Answer:
To describe the shape of a cubic function, we need to consider its end behavior, any turning points it has, and any intervals over which it is increasing or decreasing.
For the cubic function 32. y = 3x³ - x - 3, the leading coefficient is 3, which means that the end behavior of the function will be determined by the sign of the term x³. Since the coefficient of x³ is positive, the function will approach positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity. This means that the graph of the function will curve upwards at both ends.
To find the turning points of the function, we can find the zeros (or roots) of the function. In this case, we can solve the equation 3x³ - x - 3 = 0 using the cubic formula or a numerical method. This will give us three real roots, which correspond to the x-values of the turning points of the function. The function will be concave upwards at the points between these roots and concave downwards at the points between the roots and the ends of the graph.
The function will be increasing on the intervals between the roots where the function is concave upwards and decreasing on the intervals between the roots where the function is concave downwards.
For the cubic function 33. y = -9x³ - 2x² + 5x + 3, the leading coefficient is -9, which means that the end behavior of the function will be determined by the sign of the term -9x³. Since the coefficient of x³ is negative, the function will approach negative infinity as x approaches positive infinity and positive infinity as x approaches negative infinity. This means that the graph of the function will curve downwards at both ends.
As with the previous function, we can find the zeros (or roots) of the function by solving the equation -9x³ - 2x² + 5x + 3 = 0 using the cubic formula or a numerical method. This will give us three real roots, which correspond to the x-values of the turning points of the function. The function will be concave upwards at the points between these roots and concave downwards at the points between the roots and the ends of the graph.
The function will be decreasing on the intervals between the roots where the function is concave upwards and increasing on the intervals between the roots where the function is concave downwards.
Explanation: