To build this polynomial that models a basic roller coaster, we will start with a quadratic function of the form:
[tex]\[ y = a \cdot x (x - 1000) \][/tex]
### Step-by-Step Solution
1. Identify the x-intercepts:
This quadratic polynomial [tex]\( y = a \cdot x (x - 1000) \)[/tex] has its x-intercepts at the points where [tex]\( y = 0 \)[/tex].
- Set [tex]\( y = 0 \)[/tex]:
[tex]\[ 0 = a \cdot x (x - 1000) \][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 0 \quad \text{or} \quad x = 1000 \][/tex]
Thus, the x-intercepts are at [tex]\( x = 0 \)[/tex] and [tex]\( x = 1000 \)[/tex].
2. Vertex (Turning Point):
The vertex of a parabola in this form lies exactly midway between the x-intercepts. Since the intercepts are at [tex]\( x = 0 \)[/tex] and [tex]\( x = 1000 \)[/tex], the x-coordinate of the vertex can be found by averaging these intercepts:
[tex]\[ x_{\text{vertex}} = \frac{0 + 1000}{2} = 500 \][/tex]
3. Calculate the y-coordinate of the Vertex:
Substitute [tex]\( x = 500 \)[/tex] into the polynomial to find the y-coordinate of the vertex:
[tex]\[ y = a \cdot 500 \cdot (500 - 1000) \][/tex]
Simplify the expression:
[tex]\[ y = a \cdot 500 \cdot (-500) \][/tex]
[tex]\[ y = a \cdot (-250000) \][/tex]
Given [tex]\( a = 1 \)[/tex] (for simplicity):
[tex]\[ y_{\text{vertex}} = 1 \cdot (-250000) = -250000 \][/tex]
Therefore, the coordinates of the vertex are [tex]\( (500, -250000) \)[/tex].
4. Summary of Polynomial Characteristics:
- The coefficient [tex]\( a \)[/tex] is 1.
- The x-intercepts are at [tex]\( x = 0 \)[/tex] and [tex]\( x = 1000 \)[/tex].
- The vertex is at [tex]\( (500, -250000) \)[/tex].
Putting it all together:
- Polynomial Form: [tex]\( y = 1 \cdot x (x - 1000) = x (x - 1000) \)[/tex]
- Intercepts and Turning Points:
- [tex]\( x \)[/tex]-intercepts: [tex]\( x = 0 \)[/tex], [tex]\( x = 1000 \)[/tex]
- Vertex: [tex]\( (500, -250000) \)[/tex]
Thus, the polynomial [tex]\( y = x (x - 1000) \)[/tex] describes a roller coaster that rises to a maximum, falls as it crosses the x-axis, reaching a minimum at [tex]\(x = 500\)[/tex], and then rises again as [tex]\( x \)[/tex] approaches 1000.
