Cubic Functions and Their Roots

This video tutorial explores translations of cubic functions, focusing on determining values of k for which the function has three distinct roots or one root. It covers both graphical and algebraic methods to solve these problems, including the use of inequalities. Additionally, the video explains how to translate graphs and write new equations for shifted functions.

What is the main focus of the video?

• Translations of linear functions

• Translations of cubic functions

• Translations of exponential functions

• Translations of quadratic functions

What information is provided about the cubic function in the video?

• Equation, sketch, and stationary points

• Only the sketch

• Only the stationary points

• Only the equation

What is the task related to the value of k in the video?

• Find k for no roots

• Find k for three distinct roots

• Find k for four distinct roots

• Find k for two distinct roots

What happens when the cubic function is shifted up by more than 5 units?

• It has only one root

• It has four distinct roots

• It has two equal roots and one distinct root

• It has no roots

In the graphical method, what is the range of k for three distinct roots?

• k > 0 and k < 10

• k < 5 and k > 37

• k > 5 and k < 37

• k = 5 or k = 37

What is the algebraic method used for in the video?

• To find the x-intercept

• To determine k for three distinct roots

• To find the y-intercept

• To determine k for no roots

What is the condition for k to have only one distinct root?

• k < 5 or k > 37

• k < 0 or k > 10

• k > 5 and k < 37

• k = 5 or k = 37

How is the graph of f translated to find the new graph h?

• Shifted 3 units right and 2 units down

• Shifted 2 units left and 1 unit up

• Shifted 1 unit right and 2 units up

• Shifted 2 units right and 1 unit down