QUESTION THREE The graphs below represent the functions \( f(x)=a x^{3}+b x-2 \) and \( g(x)=x-2 \). Both functions \( f \) and \( g \) have a root at the point A . \( f \) touches the \( x \)-axis at -1 and C is a common y -intercept of the two graphs. B is a turning point of \( f \).

The function \( f(x) = a x^3 + b x - 2 \) is a polynomial of degree three, which generally indicates it can have a variety of turning points and roots. In this particular case, at point A, both \( f \) and \( g \) intersect at the same x-value, indicating they share a common root. Moreover, since \( f \) touches the x-axis at -1, it implies that -1 is a root of multiplicity 2, making it a turning point, as it does not cross the x-axis but merely turns around there. When determining the common y-intercept \( C \), both functions should equal -2 when \( x = 0 \). Thus, we can see \( g(0) = 0 - 2 = -2 \) and we need \( f(0) \) to equal -2 as well, leading to \( f(0) = -2 \). This can guide you in finding the coefficients \( a \) and \( b \) by constructing equations based on these conditions.