Understanding Functions and Context

This lesson explores how context influences functions by examining graphs. It begins with a review of x-intercepts, explaining their significance and common misunderstandings. The lesson then introduces a volume function derived from a cardboard box model, graphing it to find zeros and understand volume changes. Algebraic and contextual explanations are provided for why the volume is zero at certain heights. Finally, the lesson identifies positive and negative intervals of the volume function, emphasizing the importance of context in understanding functions.

What are x-intercepts and how can they be identified on a graph?

Explain the significance of the y-intercept in relation to the x-intercept.

Describe the process of creating a volume function from a piece of cardboard.

How do you determine the zeros of the function V(h) and what do they represent?

What happens to the volume of the box when the height is set to 0 inches?

How does cutting squares from the corners of the cardboard affect the dimensions of the box?

Identify the intervals where the volume of the box is positive and explain why.