the domain of a quadratic function is all real numbers and the range is y ≤ 2. how many x - intercepts does…

Explanation:

Step1: Recall quadratic - function properties

A quadratic function is of the form $y = ax^{2}+bx + c$, and its graph is a parabola. Given the range $y\leq2$, the parabola opens downwards ($a\lt0$) and has a maximum value of $y = 2$.

Step2: Analyze x - intercepts

The x - intercepts are the solutions of the equation $ax^{2}+bx + c=0$, which are found by setting $y = 0$. Since the maximum value of the function is $y = 2$ and the parabola opens downwards ($y\leq2$), the line $y = 0$ (the x - axis) can intersect the parabola at 0, 1, or 2 points. If the vertex of the parabola is above the x - axis and the parabola opens downwards, it can cross the x - axis at two points. If the vertex is on the x - axis, it has one x - intercept. If the vertex is above the x - axis and the parabola is too "narrow" or "wide" in a way that it never reaches the x - axis, it has 0 x - intercepts. But since the range allows values less than or equal to 2 and the x - axis ($y = 0$) is in the set of possible $y$ - values for the function, the number of x - intercepts can be 0, 1, or 2.

Answer:

0, 1, or 2