Final answer:
To sketch the curves x²=2y and y²=16x on the same axes, rewrite the equations in terms of y and plot the curves. To find the coordinates of the points of intersection, set the two equations equal to each other and solve for x.
Step-by-step explanation:
To sketch the curves x²=2y and y²=16x on the same axes, we can start by rewriting the equations in terms of y:
x²=2y becomes y = (1/2)x²
y²=16x becomes y = ±sqrt(16x) = ±4sqrt(x)
Now we can plot the two curves on the same axes. The curve y=(1/2)x² is a parabola that opens upward and is symmetric with respect to the y-axis. The curve y=±4sqrt(x) is made up of two branches, one opening to the right and the other opening to the left.
To find the coordinates of the points of intersection, we need to set the two equations equal to each other:
(1/2)x² = ±4sqrt(x)
By squaring both sides and solving for x, we can find the x-coordinates of the intersection points. Then we can substitute those values back into either equation to find the corresponding y-coordinates.