The graph of which of the following functions in the xy-plane has more than one x-intercept?

Rationale

f(x) = - 4x ^ 2 + 5 has more than one x-intercept.

This quadratic function opens downwards and can intersect the x-axis at two points due to its parabolic shape, resulting in two real roots when the discriminant is positive.

A (f(x) = - 4x - 5) YOUR ANSWER

This is a linear function with a slope of -4 and a y-intercept of -5. As a linear function, it can only have one x-intercept, where the line crosses the x-axis, meaning it does not meet the criteria of having more than one x-intercept.

B (f(x) = - 4x ^ 2 + 5) CORRECT

This quadratic function can have two x-intercepts because it forms a downward-opening parabola. The vertex is above the x-axis, and the graph intersects the x-axis at two points, confirming that it has more than one x-intercept.

C (f(x) = 4) YOUR ANSWER

This is a constant function that represents a horizontal line situated above the x-axis at y = 4. Since it does not intersect the x-axis at all, it has no x-intercepts.

D (f(x) = 4x ^ 2 + 5) YOUR ANSWER

This quadratic function opens upwards and has a minimum value at its vertex, which is also above the x-axis. Therefore, it does not intersect the x-axis and has no x-intercepts.

E (f(x) = 4x ^ 3 + 5) YOUR ANSWER

This is a cubic function that can have one or three x-intercepts, but it does not guarantee more than one. In this case, the function does not cross the x-axis, as its minimum value is greater than zero, resulting in no x-intercepts.

Conclusion

Among the provided choices, only the function f(x) = - 4x ^ 2 + 5 possesses more than one x-intercept due to its parabolic nature and downward opening. The other functions either have no x-intercepts or only one, illustrating the distinctive characteristics of linear and constant functions compared to quadratic ones.

The graph of which of the following functions in the xy-plane has more than one x-intercept?

f(x) = - 4x ^ 2 + 5 has more than one x-intercept.

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