Problem 41 - Algebra and Trigonometry

Question

33–48 ? 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, left-shifted, square root function, moved down 3 units.

Step 1: Identify the Base Function

Begin by identifying the base function we are working with. In this case, the base function is the square root function, represented as \( y = \sqrt{x} \). This is the parent function that we will transform.

Step 2: Horizontal Shift

The expression inside the square root, \( x + 4 \), indicates a horizontal shift. A function \( f(x+c) \) shifts \( |c| \) units horizontally. Here, \(+4\) means we shift the graph 4 units to the left. The new function is \( y = \sqrt{x + 4} \).

Step 3: Vertical Scaling

The coefficient \( \frac{1}{2} \) in front of the square root represents a vertical scaling. It compresses the graph vertically by a factor of \( \frac{1}{2} \). So now our function becomes \( y = \frac{1}{2} \sqrt{x + 4} \).

Step 4: Vertical Shift

The \(-3\) at the end of the function \( y = \frac{1}{2} \sqrt{x + 4} - 3 \) represents a downward shift of the graph by 3 units. This completes the transformation process.

Step 5: Sketch the Graph

To sketch the graph, begin by drawing the square root graph \( y = \sqrt{x} \). Apply the 4-unit shift to the left, the vertical compression by half, and finally, the 3-unit downward shift. The resulting graph shows all transformations applied to \( y = \frac{1}{2} \sqrt{x+4} - 3 \).

Key Concepts

Square Root Function

The square root function is one of the basic functions in algebra and functions. It is denoted as \( y = \sqrt{x} \) and typically forms a curve starting from the origin (0,0). The function is only defined for non-negative values of \( x \) because taking the square root of a negative value is not possible in the set of real numbers.
Some characteristics of the square root function include:

• It passes through the point (0,0)
• It increases more slowly than linear functions, resulting in a curve that rises gradually and never decreases
• The domain is \( x \geq 0 \)
• The range is \( y \geq 0 \)

Understanding these traits helps in identifying and transforming the square root function during graph transformations.

Horizontal Shift

Horizontal shifts in functions involve moving a graph left or right on the coordinate plane. This is typically represented inside the function in the form \( f(x+c) \). A positive \( c \) value means the graph shifts to the left, while a negative \( c \) means it shifts to the right.
For the problem at hand, we have \( y = \sqrt{x+4} \). The \(+4\) inside the square root causes a horizontal shift of the graph 4 units to the left. This may seem counterintuitive, but you can think of it in terms of solving \( x + 4 = 0 \):

• The solution \( x = -4 \) indicates the value of \( x \) that results in a square root of 0, meaning the curve now starts at (-4, 0) rather than (0, 0).

This concept of shifting helps create long-term dependency in function operations and is crucial when studying graph transformations.

Vertical Scaling

Vertical scaling alters the steepness or narrowness of the graph of a function. It occurs when a function is multiplied by a constant, causing it to stretch or compress vertically. In the given equation \( y = \frac{1}{2} \sqrt{x+4} \), the coefficient \( \frac{1}{2} \) causes a vertical compression.
This process:

• Reduces the vertical distance the graph travels as \( x \) increases
• Makes the graph appear flatter, as each output value is halved

Vertical scaling does not affect the shape of the roots of the equation; it only alters how the output values correspond to input values, maintaining the same domain but altering the range.

Vertical Shift

Vertical shifts move a graph up or down along the y-axis, impacting the entire graph's position relative to the standard axes. This is achieved by adding or subtracting a constant to the entire function.
In our example, the transformation \( -3 \) in \( y = \frac{1}{2} \sqrt{x+4} - 3 \) shifts the graph downwards by 3 units. This adjustment:

• Lowers every point on the graph by 3
• Modifies the range of the function to begin from \(-3\) instead of \(0\)

Vertical shifts, much like horizontal, do not alter the domain of the function but adjust the y-values, providing flexibility in function analysis and practical applications.