Student Page 4.4.2 Quadratic Quilts and Rates of Change for \(y = ax^2 + bx + c\)
Because the area of bedding is measured in square inches, it might not be surprising that quilts provide a context in which to study quadratic functions. We use the context of making quilts to also study rate of change for quadratic functions. Recall that the rate of change of a linear function was always constant. We will notice that quadratic functions change differently. Explore the student page, Quadratic Quilts and Rates of Change for \(y = ax^2 + bx + c\), to explore quadratic rate of change. 46
1.
Esther designs and makes quilts. The completed square in Figure 4.4.2.1 is called a quilt block. She makes different sizes of square quilts using the same pattern and colors for each block in each quilt. To ensure that each quilt block matches, she purchases all of the fabric she needs at the same time.
(a)
The quilt block at right has dimensions 1 × 1. Determine the number of same-size small square pieces of each color Esther needs for the quilt block.
(b)
How many small square pieces of each color does she need for each of the different projects listed? Complete Table 4.4.2.2.
| Quilt Dimensions in Blocks |
Project Type |
\(x\) | Number of Quilt Blocks |
Number of Yellow (Y) Squares |
Number of Blue (B) Squares |
Number of Red (R) Squares |
Number of White (W) Squares |
|---|---|---|---|---|---|---|---|
| 1 × 1 | Potholder | 1 | 1 | 1 | 3 | 4 | 8 |
| 2 × 2 | Nightstand cover |
2 | 4 | 12 | |||
| 3 × 3 | Doll blanket |
3 | 9 | ||||
| 4 × 4 | Baby blanket |
4 | |||||
| 5 × 5 | Lap blanket |
5 | |||||
| 6 × 6 | Tablecloth | 6 | |||||
| 7 × 7 | Twin bed cover |
7 |
(c)
Consider one color at a time. What is the relationship between the quilt's dimension, \(x\text{,}\) and the number of individual squares needed to make the quilt?
(d)
Determine an equation for each relationship in Task 4.4.2.1.c.
2.
(a)
Complete Table 4.4.2.3. Let \(f(x) = x^2\text{.}\)
| \(x\) | \(y = f(x)\) | \(g(x) = f(x) - f(x - 1)\) | \(g(x) - g(x - 1)\) |
|---|---|---|---|
| \(-3\) | X | X | |
| \(-2\) | X | ||
| \(-1\) | |||
| \(0\) | |||
| \(1\) | |||
| \(2\) | |||
| \(3\) | |||
| \(4\) |
(b)
Describe how \(f(x) = x^2\) is changing as \(x\) increases.
(c)
Describe any patterns or symmetry you see in the table.
(d)
Without graphing the function, state what you know about the graph from your table and responses to Task 4.4.2.2.b and Task 4.4.2.2.c.
(e)
Sketch the graph without plotting points.
3.
(a)
Complete a table like Table 4.4.2.4 for each equation you found in Task 4.4.2.2.d.
| \(x\) | \(y = f(x)\) | \(g(x) = f(x) - f(x - 1)\) | \(g(x) - g(x - 1)\) |
|---|---|---|---|
| \(-3\) | X | X | |
| \(-2\) | X | ||
| \(-1\) | |||
| \(0\) | |||
| \(1\) | |||
| \(2\) | |||
| \(3\) | |||
| \(4\) |
(b)
Complete a table like the one in Table 4.4.2.4 for \(f(x) = ax^2 + bx + c\text{.}\)
(c)
What patterns do you see in the first differences, \(f(x) - f(x - 1)\text{?}\)
(d)
What patterns do you see in the second differences, \(g(x) - g(x - 1)\text{?}\)
(e)
What do these patterns suggest about the rates of change of all quadratic functions?
4.
(a)
Graph the parent function, \(f(x) = x^2\text{.}\)
(b)
What are the coordinates of the \(y\)-intercept? \(x\)-intercept?
(c)
Describe the shape of the graph.
(d)
Why does the graph look as it does? Compare the table values to the graph.
(e)
What new information is provided by the graph that is not evident in the table?
(f)
What information is provided in the table that is not evident in the graph?
5.
The vertex of a quadratic function is the point at which it changes direction.
(a)
What are the coordinates of the vertex of \(f(x)=x^2\text{?}\)
(b)
Graph each equation from Task 4.4.2.2.d. How are the graphs related to \(f(x)=x^2\text{?}\)
(c)
For each equation in Task 4.4.2.2.d, what is the vertex for each graph?
6.
(a)
Two other important points on the graph of \(f(x)=x^2 \) are \((1,1)\) and \((-1,1)\text{.}\) Graph each equation from Task 4.4.2.2.d for \(x\) in \([-5, 5]\text{.}\) What points correspond with \((1, 1)\) and \((-1, 1)\) for these quadratic function family members? Why do you think so?
(b)
For functions in the quadratic function family, why are the points that correspond to \((0, 0)\text{,}\) \((1, 1)\text{,}\) and \((-1, 1)\) on \(f(x) = x^2\) important?
(c)
How might these points help you determine the equation of the function represented by the table or graph? Explain.