10.1 Attributes & Transformations of Cubic Functions

Front

What is a cubic function?

Back

A cubic function is a polynomial function of degree three, typically expressed in the form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants and a ≠ 0.

Front

What is the general shape of the graph of a cubic function?

Back

The graph of a cubic function has an S-shape, which can have one or two turning points, depending on the coefficients.

Front

What does the y-intercept of a function represent?

Back

The y-intercept is the point where the graph of the function crosses the y-axis, found by evaluating f(0).

Front

How do you find the y-intercept of f(x) = (x + 2)^3 - 6?

Back

Evaluate f(0): f(0) = (0 + 2)^3 - 6 = 8 - 6 = 2. The y-intercept is (0, 2).

Front

What is the maximum value of a cubic function on a given interval?

Back

The maximum value is the highest point on the graph within that interval, found by evaluating the function at critical points and endpoints.

Front

How do you determine the maximum value of f(x) = (x + 2)^3 - 6 on [-1, 1]?

Back

Evaluate f(-1), f(0), and f(1). The maximum value is f(1) = 11.

Front

What does it mean for a function to be reflected across the x-axis?

Back

Reflecting a function across the x-axis means multiplying the function by -1, changing the sign of all y-values.

Front

How does the transformation f(x) = - (x + 3)^3 + 2 affect the graph of f(x) = x^3?

Back

The graph is shifted 3 units left, 2 units up, and reflected across the x-axis.

Front

What is the effect of changing the coefficient of a cubic function?

Back

Changing the coefficient affects the steepness and direction of the graph. A larger coefficient makes the graph steeper.

Front

What is the significance of the leading coefficient in a cubic function?

Back

The leading coefficient determines the end behavior of the graph: if positive, the graph rises to the right; if negative, it falls to the right.