Problem 52 - Precalculus: Functions and Graphs

Question

Determine whether the graph of the equation is the upper, lower, left, or right half of a parabola, and find an equation for the parabola. $$x=\sqrt{y-4}+8$$

Step-by-Step Solution

Answer

The graph is the right half of an upward-opening parabola: \( y = (x - 8)^2 + 4 \).

Step 1: Identify the given equation

The given equation is \( x = \sqrt{y-4} + 8 \). This equation expresses \( x \) in terms of \( y \), suggesting it might represent a parabola that opens either to the right or the left.

Step 2: Isolate the square root term

To begin analyzing the parabola, isolate the square root term by subtracting 8 from both sides of the equation.\[ x - 8 = \sqrt{y-4} \]

Step 3: Square both sides

Square both sides of the equation from Step 2 to eliminate the square root. This will help in finding the standard form of the parabola equation.\[ (x - 8)^2 = y - 4 \]

Step 4: Solve for y

Finally, solve for \( y \) by adding 4 to both sides to express the equation explicitly in terms of \( y \). \[ y = (x - 8)^2 + 4 \]

Step 5: Determine the orientation of the parabola

The equation \( y = (x - 8)^2 + 4 \) is in the standard form of a parabola \( y = a(x-h)^2 + k \), which opens upwards since the coefficient \( a = 1 \) is positive.

Step 6: Interpret the graph in terms of x

Originally, from \( x = \sqrt{y-4} + 8 \), we had a positive square root, indicating this equation represented only the right half of the parabola. The \( +8 \) suggests a horizontal shift. Therefore, the graph is the right half of the parabola opening upwards.

Key Concepts

Graph Orientation

When analyzing a parabola, understanding its orientation is key. Parabolas can open upwards, downwards, to the left, or to the right. The direction in which a parabola opens depends on its equation. For instance, if it's written in the form \( y = a(x-h)^2 + k \), the parabola opens either up or down. If \( a \) is positive, it opens upwards. If \( a \) is negative, it opens downwards.
However, if a parabola is expressed such that \( x = a(y-k)^2 + h \), then the parabola opens left or right. Specifically:

• If \( a > 0 \), the parabola opens to the right.
• If \( a < 0 \), it opens to the left.

Understanding these basic principles can help you determine the section of the parabola represented by an equation. In our exercise, we started with the equation \( x = \sqrt{y-4} + 8 \), which involved a square root, suggesting a half-parabola. In particular, we determined it represents the right half, as the square root terms usually indicate the right or upper half, depending on context.

Square Roots

Square roots play an essential role in describing half-parabolas in mathematical expressions. When a parabola's equation includes a square root, like \( x = \sqrt{y-4} + 8 \), it typically indicates that the graph only represents one half of the complete parabola.
For example, in the equation given in the exercise, the presence of the square root implies that the value of \( x \) is considered only for positive values coming from the square root. Consequently, the graph will be limited to one direction or side:

• In the context \( x = \sqrt{y} \), the graph will reflect positive \( y \) values, representing what is typically the right half of a horizontally oriented parabola.
• If it were \( x = -\sqrt{y} \), it would describe the left half.

This understanding helps you interpret how and why certain equations only showcase part of the overall parabolic structure.

Horizontal Translation

Horizontal translation refers to shifting a graph left or right along the x-axis. This concept is vital for graph transformations of parabolas. Let's explore how this applies to our equation.
The mathematical form \( x = \sqrt{y-4} + 8 \) tells us about the position of the parabola on the graph. The \(+8\) outside the square root adjusts the graph horizontally. Specifically:

• A \(+8\) shifts the entire graph to the right by 8 units.
• Conversely, a \(-8\) would shift it left by the same amount.

This horizontal shift is a translation because it does not change the shape of the parabola, only its position. It is important to note that other translations, such as vertical shifts, change the \( k \) value, affecting the apex of the parabola on the \( y \)-axis. In our exercise, recognizing the \(+8\) helped us understand why the graph was shifted right.

Quadratic Equations

At the heart of parabolic graphs lies the quadratic equation. Quadratic equations are expressed in the form \( ax^2 + bx + c = 0 \) or similar. They are paramount in graphing parabolas.
In our case, the exercise transitions to quadratic expression form through manipulation. We commenced with \( x = \sqrt{y-4} + 8 \), which involved isolating \( y \) for a complete squared equation. By squaring both sides, we arrived at \( (x-8)^2 = y - 4 \). When rearranged, this gives \( y = (x-8)^2 + 4 \).
This is a classic quadratic equation showing a parabolic structure. Here, the parabola opens upwards due to the positive coefficient \( a = 1 \). Dealing with quadratic equations allows us to:

• Determine critical points like the vertex \((h, k)\).
• Understand the symmetry and direction of the graph.
• Evaluate the domain and range for more complex analyses.

Mastering quadratics equips you with skills to interpret various polynomial scenarios involving parabolas efficiently.