Problem 35 - A Graphical Approach to Precalculus with Limits

Question

Give the equation of each function whose graph is described. The graph of \(y=\sqrt{x}\) is shifted 3 units to the right. This graph is then vertically stretched by applying a factor of \(4.5 .\) Finally, the graph is shifted 6 units downward.

Step-by-Step Solution

Answer

The equation is \( y = 4.5 \sqrt{x - 3} - 6 \).

Step 1: Shift the graph to the right

The graph of \( y = \sqrt{x} \) is shifted 3 units to the right. To achieve this, replace \( x \) with \( x - 3 \). The equation becomes \( y = \sqrt{x - 3} \).

Step 2: Apply vertical stretch

The graph is vertically stretched by a factor of 4.5. Multiply the entire function by 4.5: \( y = 4.5 \times \sqrt{x - 3} \).

Step 3: Shift the graph downwards

The graph is shifted 6 units downward. Subtract 6 from the entire function: \( y = 4.5 \times \sqrt{x - 3} - 6 \).

Key Concepts

Graph Shifts

Graph shifts refer to moving the entire graph of a function up, down, left, or right on a coordinate plane. When shifting the graph of a function horizontally, you are adjusting the function's input or the variable inside the equation.

• Horizontal Shifts: To shift a graph horizontally, you modify the input of the function. For example, replacing every occurrence of "\(x\)" with "\(x - a\)" shifts the graph "\(a\)" units to the right. Conversely, using "\(x + a\)" would shift it "\(a\)" units to the left.
• Vertical Shifts: These are achieved by adding or subtracting a constant to the entire function. Adding a number moves the graph upward, while subtracting brings it downward.

In our problem, we first shifted the graph of the square root function 3 units to the right by replacing \(x\) with \(x-3\). Later, after stretching, we shifted the graph 6 units down by subtracting 6 from the function.

Vertical Stretching

Vertical stretching refers to manipulating the output value of a function, causing the graph to stretch or compress along the y-axis. This is achieved by multiplying the entire function by a constant factor.

• Stretching: If the factor is greater than 1, it pulls the graph away from the x-axis, making it taller and narrower.
• Compressing: Conversely, if the factor is between 0 and 1, it pushes the graph towards the x-axis, making it shorter and wider. A negative factor would also flip the graph upside down.

In this exercise, we applied a vertical stretch to the function \(y = \sqrt{x - 3}\) by multiplying the entire function by 4.5. This transformation amplifies the heights of the graph's peaks and valleys by 4.5 times, making the graph significantly steeper compared to its original version.

Square Root Functions

Square root functions are a type of radical function that includes the square root of a variable. These functions are typically represented as \(y = \sqrt{x}\) and feature a characteristic curve that starts at a certain point and increases slowly, forming a gentle slope that never decreases.

• They define the vertical and horizontal movement on a coordinate plane, playing a critical role in transformations.
• Such functions are only defined for non-negative values of \(x\), since the square root of a negative number is not real (unless complex numbers are considered).

In the problem, we start with the base square root function \(y = \sqrt{x}\), and apply successive transformations such as horizontal shifts, vertical stretches, and vertical shifts to effectively grasp how these changes can modify the behavior and appearance of square root functions. This insight is essential for understanding more complex graph transformations.