Question

Write a linear equation that intersects y = x2 at two points. Then write a second linear equation that intersects y = x2 at just one point, and a third linear equation that does not intersect y = x2. Explain how you found the linear equations.

Answer 1

A linear equation that intersects at two points would be y = 1x. A linear equation that only has one intersect point would be y=0. finally, a linear equation that has no intersect point would be y = x - 4. I could these by plotting them on a graph and finding out that these will work.

Explanation:

Answer 2

1. y = x
2. y = x - 1/4
3. y = x - 1/2

Explanation:

1. Any linear equation that describes a line with non-zero slope through the vertex of the parabola will intersect the parabola at two points (the vertex being one of them). A simple equation for such a line is y=x.

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2. Differentiating the equation, you find that the slope of the curve y = x^2 is 2x, so if we choose a line with a slope of 1, it will go through the point on the curve with x-value equal to 1/2. The y-value at that point is y = (1/2)^2 = 1/4, so the y-intercept of the line must be -1/4.

The line that intersects the curve at one point (1/2, 1/4) is tangent at that point. It has equation y = x -1/4.

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3. Any line with the same slope as the tangent line, but a more negative y-intercept, will not intersect the parabola at all. Such a line is y = x -1/2.

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Truth be told, I found the line y = x -1/2 did not intersect the parabola at all when I thought I was writing the equation for the tangent line. It was an answer to part of your question, just not the part I originally intended.