Word Problem:

A local park is planning to build a parabolic-shaped flower bed along the path. The landscapers measured the width and height of the flower bed at several points. Their measurements are listed in the table below. They need to determine a quadratic equation to model the shape of the flower bed. The quadratic equation should be in factored form, where the x-values represent the distance from the center of the flower bed (in meters), and the y-values represent the height of the bed (in meters).

Using the table of values provided, develop a quadratic equation in factored form that best fits the data.

Distance from Center (x)	Height of Bed (y)
-3.0	0.50
-2.0	1.80
-1.0	2.50
0.0	3.20
1.0	2.50
2.0	1.80
3.0	0.50

Solution:

We can observe that the data forms a symmetrical parabola, with the vertex at $(0, 3.2)$. To create a quadratic model in factored form, we can use the general factored form of a quadratic equation:

$y = a(x - r_1)(x - r_2)$

Where $r_1$ and $r_2$ are the roots of the parabola. From the table, we observe that the height $y$ is 0.50 when $x = -3.0$ and $x = 3.0$, suggesting the roots are $r_1 = -3$ and $r_2 = 3$.

Substituting $r_1 = -3$ and $r_2 = 3$ into the equation:

$y = a(x + 3)(x - 3)$

This simplifies to:

$y = a(x^2 - 9)$

Now, substitute the vertex point $(0, 3.2)$ to solve for $a$:

$3.2 = a(0^2 - 9)$ $3.2 = -9a$ $a = -\frac{3.2}{9} = -\frac{8}{25}$

Thus, the quadratic equation is:

$y = -\frac{8}{25}(x^2 - 9)$

Finally, the quadratic equation in factored form is:

$y = -\frac{8}{25}(x + 3)(x - 3)$

Follow-up Questions:

1. What is the vertex of the quadratic equation and how does it relate to the maximum height of the flower bed?
2. How would you use this model to estimate the height of the flower bed at $x = 1.5$?
3. If the roots of the equation were different, how would it affect the shape of the curve?
4. Can you find the equation in standard form by expanding the factored form equation?
5. How would you interpret the value of $a$ in terms of the shape of the parabola?

Tip:

In a quadratic equation, the sign of $a$ determines whether the parabola opens upwards or downwards. A negative $a$ value indicates a downward-facing parabola, while a positive $a$ results in an upward-facing parabola.