Cubic Functions and Their Properties

The video tutorial explains how to sketch cubic functions, starting with the basic form y = ax^3 + bx^2 + cx + d. It covers the special case of y = x^3, which has no turning points, and then discusses more complex cubic functions with turning points. The tutorial divides cubic functions into cases based on the sign of the leading coefficient and provides examples of sketching cubic functions in factored form, with negative coefficients, repeated roots, and a single root. Each example demonstrates how to find x-intercepts, y-intercepts, and the general shape of the graph.

What is the general form of a cubic function?

• y = ax^3 + bx^2 + cx + d

• y = ax^2 + bx + c

• y = ax + b

• y = ax^4 + bx^3 + cx^2 + d

How does the graph of y = x^3 behave as x becomes very large and positive?

• The graph oscillates.

• The graph approaches positive infinity.

• The graph remains constant.

• The graph approaches negative infinity.

What happens to the graph of a cubic function when the leading coefficient 'a' is negative?

• The graph becomes a straight line.

• The graph opens downwards.

• The graph remains unchanged.

• The graph opens upwards.

In the example y = (x - 2)(x - 1)(x + 1), what are the x-intercepts?

• x = 2, x = 1, x = -1

• x = 0, x = 1, x = -1

• x = 2, x = 0, x = -1

• x = 2, x = 1, x = 0

For the function y = (x - 2)(x - 1)(x + 1), what is the y-intercept?

• y = 0

• y = -2

• y = 1

• y = 2

In the example y = (x - 3)(x - 1)(x + 1), what is the y-intercept?

• y = -3

• y = 3

• y = -1

• y = 1

What is a repeated root in a cubic function?

• A root that appears three times.

• A root that appears twice.

• A root that appears once.

• A root that does not appear.

How does the graph of y = (x + 2)^3 differ from y = x^3?

• It is shifted to the right by 2.

• It is shifted downwards by 2.

• It is shifted to the left by 2.

• It is shifted upwards by 2.

What is the y-intercept of the function y = (x + 2)^3?

• y = 8

• y = -8

• y = 2

• y = -2

How many roots does a cubic function with a single factor raised to the third power have?

• Three roots

• Two roots

• One root

• No roots