Lesson 2 I’ve Been Framed Develop Understanding

Learning Focus

Model a story context with a table, graph, and equation.

Identify features of a function from a graph.

Why are quadratic equations often used to model area?

Open Up the Math: Launch, Explore, Discuss

Maya has a large $8 \times 10 inches$ prom photo that she would like to frame. She wants to make her own wood frame using the tools in the school’s wood shop. Maya is trying to determine how much wood she should buy, but she is not sure how wide to make the frame.

1.

What would be the area of the frame (just the part made of wood) if Maya made it $1 inch$ wide?

2.

Use a table, graph, and equation to write the relationship between the area of the wood frame $A (x)$ and the width of the frame $x .$

3.

Describe each part of your equation in terms of the context.

4.

What kind of function is $A (x) ?$ How do you know?

5.

What are the domain and range in this context?

6.

If the function $A (x)$ was given without the context, what would be the features of this function? Graph the function and describe the domain, range, intervals of increase and decrease, maximum or minimum, line of symmetry, and any other features that help to describe $A (x)$ .

Ready for More?

What is the width of the frame that has an area of $760 {in}^{2}$ ?

Takeaways

Common features of quadratic functions:

Key features:

Lesson Summary

In this lesson, we examined a quadratic function that was a model for area but had many features that were different than those we have seen previously. We learned that all quadratic functions have a linear rate of change and constant second difference, but some may be continuous and have intervals of increase and decrease depending on the domain.

Retrieval

Multiply the given binomials.

1.

$(x + 4) (x - 13)$

2.

$(- 5 x + 1) (x + 2)$

Write the equation of a line perpendicular to the given line and through the given point.

3.

$y = \frac{1}{2} x + 7$ ; point: $(6, - 4)$

4.

$y = - \frac{3}{5} x - 12$ ; point: $(- 11, 32)$