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 in a graph?

• Points where the graph crosses the x-axis

• Points where the graph is at its minimum

• Points where the graph is at its maximum

• Points where the graph crosses the y-axis

In the function V(h) = h(10 - 2h)(14 - 2h), what does V(h) represent?

• The height of the box

• The width of the box

• The volume of the box

• The surface area of the box

What does it mean when the volume of the box is zero at certain heights?

• The box is at its maximum volume

• The box is at its minimum volume

• The box is not formed properly

• The box has a negative volume

Why is the volume zero when the height is 5 inches?

• The length of the box becomes zero

• The width of the box becomes zero

• The height of the box becomes zero

• The box is perfectly formed

What happens to the box when the height is 7 inches?

• The volume becomes maximum

• The width becomes negative

• The height becomes negative

• The length becomes negative

In which interval is the volume of the box positive?

• Between 0 and 5 inches

• Between 5 and 7 inches

• For all heights less than 0

• For all heights greater than 7

What does a negative volume indicate in the context of the box?

• The box is perfectly formed

• The box has a negative dimension

• The box is at its maximum volume

• The box is at its minimum volume