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Question 1170844: I have a parabola with a length of 72 feet and a height of 64 ft, the goal is to find the vertex as well as the x-intercepts and then write the parabola in standard, vertex and factored form; thanks a ton!
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step by step.
**1. Set up a Coordinate System**
To make the calculations easier, let's place the parabola in a coordinate system. We'll place the vertex on the y-axis, which will simplify the equations.
* Let the vertex be at (0, 64).
* The parabola opens downwards since it has a maximum height of 64 ft.
* The length of 72 feet means the x-intercepts are at (-36, 0) and (36, 0).
**2. Vertex Form**
The vertex form of a parabola is:
y = a(x - h)² + k
where (h, k) is the vertex.
* We know the vertex is (0, 64), so h = 0 and k = 64.
* y = a(x - 0)² + 64
* y = ax² + 64
**3. Find the Value of 'a'**
We can use one of the x-intercepts to find 'a'. Let's use (36, 0):
* 0 = a(36)² + 64
* 0 = 1296a + 64
* -64 = 1296a
* a = -64 / 1296
* a = -1 / 20.25
* a = -4 / 81
Therefore, the vertex form is:
y = (-4/81)x² + 64
**4. Standard Form**
The standard form of a parabola is:
y = ax² + bx + c
We already have the vertex form:
y = (-4/81)x² + 64
Since there's no 'x' term, b = 0. So the standard form is:
y = (-4/81)x² + 64
**5. Factored Form**
The factored form of a parabola is:
y = a(x - r₁)(x - r₂)
where r₁ and r₂ are the x-intercepts.
* We know the x-intercepts are (-36, 0) and (36, 0), so r₁ = -36 and r₂ = 36.
* We know a = -4/81.
Therefore, the factored form is:
y = (-4/81)(x - (-36))(x - 36)
y = (-4/81)(x + 36)(x - 36)
**Summary**
* **Vertex:** (0, 64)
* **X-intercepts:** (-36, 0) and (36, 0)
* **Vertex Form:** y = (-4/81)x² + 64
* **Standard Form:** y = (-4/81)x² + 64
* **Factored Form:** y = (-4/81)(x + 36)(x - 36)
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