Problem 44 - A Graphical Approach to College Algebra

Question

Use translations of one of the basic functions \(y=x^{2}, y=x^{3},\) \(y=\sqrt{x},\) or \(y=|x|\) to sketch a graph of \(y=f(x)\) by hand. Do not use a calculator. $$y=\sqrt{x+2}$$

Step-by-Step Solution

Answer

Graph starts at (-2, 0) and follows the pattern of \( y = \sqrt{x} \) shifted 2 units left.

Step 1: Identify the Base Function

The base function for this graph is the square root function, denoted by \( y = \sqrt{x} \). This function typically starts at the origin (0,0) and has a shape that extends to the right and upwards.

Step 2: Determine the Translation

The function \( y = \sqrt{x+2} \) can be obtained by translating the base function \( y = \sqrt{x} \). The \( +2 \) inside the square root indicates a horizontal shift. Specifically, \( y = \sqrt{x+2} \) represents a shift of the base graph \( y = \sqrt{x} \) 2 units to the left on the x-axis.

Step 3: Find Key Points of the Base Function

For \( y = \sqrt{x} \), some key points are: (0, 0), (1, 1), and (4, 2), which reflects the basic property \( y = \sqrt{x} \). These points will shift according to the translation determined earlier.

Step 4: Apply Translation to Key Points

We apply the translation 2 units to the left to each key point:1. (0, 0) shifts to (-2, 0).2. (1, 1) shifts to (-1, 1).3. (4, 2) shifts to (2, 2).This gives the key points for the transformed function \( y = \sqrt{x+2} \).

Step 5: Sketch the Graph

Using the shifted key points (-2, 0), (-1, 1), and (2, 2), sketch a smooth curve that connects these points and follows the pattern of \( y = \sqrt{x} \), extending to the right as it moves away from the starting point (-2, 0).

Key Concepts

Base Function Identification

Identifying the base function is the first step to understanding any transformation or translation. In the given exercise, the function is \( y = \sqrt{x+2} \). The base function here is the square root function, \( y = \sqrt{x} \). Knowing the base function is crucial because it serves as the reference point for any transformations.The square root function, \( y = \sqrt{x} \), is characterized by its distinct shape. It begins at the origin (0,0) and continues to grow steadily as \( x \) increases. This can be visualized as a curve that shoots up and to the right, never looping back or dipping down. Understanding the base function's characteristics, such as starting point and general shape, are building blocks for predicting how transformations will alter the graph.

Horizontal Shifts

The function \( y = \sqrt{x+2} \) involves a horizontal shift, which occurs when a constant is added to or subtracted from the variable within the function. In this case, the \(+2\) within the square root signifies a shift to the left by 2 units on the x-axis.A useful tip is to remember that adding a number inside the square root or parentheses results in a left shift, while subtracting would cause a shift to the right. It's like setting a new starting point for the base function.

• For our function \( y = \sqrt{x+2} \), this leftward shift means all points on the base graph move directly left by 2 units.

Perceiving the function’s shift is important for graphing, as it repositions the function while maintaining its other characteristics.

Key Points in Graphing

Identifying and graphing key points help bridge the base function and the transformed function. The key points for the base function \( y = \sqrt{x} \) include familiar coordinates:

• (0, 0) – where the graph starts
• (1, 1) – where the graph has equal x and y values
• (4, 2) – where the x value quadruples compared to y

These points offer a map to plot the original curve accurately. For \( y = \sqrt{x+2} \), we apply the shift determined earlier.

• (0, 0) shifts to (-2, 0)
• (1, 1) shifts to (-1, 1)
• (4, 2) shifts to (2, 2)

Once these points are accurately plotted, drawing a smooth curve passing through these points mimics the shape of the square root function but starts from the new point (-2, 0). Understanding and plotting these key points simplifies sketching complex transformations by providing precise guidance where the curve should go and how it should look.