Question
Which of the following descriptions accurately characterizes a graph that has no real roots?
Answers
Answered by
GPT-4o mini
A graph that has no real roots typically has specific characteristics. Here are some descriptions that accurately characterize such a graph:
1. **Does not intersect the x-axis:** The graph does not touch or cross the x-axis at any point. This means there are no x-values for which the function equals zero.
2. **Parabola Opening Upward or Downward Above/Below the x-axis:** If the graph is a parabola that opens upwards (the coefficient of the \(x^2\) term is positive) and its vertex is above the x-axis, or if it opens downwards (the coefficient of the \(x^2\) term is negative) and its vertex is below the x-axis, it will have no real roots.
3. **Complex Roots:** For polynomial functions, if the discriminant (for quadratic equations, \(b^2 - 4ac\)) is less than zero, it indicates that the equation has no real solutions (roots), only complex conjugate roots.
4. **Positive or Negative Value for All x:** The function always maintains a positive value (for example, \(f(x) = x^2 + 1\)) or negative value (for example, \(f(x) = -x^2 - 1\)) for all real numbers, indicating it does not meet the x-axis.
In summary, a graph without real roots can be characterized by its inability to intersect the x-axis, often due to its orientation and the location of its vertex relative to the x-axis.
1. **Does not intersect the x-axis:** The graph does not touch or cross the x-axis at any point. This means there are no x-values for which the function equals zero.
2. **Parabola Opening Upward or Downward Above/Below the x-axis:** If the graph is a parabola that opens upwards (the coefficient of the \(x^2\) term is positive) and its vertex is above the x-axis, or if it opens downwards (the coefficient of the \(x^2\) term is negative) and its vertex is below the x-axis, it will have no real roots.
3. **Complex Roots:** For polynomial functions, if the discriminant (for quadratic equations, \(b^2 - 4ac\)) is less than zero, it indicates that the equation has no real solutions (roots), only complex conjugate roots.
4. **Positive or Negative Value for All x:** The function always maintains a positive value (for example, \(f(x) = x^2 + 1\)) or negative value (for example, \(f(x) = -x^2 - 1\)) for all real numbers, indicating it does not meet the x-axis.
In summary, a graph without real roots can be characterized by its inability to intersect the x-axis, often due to its orientation and the location of its vertex relative to the x-axis.