Problem 10 - Precalculus : Building Concepts and Connections

Question

Identify the underlying basic function, and use transformations of the basic function to sketch the graph of the given function. $$f(x)=\sqrt{x+3}$$

Step-by-Step Solution

Answer

The function \(f(x) =\sqrt{x+3}\) is a transformation of the basic function \(f(x) =\sqrt{x}\), specifically a shift of 3 units to the left.

Step 1: Identify Basic Function

The basic function in this case is \( f(x) =\sqrt{x}\). We can identify this by removing the transformations from the given function. \(f(x) =\sqrt{x+3}\) becomes \(f(x) =\sqrt{x}\) when we remove the shift of 3 units to the left.

Step 2: Identify Transformations

Now, the task is to identify the transformations applied to the basic function. Looking at the function \(f(x) =\sqrt{x+3}\), we can see that the transformation is a horizontal shift of 3 units to the left. This is due to the '+3' inside the function sqrt, which moves the graph to the left.

Step 3: Sketch the Graph

The last step is to sketch the graph of the function using the knowledge of the basic function and the transformations. Starting with the basic function \(f(x) =\sqrt{x}\), which is the graph of a square root starting at the origin (0,0) and extending to the right, we then apply the transformation. The transformation is a shift of 3 units to the left, so move the entire curve of the basic function 3 units to the left. This results in the graph of the function \(f(x) =\sqrt{x+3}\)

Step 4: Validate the result

To make sure the graph is correct, you can plug in some values for x into the function and see if the output matches the y-value at that point on the graph. For example, if you plug in x = 1 into \(f(x) =\sqrt{x+3}\), the output is \(f(1) =\sqrt{1+3} = 2\). Therefore, the graph should pass through the point (1,2).

Key Concepts

Basic Function Identification

Understanding the foundation of any mathematical graph begins with identifying the basic function. In the realm of algebra, the basic function is the simplest form of the function without any modifications or transformations. For the function \[\begin{equation}f(x) = \[\math\]sqrt{x + 3}\end{equation}\], the underlying basic function is \[\begin{equation}f(x) = \[\math\]sqrt{x}\end{equation}\]. The square root function is a fundamental part of algebra that students encounter early on. It represents the set of all points \[\(x, y\)\] such that \[\y = \sqrt{x}\] for non-negative \[\x\] values. This basic function starts at the origin (0,0) and extends infinitely to the right, creating a curve that gradually flattens as \[\x\] increases. By stripping away additional terms and focusing on the core component, \[\sqrt{x}\], students can begin to understand the nature and shape of the graph before diving into the effects of any transformations.

Function Transformations

Once the basic function is identified, the next step is to explore function transformations. Transformations can be thought of as movements or changes applied to the basic graph. The original function \[\f(x) = \sqrt{x + 3}\] involves a specific transformation known as a horizontal shift.

Understanding Horizontal Shifts

The '+3' inside the square root indicates that the graph of the basic square root function is to be shifted 3 units to the left. This type of transformation is vital for altering the position of the graph on the coordinate plane without affecting its shape. In general, for the basic function \[\f(x) = \sqrt{x}\], a function of the form \[\f(x) = \sqrt{x + a}\] will shift the graph \[\a\] units to the left if \[\a\] is positive, and \[\a\] units to the right if \[\a\] is negative.

• A positive inside the square root shifts left.
• A negative inside the square root shifts right.

By visualizing these movements, students can grasp how each transformation manipulates the graph.

Sketching Function Graphs

Sketching function graphs is the art of bringing the effects of transformations on the basic function to life on paper. With \[\f(x) = \sqrt{x + 3}\], after recognizing the horizontal shift, one can start sketching.

Start with the Basic Graph:

First, sketch the basic square root graph, which begins at the origin and rises, while curving to the right.

Apply the Transformation:

Next, apply the identified transformation by shifting the graph 3 units to the left.

Check Points:

Finally, it's important to validate the sketch. For instance, calculating \[\f(1)\] should give you 2, meaning the point (1,2) must be on your graph. It's this detail-oriented approach to sketching that can solidify a student's comprehension of graphing square root functions. Ensuring that the curve appropriately reflects the transformation will help students not only in sketching but also in understanding the broader implications of function behaviors.