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 extend from the lower left to the upper right or vice versa, depending on the leading coefficient.

Front

What does the y-intercept of a function represent?

Back

The y-intercept of a function is the point where the graph intersects the y-axis, which occurs when x = 0.

Front

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

Back

To find the y-intercept, substitute x = 0: f(0) = (0 + 2)^3 - 6 = 8 - 6 = 2. Thus, the y-intercept is (0, 2).

Front

What is the minimum value of a cubic function?

Back

The minimum value of a cubic function is the lowest point on the graph, which can be found by analyzing critical points and the behavior of the function.

Front

How do transformations affect the graph of a cubic function?

Back

Transformations such as shifts, reflections, and stretches/compressions change the position and shape of the graph. For example, f(x) = - (x + 3)^3 + 2 reflects the graph across the x-axis and shifts it left and up.

Front

What is rotational symmetry in the context of cubic functions?

Back

Rotational symmetry refers to the property of a graph where it can be rotated around a point and still look the same. For cubic functions, this point is often the inflection point.

Front

What is the domain of a cubic function?

Back

The domain of a cubic function is all real numbers, expressed as (-∞, ∞), since cubic functions can take any real value for x.

Front

What is the range of a cubic function?

Back

The range of a cubic function is also all real numbers, expressed as (-∞, ∞), as cubic functions can output any real value.

Front

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

Back

Reflecting a function across the x-axis means that for every point (x, y) on the graph, there is a corresponding point (x, -y). This changes the sign of the output values.