Problem 46 - Precalculus: Functions and Graphs

Question

Find an equation for the indicated half of the parabola. Lower half of \((y+4)^{2}=x-3\)

Step-by-Step Solution

Answer

The lower half of the parabola is given by \(y = -4 - \sqrt{x-3}\).

Step 1: Understand the Given Equation

The given equation \((y+4)^2 = x-3\) represents a parabola. This comes in the form of \((y-k)^2 = 4p(x-h)\) which opens to the right or left because \(y\) is squared. Here, \((h, k)\) is the vertex. Comparing, the vertex \((h, k)\) is \((3, -4)\).

Step 2: Solving for x

Since we are looking for \(x\) as a function of \(y\), rearrange the equation: \((y+4)^2 + 3 = x\). This simplifies to \(x = (y+4)^2 + 3\).

Step 3: Determine the Lower Half

The equation \((y+4)^{2}=x-3\) represents a parabola symmetric with respect to the line \(y = -4\). To get the lower half, we recognize that for a parabolic equation opening sideways, the 'lower' part corresponds to when \(y+4\) is non-positive, meaning \(y+4 ≤ 0\). Thus, take \(y+4 = -\sqrt{x-3}\), giving us \(x = (y+4)^2 + 3\) with \(y = -4 -\sqrt{x-3}\).

Step 4: Final Equation

The equation that represents the lower half of the given parabola is \(y = -4 - \sqrt{x-3}\), ensuring \(y+4 ≤ 0\).

Key Concepts

Understanding the Vertex Form

Vertex form is an essential concept when dealing with parabolas. It's generally given by the equation \((y-k)^2 = 4p(x-h)\) for horizontal parabolas. Here, \(h\) and \(k\) represent the vertex coordinates of the parabola. This form is particularly useful because it tells us exactly where the vertex of the parabola is located, and it indicates the direction the parabola opens. For horizontal parabolas, the graph opens to the left or the right, not up or down.

• \(h\) is the x-coordinate of the vertex.
• \(k\) is the y-coordinate of the vertex.
• 4p indicates the distance that determines how "wide" or "narrow" the parabola will be.

In our given problem, \((y+4)^2 = x-3\), by matching it to the vertex form, we can determine that \(h = 3\) and \(k = -4\). This tells us that the vertex of the given equation is at the point \( (3, -4) \).

Exploring Parabola Symmetry

Parabolas have a unique property: symmetry. For horizontal parabolas, this symmetry is about a horizontal line, specifically a line parallel to the x-axis. In the equation \((y+4)^2 = x-3\), the line of symmetry is given by \((k = -4)\). This means that the parabola is mirrored on either side of the horizontal line \(y = -4\).
This symmetry is significant because it allows us to separate the parabola into two halves.

• The upper half, where \(y > -4\), and
• The lower half, where \(y <= -4\).

Understanding symmetry helps in identifying these halves and in graphing the parabola accurately.

Identifying Horizontal Parabolas

Unlike the more commonly visualized vertical parabolas, horizontal parabolas are defined by an equation where \(y\) is squared, such as in \((y-k)^2 = 4p(x-h)\).

• Horizontal parabolas open to the right if \(4p > 0\).
• They open to the left if \(4p < 0\).

In solving the equation for the given parabola, understanding that it is horizontal helps us determine which direction it opens. In this exercise, the parabola opens to the right because when rearranged, \(x = (y+4)^2 + 3\) is a standard positive quadratic expression for \(x\). Hence, it opens towards increasing \(x\) values.

Approach to Solving Quadratic Equations

To solve quadratic equations effectively, it is often useful to rearrange them into a form that makes finding the solutions straightforward. Here, with \((y+4)^2 = x-3\), we rearranged it to find \(x\) in terms of \(y\): \(x = (y+4)^2 + 3\). Doing this helps in clearly delineating the function and guides us towards solving for certain conditions, like the lower half of the parabola.
When dealing with parabolic splitting (upper and lower halves), understanding the domain and range conditions is vital. Since the task at hand was to determine the lower half, ensuring that \(y+4\leq 0\) was a crucial step to express it effectively. Thus, \(y = -4 - \sqrt{x-3}\) aligns perfectly with maintaining \(y \le -4\), giving us the needed section of the parabola.