Key Features 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 are the key features of cubic functions?

Back

Key features include: 1) Up to 3 real roots (x-intercepts), 2) At most 2 turning points (relative max/min), 3) Continuous and smooth graph, 4) Domain and range are all real numbers.

Front

What is an x-intercept?

Back

An x-intercept is a point where the graph of a function crosses the x-axis, meaning f(x) = 0 at that point.

Front

How do you find the roots of a cubic function?

Back

To find the roots, set the cubic function equal to zero and solve for x, either by factoring, using the Rational Root Theorem, or applying synthetic division.

Front

What is the difference between relative and absolute extrema?

Back

Relative extrema are the highest or lowest points in a specific interval, while absolute extrema are the highest or lowest points over the entire domain of the function.

Front

What does it mean for a function to be increasing?

Back

A function is increasing on an interval if, for any two points x1 and x2 in that interval, if x1 < x2 then f(x1) < f(x2).

Front

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

Back

The leading coefficient (the coefficient of x^3) determines the end behavior of the cubic function: if positive, the graph rises to the right; if negative, it falls to the right.

Front

What is a turning point in a cubic function?

Back

A turning point is a point on the graph where the function changes direction from increasing to decreasing or vice versa.

Front

How many x-intercepts can a cubic function have?

Back

A cubic function can have up to 3 real x-intercepts, depending on the nature of its roots.

Front

What is the domain of a cubic function?

Back

The domain of a cubic function is all real numbers, represented as (-∞, ∞).