Problem 86 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the standard form of the equation of the parabola with the given characteristics. Vertex: (-1,2)\(;\) focus: (-1,0)

Step-by-Step Solution

Answer

The standard form equation of the parabola is \(y=-2(x+1)^2+2\).

Step 1: Identify the Orientation and Vertex of the Parabola

From the given problem, we see that the parabola has a vertex at (-1,2) and a focus at (-1,0). Since only the y-coordinate changes, this tells us that the parabola opens either up or down. This is an important hint that to confirm the orientation.

Step 2: Determine the Direction of the Parabola

A parabola always opens towards its focus. Here our focus is below the vertex, so the parabola opens downwards.

Step 3: Find the Value of p

The value of p is the distance from the vertex to the focus, which in this instance is 2 units (2 - 0). But since the parabola opens downward, we use -2.

Step 4: Write the Equation

Now, by plugging the values h = -1, k = 2 and p = -2 to the standard form equation \(y=k(x-h)²+k\) for a vertically oriented parabola, we arrive at the equation: \(y=-2(x+1)^2+2\).