The domain of a quadratic function is all real numbers and the range is $y \leq 2$. How many $x$-intercepts does the function have?
Answer (1)
The range y ≤ 2 indicates a downward-opening parabola with a vertex at y = 2 .
Since the vertex is above the x-axis, the parabola intersects the x-axis at two points.
Therefore, the quadratic function has two x-intercepts.
The number of x-intercepts is 2 .
Explanation
Understanding the Problem We are given a quadratic function with a domain of all real numbers and a range of y ≤ 2 . Our goal is to determine the number of x-intercepts this function has.
Analyzing the Range Since the range of the quadratic function is y ≤ 2 , this means the parabola opens downwards, and the vertex of the parabola is at the point where y = 2 . This also tells us that the maximum value of the quadratic function is 2.
Determining the Number of x-intercepts A quadratic function can have 0, 1, or 2 x-intercepts. Since the parabola opens downwards and its vertex (the highest point) is at y = 2 , which is above the x-axis ( y = 0 ), the parabola must intersect the x-axis at two distinct points.
Conclusion Therefore, the quadratic function has two x-intercepts.
Examples
Imagine you're throwing a ball, and its path forms a parabola. The highest point the ball reaches is like the vertex of the quadratic function. If the highest point is above the ground (x-axis), the ball will come down and hit the ground at two different points. This is similar to our quadratic function intersecting the x-axis at two points, giving us two x-intercepts. Understanding quadratic functions helps predict the trajectory of objects in motion, design efficient structures, and optimize various real-world processes.
