Question
The domain of a quadratic function is all real numbers and the range is y ≤ 2. How many x-intercepts does the function have?
Answers
Answered by
GPT-4o mini
A quadratic function can be expressed in the standard form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is non-zero.
Given that the domain of the quadratic function is all real numbers, this is a characteristic of all quadratic functions since they are defined for every real input. However, the crucial aspect in your question is regarding the range of the function, which is stated as \( y \leq 2 \).
The range \( y \leq 2 \) indicates that the quadratic function has a maximum value of 2. This occurs when the function opens downwards, which is the case when \( a < 0 \).
Since the maximum value of the function is at \( y = 2 \) and the quadratic opens downwards, we need to determine how many times the graph of the function intersects the x-axis, which corresponds to the function having x-intercepts.
- If the vertex of the parabola (maximum point) is exactly at \( (h, 2) \), then the function touches the x-axis at one point, giving it exactly one x-intercept (this is known as a double root).
- If the maximum value \( y = 2 \) occurs above the x-axis, the parabola will not intersect the x-axis at all and would have zero x-intercepts.
- If the vertex is at \( y = 2 \) (the maximum) and the parabola goes downwards, it will cross the x-axis, resulting in two distinct x-intercepts.
Therefore, in the scenario where the range is \( y \leq 2 \), if the vertex of the parabola is at \( y = 2 \), it will have exactly one x-intercept. If the maximum value was less than 2, it would have two x-intercepts.
In the case mentioned, because the vertex must be at or below 2 and is at that maximum, we conclude that there is one x-intercept if the vertex is at that maximum.
Thus, the answer for the number of x-intercepts is \( \boxed{1} \).
Given that the domain of the quadratic function is all real numbers, this is a characteristic of all quadratic functions since they are defined for every real input. However, the crucial aspect in your question is regarding the range of the function, which is stated as \( y \leq 2 \).
The range \( y \leq 2 \) indicates that the quadratic function has a maximum value of 2. This occurs when the function opens downwards, which is the case when \( a < 0 \).
Since the maximum value of the function is at \( y = 2 \) and the quadratic opens downwards, we need to determine how many times the graph of the function intersects the x-axis, which corresponds to the function having x-intercepts.
- If the vertex of the parabola (maximum point) is exactly at \( (h, 2) \), then the function touches the x-axis at one point, giving it exactly one x-intercept (this is known as a double root).
- If the maximum value \( y = 2 \) occurs above the x-axis, the parabola will not intersect the x-axis at all and would have zero x-intercepts.
- If the vertex is at \( y = 2 \) (the maximum) and the parabola goes downwards, it will cross the x-axis, resulting in two distinct x-intercepts.
Therefore, in the scenario where the range is \( y \leq 2 \), if the vertex of the parabola is at \( y = 2 \), it will have exactly one x-intercept. If the maximum value was less than 2, it would have two x-intercepts.
In the case mentioned, because the vertex must be at or below 2 and is at that maximum, we conclude that there is one x-intercept if the vertex is at that maximum.
Thus, the answer for the number of x-intercepts is \( \boxed{1} \).