write a linear equation that intersects y=x^2 at two points. Then write a second linear equation that intersects y=x^2 at just one point, and a third linear that does not intersect y=x^2. Explain how you found the linear equations.

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10-09-2018
Mathematics

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write a linear equation that intersects y=x^2 at two points. Then write a second linear equation that intersects y=x^2 at just one point, and a third linear that does not intersect y=x^2. Explain how you found the linear equations.

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Аноним Аноним · Answer 1 · 2018-09-10T11:39:38+02:00

We know that [tex] y = x^2 [/tex] is a parabola, concave up, with vertex in the origin [tex] (0,0) [/tex]

So, we may use three horizontal lines for our purpose: any horizontal line above the x axis will intersect the parabola twice. The x axis itself intersects the parabola once on the vertex, while any horizontal line below the x axis won't intercept the parabola.

Here's the examples:

The horizontal line [tex] y = 4 [/tex] intercepts the parabola twice: the system [tex] y = x^2,\ y = 4 [/tex] is solved by [tex] x^2=4 \implies x = \pm 2 [/tex]
The horizontal line [tex] y=0 [/tex] intercepts the parabola only once: the system is [tex] y=x^2,\ y=0 [/tex] which yields [tex]x^2=0\implies x=0 [/tex] which is a repeated solution
The horizontal line [tex] y=-5 [/tex] intercepts the parabola only once: the system is [tex] y=x^2,\ y=-5 [/tex] which yields [tex]x^2=-5[/tex] which is impossible, because a squared number can't be negative.