Problem 43 - Precalculus: Functions and Graphs

Question

Find an equation for the indicated half of the parabola. Right half of \((x+1)^{2}=y-4\)

Step-by-Step Solution

Answer

The equation for the right half is \(x = \sqrt{y - 4} - 1\) with \(y \geq 4\).

Step 1: Identify the original equation

The given equation is \((x+1)^{2} = y-4\). This equation can be rewritten in the form \(y = (x+1)^{2} + 4\) to represent a parabola that opens upwards.

Step 2: Determine the vertex of the parabola

The vertex form of the equation follows \((x - h)^{2} = y - k\), where \((h, k)\) is the vertex of the parabola. From the equation \((x+1)^{2} = y-4\), we identify the vertex as \((-1, 4)\).

Step 3: Analyze the direction of the parabola

Since the equation is in the form \((x+1)^{2} = y-4\), it resembles \(x^{2} = y\), which indicates the parabola opens upwards with the axis of symmetry being vertical.

Step 4: Find the equation for the right half of the parabola

The right half of the parabola includes the portion where \(x\geq -1\), which is the vertex. In terms of the parabola's orientation, the right half corresponds to \((x+1) \geq 0\), or \(x \geq -1\). Thus, for the right half, we will solve for \(x\) in terms of \(y\).

Step 5: Solve for x for the right side

Starting from \((x+1)^{2} = y-4\), solve for \(x\): \(x + 1 = \sqrt{y - 4}\) since we want the right half of the parabola and thus use the positive square root. Therefore, \(x = \sqrt{y - 4} - 1\).

Step 6: Incorporate domain limitations

Since we are considering the right half, make sure that \(y\geq 4\), because \(\sqrt{y-4}\) is only defined for \(y - 4 \geq 0\). The equation for the right half becomes \(x = \sqrt{y - 4} - 1\) with \(y\geq 4\).

Key Concepts

Vertex form of a parabola

The vertex form of a parabola is a special way of expressing the equation of a parabola that makes it easy to find the vertex, which represents the highest or lowest point of the parabola, depending on its orientation.
The vertex form is written as:

• \( (x-h)^2 = y-k \)

In this form, \( (h, k) \) is the vertex of the parabola. It provides a quick way to identify the vertex without reshuffling the equation or graphing the function.
By transforming any parabola equation into its vertex form, you can immediately determine the vertex point, a crucial piece of information for graphing or solving real-world problems.
In our example, the vertex form \((x + 1)^2 = y - 4\) shows us that the vertex is at \((-1, 4)\). This information helps us understand how the parabola is positioned in the coordinate plane.

Equation of a parabola

The equation of a parabola is the mathematical description of the curve, and it can be expressed in multiple forms, such as standard form, vertex form, or factored form. Each form provides different but useful information about the properties of the parabola.
The standard form of a parabola that opens upwards or downwards is written as:

• \(y = ax^2 + bx + c\)

While the vertex form, as discussed earlier, emphasizes the vertex, the standard form allows for easy identification of the parabola's direction and width.
Our given parabola equation, \((x+1)^2 = y-4\), is structurally similar to \(x^2 = y\), indicating that it opens upwards. By reworking this into the vertex form \(y = (x+1)^2 + 4\), we learn its orientation in the plane.
Finding these forms is crucial for solving problems or adjustments in graphing, and is essential in converting between each to suit the particular needs of the situation.

Axis of symmetry

The axis of symmetry in a parabola is an imaginary vertical line which divides the parabola into two mirror-image halves. This line is key to understanding the parabola's shape and direction.
For a parabola in the vertex form \( (x-h)^2 = y-k \), the axis of symmetry can be easily read from the equation as the line \(x = h\).
It always passes through the vertex of the parabola, ensuring the parabola is perfectly balanced around it. Understanding this symmetry helps with sketching the parabola and calculating points on the parabola's curve.
In the given equation \((x+1)^2 = y-4\), the axis of symmetry is \(x = -1\). This knowledge is vital when determining which part of the parabola we are working with, and for ensuring proper calculations and solutions involving specific regions of a parabola.
Understanding the axis of symmetry also aids in predicting the parabola's behavior and can be useful when working with quadratic relations in physics and economics.