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

Back

Reflecting a cubic function across the x-axis means that the output values (y-values) are multiplied by -1, resulting in a vertical flip of the graph.

Front

How does a horizontal shift affect the graph of a cubic function?

Back

A horizontal shift moves the graph left or right. For example, f(x) = (x - h)^3 shifts the graph h units to the right, while f(x) = (x + h)^3 shifts it h units to the left.

Front

What is the effect of a vertical shift on a cubic function?

Back

A vertical shift moves the graph up or down. For example, f(x) = x^3 + k shifts the graph k units up, while f(x) = x^3 - k shifts it k units down.

Front

What is the range of all cubic functions?

Back

The range of all cubic functions is (-∞, ∞), meaning they can take any real number as an output.

Front

What is the point of rotational symmetry for cubic functions?

Back

The point of rotational symmetry for cubic functions is typically at the origin (0,0) unless the function is transformed.

Front

How do you determine the domain of a cubic function?

Back

The domain of a cubic function is always (-∞, ∞) since cubic functions are defined for all real numbers.

Front

What does it mean for a cubic function to have a local maximum or minimum?

Back

A local maximum is a point where the function value is higher than all nearby points, while a local minimum is where the function value is lower than all nearby points.

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 it is positive, the graph rises to the right; if negative, it falls to the right.