The function y=(x-1)^3+2 is a cubic function that has been translated one unit to the right, shifted two units up, and reflected across the x-axis. The inflection point of the function is at the point (1,1).
a. Family of functions
The function y=(x-1)^3+2 belongs to the family of cubic functions. These functions are characterized by their third-degree polynomial form, which can be written as ax^3+bx^2+cx+d, where a, b, c, and d are constants. Cubic functions can exhibit a variety of behaviors, depending on the values of the coefficients. However, they all share some common features, such as having at most three turning points and at most two inflection points.
b. Transformations
The function y=(x-1)^3+2 can be obtained from the basic cubic function y=x^3 by applying the following transformations:
1. Translation: The function is shifted one unit to the right, since the value of x is subtracted by 1 in the exponent.
2. Vertical shift: The function is shifted two units up, since the constant term in the polynomial is 2.
3. Reflection: The function is reflected across the x-axis, since the coefficient of the third-degree term is negative.
c. New inflection point
The inflection point of a function is the point where its second derivative changes sign. The second derivative of y=(x-1)^3+2 is y''=6(x-1). This derivative is equal to 0 when x=1, which means that the inflection point is at the point (1,1).
d. Graph of the function
The following graph shows the function y=(x-1)^3+2, with the inflection point labeled:
Analysis of the graph
The graph of y=(x-1)^3+2 is a cubic curve with a downward-facing parabola shape. The curve passes through the point (1,1), which is the inflection point. The function is increasing for all values of x, and it approaches infinity as x approaches positive or negative infinity.