To determine the relationship between the two equations, we can analyze their graphs and the points where they intersect.
The first equation, y = 2x^2, is a parabola that opens upwards. It will have its vertex at the origin (0, 0) and will be symmetric about the y-axis.
The second equation, y = 2x, is a straight line with a slope of 2 that passes through the origin as well. This means it will intersect the parabola at certain points.
To find the points of intersection, we can set the two equations equal to each other:
2x^2 = 2x
By simplifying this, we can factor out the common terms:
2x^2 – 2x = 0
2x(x – 1) = 0
This gives us the solutions:
x = 0 and x = 1
Now we can find the corresponding y-values for these x-values:
For x = 0:
y = 2(0) = 0
For x = 1:
y = 2(1) = 2
Thus, the points of intersection are (0, 0) and (1, 2).
From this analysis, we can conclude that:
- The line intersects the parabola at two points: (0, 0) and (1, 2).
- For values of x between 0 and 1, the line is above the parabola, while for values of x greater than 1, the parabola begins to rise more steeply than the line.
Therefore, the true statement about the relationship between these two equations is that the line y = 2x intersects the parabola y = 2x^2 at two points, namely (0, 0) and (1, 2).