A graph that has no real roots means that the function does not intersect the x-axis. Here are some characteristics that can accurately describe such a graph:
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Quadratic Functions: If the graph is a parabola that opens upwards and the vertex of the parabola is above the x-axis (the minimum point is above zero), or if the parabola opens downwards and the vertex is below the x-axis (the maximum point is below zero), then the graph has no real roots.
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Exponential Functions: For functions like \(f(x) = a^x\) where \(a > 0\), the graph does not touch or cross the x-axis, hence it has no real roots.
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Sinusoidal Functions: A sinusoidal function like \(f(x) = a \sin(bx) + c\) can also be adjusted (by changing the amplitude and vertical shift) to have no real roots if \(c\) is greater than or less than the maximum or minimum value of the sine function (e.g., the entire curve is above or below the x-axis).
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Polynomial Functions: In higher-degree polynomials, if all the coefficients are of the same sign and the leading coefficient determines the end behavior (either both ends go up or both go down), there can be scenarios where the polynomial does not cross the x-axis.
In summary, look for a graph that does not intersect the x-axis and is either entirely above or entirely below it. If you have more specific descriptions or options, I could help pinpoint which one characterizes graphs with no real roots.