Features & Transformations of Cubic Functions (March 2))

Front

What is a cubic function?

Back

A cubic function is a polynomial function of degree three, typically expressed in the form f(x) = ax³ + bx² + cx + d, where a, b, c, and d are constants and a ≠ 0.

Front

What does the graph of a cubic function generally look like?

Back

The graph of a cubic function can have one or two turning points and can extend to positive or negative infinity in both directions, resembling an 'S' shape.

Front

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

Back

The leading coefficient (a) determines the end behavior of the graph: if a > 0, the graph rises to the right; if a < 0, it falls to the right.

Front

How does the value of 'd' in f(x) = ax³ + bx² + cx + d affect the graph?

Back

The value of 'd' represents the y-intercept of the graph, indicating where the graph crosses the y-axis.

Front

What transformation occurs when a cubic function is reflected over the x-axis?

Back

Reflecting a cubic function over the x-axis changes the sign of the function, resulting in f(x) = -ax³ - bx² - cx - d.

Front

What does it mean to shift a cubic function to the right by 'h' units?

Back

Shifting a cubic function to the right by 'h' units involves replacing x with (x - h) in the function, resulting in f(x) = a(x - h)³ + b(x - h)² + c(x - h) + d.

Front

What is the effect of a vertical stretch by a factor of 'k' on a cubic function?

Back

A vertical stretch by a factor of 'k' multiplies the entire function by 'k', resulting in f(x) = k(ax³ + bx² + cx + d).

Front

How do you identify the vertex of a cubic function?

Back

The vertex of a cubic function is not defined like in quadratic functions, but the turning points can be found using calculus or by factoring.

Front

What is the role of the discriminant in determining the nature of roots of a cubic function?

Back

The discriminant helps determine the nature of the roots: if positive, there are three distinct real roots; if zero, there is one real root; if negative, there is one real root and two complex roots.

Front

What does it mean for a cubic function to have a double root?

Back

A double root occurs when the function touches the x-axis at that root but does not cross it, indicating a change in direction at that point.