When graphing two quadratic functions on the same xy-plane, the number of intersection points can vary depending on the specific functions. A quadratic function can be represented in the general form \( f(x) = ax^2 + bx + c \).
The intersection points of two quadratic functions, say \( f(x) = a_1x^2 + b_1x + c_1 \) and \( g(x) = a_2x^2 + b_2x + c_2 \), occur where \( f(x) = g(x) \). This can be rearranged to form a new quadratic equation:
\[ (a_1 - a_2)x^2 + (b_1 - b_2)x + (c_1 - c_2) = 0 \]
The number of solutions (and hence intersection points) to this equation depends on the values of the coefficients.
The possible scenarios for the number of intersection points (real solutions) are:
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0 Intersection Points: The quadratics do not touch or intersect at all (they could be completely separate or one could be entirely above/below the other).
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1 Intersection Point: The quadratics are tangent to each other, meaning they touch at one point.
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2 Intersection Points: The quadratics intersect at two points, which is the typical case when they cross each other.
Thus, the number of intersection points between two quadratic functions can be 0, 1, or 2.