Consider a parabola P that is congruent to y=x^2, opens upward,and has its vertex at (2,-4). Now find the equation of a new parabola that results if P is:
A. stretched vertically by a factor of 5.
B. compressed by a factor of 2.
C. translated 2 units to the left.
D. translated 3 units up.
E. reflected about the x-axis and translated 2 units to the right and 4 units down.
A. To stretch the parabola vertically by a factor of 5, we can multiply the equation y = x^2 by 5. So the new equation will be y = 5x^2.
B. To compress the parabola by a factor of 2, we can multiply the equation y = x^2 by 1/2. So the new equation will be y = (1/2)x^2.
C. To translate the parabola 2 units to the left, we can subtract 2 from the x-coordinate of the vertex. Since the original vertex is (2,-4), the new vertex will be (0,-4). The equation of the translated parabola will still be y = x^2.
D. To translate the parabola 3 units up, we can add 3 to the y-coordinate of the vertex. Since the original vertex is (2,-4), the new vertex will be (2,-1). The equation of the translated parabola will still be y = x^2.
E. To reflect the parabola about the x-axis, we can multiply the equation y = x^2 by -1. So the new equation will be y = -x^2. To translate the parabola 2 units to the right, we can add 2 to the x-coordinate of the vertex. Since the original vertex is (2,-4), the new vertex will be (4,-4). Finally, to translate the parabola 4 units down, we can subtract 4 from the y-coordinate of the vertex. The new vertex will be (4,-8). The equation of the reflected and translated parabola will be y = -x^2 - 8.