Problem 47 - Algebra and Trigonometry

Question

\(45-54=\) A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=\sqrt{x},\) shift 2 units to the left

Step-by-Step Solution

Answer

The transformed function is \(f(x) = \sqrt{x + 2}\).

Step 1: Identify the Original Function

The original function provided is \(f(x) = \sqrt{x}\). This function represents the square root of \(x\). It is important to first understand the graph of \(f(x)\) as the starting point of any transformations.

Step 2: Understand the Transformation

We need to shift the graph of the function \(f(x) = \sqrt{x}\) 2 units to the left. A horizontal shift to the left by \(c\) units is achieved by replacing \(x\) with \(x + c\) in the function. Here, \(c = 2\).

Step 3: Apply the Shift Transformation

To shift the function \(f(x) = \sqrt{x}\) 2 units to the left, replace \(x\) with \(x + 2\) in the function. The new function becomes \(f(x) = \sqrt{x + 2}\). This represents the graph of the square root function moved 2 units to the left.

Step 4: Write the Final Transformed Function

After applying the horizontal shift, the equation of the transformed function graph is \(f(x) = \sqrt{x + 2}\). This is the equation for the final transformed graph.

Key Concepts

Horizontal Shift

A horizontal shift is a type of graph transformation that changes the position of a function on the x-axis. When you want to move a graph left or right, you perform a horizontal shift. This is achieved by adding or subtracting a number to or from the variable inside the function. For instance, in the function transformation context, modifying the variable from \(x\) to \(x + c\) results in a shift of the graph \(c\) units to the left.
In our example with the square root function \(f(x) = \sqrt{x}\), shifting the graph 2 units to the left involves replacing \(x\) with \(x + 2\). The function transforms into \(f(x) = \sqrt{x + 2}\).

• A shift to the left uses the form \(x + c\).
• A shift to the right would use \(x - c\).

Remember, changing the graph horizontally does not affect its vertical placement or its general shape. It simply moves the position along the x-axis.

Square Root Function

The square root function is represented by \(f(x) = \sqrt{x}\). It is one of the basic functions in mathematics, recognized by its distinctive shape. The graph of the square root function has a gentle curve that starts at the origin \((0,0)\) and gradually rises, moving to the right.
This function is defined only for non-negative values of \(x\), as the square root of a negative number is not defined in the set of real numbers. Thus, its domain is \([0, \infty)\) and its range is also \([0, \infty)\).
In transformations, understanding the basic graph of a function like \(f(x) = \sqrt{x}\) is crucial. Any changes or shifts to this function will primarily affect where the curve starts or in which direction it will move from its starting point.

Graph Transformation

Graph transformation involves changing the position, shape, or size of a graph. These transformations can include translation (moving the graph up, down, left, or right), dilation (stretching or compressing the graph), and reflection (flipping the graph over an axis).
For our specific example of shifting the square root function 2 units to the left, we are dealing with a **translation transformation**. This does not change the size or shape of the graph; it simply moves it horizontally.

• Understanding the type of transformation applied helps predict how the graph looks post-transformation.
• Shifts or translations adjust the starting point of a graph but keep its original form intact.

By practicing various transformations, students strengthen their grasp on function behaviors and improve their skills in predicting how graphs will change based on different mathematical operations.