From a Line Meeting a Parabola to Absolute Value as Distance
Systems of quadratic equations, systems of inequalities, and equations and inequalities with absolute value — explore all three, split into tabs, with sliders.
Set the line y=mx+k equal to the parabola y=x², and you get x²=mx+k, that is, the quadratic equation x²−mx−k=0. Depending on the sign of this equation's discriminant D=m²+4k, the two graphs meet at two points (D>0), one point (D=0), or not at all (D<0).
Line's slope m1
Line's y-intercept k2
A system of inequalities asks you to find x-values that satisfy two or more inequalities at once. Draw each inequality's solution on a number line, and the overlapping part is exactly the solution to the system.
Inequality ① boundary a-2
Inequality ② boundary b3
The absolute value |x−a| means "the distance between x and a." |x−a|=b means "the points exactly a distance of b from a" (two points), |x−a|<b means "the range closer to a than b," and |x−a|>b means "the range farther from a than b."