Graphing Cubics

Front

What are the roots of the polynomial f(x) = -x(x+1)(x-3)?

Back

The roots are 0, -1, and 3.

Front

What is the general form of a cubic function?

Back

A cubic function can be expressed as f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants and a ≠ 0.

Front

How do you determine the roots of a polynomial?

Back

To find the roots, set the polynomial equal to zero and solve for x.

Front

What does it mean for a root to be a double root?

Back

A double root occurs when a root is repeated, meaning the polynomial touches the x-axis at that point but does not cross it.

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

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 do you graph a cubic function?

Back

To graph a cubic function, identify the roots, plot the vertex, and determine the end behavior based on the leading coefficient.

Front

What is the relationship between the roots and the factors of a polynomial?

Back

The roots of a polynomial correspond to the values of x that make each factor equal to zero.

Front

What are the roots of the polynomial f(x) = -6x(x+4)(x-2)?

Back

The roots are 0, -4, and 2.

Front

What is the effect of adding a constant to a cubic function?

Back

Adding a constant shifts the graph vertically without changing its shape.