Transformations and Roots

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 end behavior of a cubic function as x approaches infinity?

Back

As x approaches infinity, the end behavior of a cubic function depends on the leading coefficient. If the leading coefficient is positive, f(x) approaches infinity; if negative, f(x) approaches negative infinity.

Front

Define domain in the context of functions.

Back

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

Front

Define range in the context of functions.

Back

The range of a function is the set of all possible output values (y-values) that the function can produce.

Front

What is the y-intercept of a function?

Back

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

Front

How do you find the y-intercept of the function y = x(x-4)(x+5)?

Back

To find the y-intercept, substitute x = 0 into the function: y = 0(0-4)(0+5) = 0. Thus, the y-intercept is (0, 0).

Front

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

Back

The leading coefficient determines the end behavior of the polynomial function. It affects whether the graph rises or falls as x approaches positive or negative infinity.

Front

What does it mean for a function to be increasing or decreasing?

Back

A function is increasing on an interval if the output values (y-values) rise as the input values (x-values) increase. It is decreasing if the output values fall as the input values increase.

Front

What is a root of a function?

Back

A root of a function is a value of x for which f(x) = 0. It represents the x-intercepts of the graph.

Front

How can you determine the roots of a cubic function?

Back

To determine the roots of a cubic function, you can factor the function, use synthetic division, or apply the Rational Root Theorem.