Question

What is a linear equation that intersects y = x2 at TWO points?

Answer 1

Answer:

• y = x intersects at 2 points
• y = -2x-1 intersects at 1 point
• y = x-2 intersects at 0 points

Explanation:

Any line with a y-intercept greater than 0 will intersect y = x² at two points, as will any line with a non-zero slope and a y-intercept of zero. I have chosen the line y = x to answer this question.

Any tangent line will intersect y = x² at one point. A tangent line will have slope 2x at the point (x, x²). Here, we have chosen to write the equation for the tangent at (x, y) = (-1, 1). That line has equation y = -2x-1.

Any line with a negative y-intercept and a sufficiently small slope will not intersect the curve y = x². (The slope needs to have a magnitude less than double the magnitude of the y-intercept.) Here, we have chosen a line with slope 1 and y-intercept -2: y = x-2.