Answer:
- y = x
- y = x - 1/4
- y = x - 1/2
Step-by-step 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.
