Transformations and Roots

Front

What is a transformation in mathematics?

Back

A transformation is a function that maps a set of points to another set of points, altering their position, size, or orientation.

Front

Define the term 'zero' of a function.

Back

A zero of a function is a value of x for which the function f(x) equals zero.

Front

What is the general form of a cubic function?

Back

The general form of a cubic function is f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants and a ≠ 0.

Front

How do you find the zeros of a cubic function?

Back

To find the zeros of a cubic function, set the function equal to zero and solve for x using factoring, synthetic division, or the Rational Root Theorem.

Front

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

Back

The leading coefficient determines the end behavior of the cubic function's graph. If it's positive, the graph rises to the right; if negative, it falls to the right.

Front

What does the term 'range' refer to in a function?

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 range of the function f(x) = x^3 - 5?

Back

The range of f(x) = x^3 - 5 is (-∞, ∞), as cubic functions can take any real value.

Front

Explain the concept of a cubic graph's inflection point.

Back

An inflection point is where the graph of a cubic function changes concavity, indicating a change in the direction of curvature.

Front

What is the vertex form of a cubic function?

Back

The vertex form of a cubic function is f(x) = a(x - h)^3 + k, where (h, k) is the vertex of the graph.

Front

How can you determine the end behavior of a cubic function?

Back

The end behavior can be determined by the leading coefficient and the degree of the polynomial. For a cubic function, if the leading coefficient is positive, the left end goes down and the right end goes up.