies%20x%5E2.%20If%20it%20is%20positive%2C%20then%20it%20is%20the%20lowest%20point%2C%20and%20it%20is%20the%20highest%20otherwise%29.%20%3C%2Fp%3E%3Cp%3EThen%2C%20we%20can%20proceed%20as%20follows.%20%3C%2Fp%3E%3Cp%3E%5Btex%5D%20f%28x%29%20%3D%20-0.11x%5E2%2B2.2x%2B1%20%3D%20-0.11%28x%5E2-20x%29%2B1)
We will complete the square for 
. In this case b=-20, so
We can distribute -0.11 to the number -100, so we can take it out of the parenthesis, then
So, the highest point in the ball's trajectory is 12 feet.
