Which statement describes the graph of f(x) = -4×3 – 28×2 – 32x + 64?

Which statement describes the graph of f(x) = -4×3 – 28×2 – 32x + 64?

A. The graph crosses the x-axis at x = 4 and touches the x-axis at x = -1.

B. The graph touches the x-axis at x = 4 and crosses the x-axis at x = -1.

C. The graph crosses the x-axis at x = -4 and touches the x-axis at x = 1.

D. The graph touches the x-axis at x = -4 and crosses the x-axis at x = 1.

Answer: B. The graph touches the x-axis at x = 4 and crosses the x-axis at x = -1.

To comprehend this graph, let us examine the function used: f(x) = -4x³ – 28x² – 32x + 64. This is a cubic function and that indicates that at most, we can have three x-intercepts of the graph of the function (points at which the graph cuts or touches the x-axis). Therefore, the idea of multiple roots is the central point that distinguishes between the crossing and touching of the x-axis.

When a graph crosses the x-axis, then there is a change in the sign of the function and this indicates a simple root. This occurs at x = -1 in the stated function and using the given values of ‘a’ and ‘b’, the function can be further calculated. Thus, the graph will enter and leave this point –going below the axes on one side and going above the axes on another. In contrast, when any graph comes in contact with the X-axis it conveys that the root of that equation is of higher multiplicity. In this case, the graph of the function v = f(x) ‘touches,’ though does not ‘cross’ the axis at x = 4 which indicates a quadratic root.

In more detail, what is happening is that the eye is taken on a journey around the curve, so to speak, that might be likened to running one’s finger over the curve in question. If at x = -1, your finger would be moving from the right side of the axe to the left side of the axe. If placing the finger at x = 4 the straight thin line of your finger will barely cross the x-axis and will be on the same side. This kind of behaviour of a function is inherent in cubic functions with a simple root and a double root. Yes, the general trend of the graph will resemble an S-curve mainly because the graph touches the x-axis at x=4.