Problem 43 - Algebra and Trigonometry

Question

\(21-44\) . Sketch the graph of the function, not by plotting points, but by starting with the graph of a standard function and applying transformations. $$ y=\frac{1}{2} \sqrt{x+4}-3 $$

Step-by-Step Solution

Answer

The graph is a vertically compressed and shifted square root function.

Step 1: Identify the standard function

The given function is built upon the square root function. The standard form to start with is the basic square root function, \( y = \sqrt{x} \). This is our starting point for transformations.

Step 2: Horizontal Shift

The function is \( y = \frac{1}{2} \sqrt{x+4} - 3 \). The expression inside the square root, \( x + 4 \), indicates a horizontal shift. Since it is \( +4 \), the graph of \( y = \sqrt{x} \) is shifted 4 units to the left. Now, the function is \( y = \sqrt{x+4} \).

Step 3: Vertical Compression

The factor \( \frac{1}{2} \) in front of the square root term indicates a vertical compression. The graph of \( y = \sqrt{x+4} \) is compressed vertically by a factor of \( \frac{1}{2} \). This changes it to \( y = \frac{1}{2}\sqrt{x+4} \).

Step 4: Vertical Shift

The final adjustment is the \( -3 \) outside the square root, indicating a vertical shift downward. Thus, every point on the graph of \( y = \frac{1}{2}\sqrt{x+4} \) is moved down by 3 units, resulting in the final graph of \( y = \frac{1}{2}\sqrt{x+4} - 3 \).

Key Concepts

square root function

A square root function is a type of function that involves the square root of a variable. The most basic form of this function is represented as \( y = \sqrt{x} \). This function produces a graph that starts at the origin (0,0) and produces a gentle curve that rises to the right. It is important to remember that the square root function only produces outputs for non-negative values of \( x \), because the square root of a negative number is not a real number.

• The graph of \( y = \sqrt{x} \) is increasing, always going up as \( x \) values increase.
• This function has the unique property of being symmetric with respect to a vertical line through the point (0,0).

The square root function is foundational in understanding more complex transformations as seen in applied math fields.

horizontal shift

A horizontal shift in the graph of a function occurs when the entire graph moves to the left or to the right on the coordinate plane. This is usually represented by adding or subtracting a constant from the \( x \)-variable inside a function. For example, in the function \( y = \sqrt{x+4} \), the graph of \( y = \sqrt{x} \) is shifted horizontally.

How it Works

When you see \( x+4 \), it means that the graph shifts 4 units to the left. This might seem the opposite of what one would expect since we are adding a number. Always remember:

• \( x + 4 \) shift to the left: 4 units left.
• \( x - 4 \) would shift to the right: 4 units right.

This concept is key for transforming the graph's position along the \( x \)-axis without altering its shape directly.

vertical compression

Vertical compression involves altering the graph of a function so that it becomes "flatter" or less steep. This is achieved by multiplying the entire function by a factor that is between 0 and 1. For example, in the function \( y = \frac{1}{2}\sqrt{x+4} \), the presence of the \( \frac{1}{2} \) causes vertical compression.

Implications of Vertical Compression

This factor influences how the graph stretches or compresses vertically:

• Because \( \frac{1}{2} \) is less than 1 and greater than 0, it causes the graph to "flatten" by half vertically.
• Every y-coordinate (output) of the function \( \sqrt{x+4} \) is halved, making the graph of \( y = \frac{1}{2}\sqrt{x+4} \) rise more gently.

Understanding vertical compression is crucial in grasping how factors affect the visual representation of functions.

vertical shift

A vertical shift moves the graph of a function up or down on the coordinate plane. This is accomplished by adding or subtracting a constant from the entire function. In the function \( y = \frac{1}{2}\sqrt{x+4} - 3 \), the subtraction of 3 shifts the graph vertically.

Vertical Shift Details

Understanding vertical shifts helps in determining how a graph is altered:

• Subtracting a number, such as \( -3 \), shifts the graph down by 3 units.
• Adding a number would shift it up.

These shifts do not change the shape of the graph. Instead, they adjust its position along the y-axis. This helps you pinpoint the new baseline level from where other transformations start.