Problem 25 - Algebra and Trigonometry

Question

In Exercises 17–30, find the standard form of the equation of each parabola satisfying the given conditions. Vertex: \((2,-3) ;\) Focus: \((2,-5)\)

Step-by-Step Solution

Answer

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

Step 1: Determine the direction of the parabola

From vertex (2,-3) and focus (2,-5), we see that the parabola opens downwards. This information will help us in choosing the precise form to use. In this case, we will use \(y=a(x-h)^2+k\).

Step 2: Determine the value of 'a' using the focus

The absolute value of 'a' is 1/4 the distance from vertex to the focus. Here, it is \(a=1/4|-5-(-3)|\), hence \(a=-1/2).\) Negative sign indicates the parabola opens downwards.

Step 3: Write out the equation of the parabola

Substitute the known values for 'a', 'h', and 'k' into the formula \(y=a(x-h)^2+k\). We get \(y=-1/2(x-2)^2-3\).

Step 4: Expand the equation

Expand and simplify the equation to the standard form \(y=ax^2+bx+c\). By doing so, the equation becomes \(y=-1/2x^2+2x-2\).

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is crucial in graphing and interpreting its shape. The vertex is the highest or lowest point on the graph of a parabola, depending on whether it opens upwards or downwards, respectively.

In the given exercise, the vertex is at \( (2,-3) \), which provides us with the values of \(h\) and \(k\) in the vertex form of a parabola's equation \( y = a(x-h)^2 + k \). When plotting this on a coordinate graph, the vertex acts as a pivotal point from which the parabola takes its shape.

Focus of a Parabola

The focus of a parabola is a fixed point that, along with the directrix, defines the curve. The distance between any point on the parabola and the focus is equal to the distance from that point to the directrix.

In our problem, the focus is located at \( (2,-5) \). The vertical distance between the vertex and the focus is an essential component in determining the value of \(a\) in the equation \( y = a(x-h)^2 + k \). This value impacts the 'width' and 'direction' of the parabola's opening, with a larger absolute value of \(a\) indicating a 'steeper' parabola.

Parabola Opens Downwards

Identifying the direction in which a parabola opens is imperative to understanding its graph. A downwards opening parabola has a vertex that is the highest point on the graph, resembling an upside-down 'U'.

The exercise demonstrates that the parabola's focus is below its vertex, which means we are indeed dealing with a parabola that opens downwards. Consequently, the coefficient \(a\) in the standard equation is negative, reflecting this orientation. The step-by-step solution clearly uses a negative sign for \(a\) after determining it from the distance between the vertex and focus.

Simplifying Quadratic Equations

Simplifying quadratic equations can often make solving or graphing them much easier. The standard form of a quadratic equation is \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. To simplify a quadratic equation, you may need to expand, factor, or complete the square to find its simplest form.

In this scenario, simplifying involves expanding and rewriting the vertex form of the equation \( y = a(x-h)^2 + k \) into the standard form. It can require distributing the \(a\) value over the square and then combining like terms, as shown in the provided step-by-step solution, which has simplified the equation to \( y = -1/2x^2 + 2x - 2 \).