Problem 67 - Precalculus Essentials

Question

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$g(x)=\sqrt{x}+2$$

Step-by-Step Solution

Answer

The graph of the function \(g(x)=\sqrt{x}+2\) is the graph of the function \(f(x)=\sqrt{x}\) shifted upwards by 2 units on the y-axis. Both graphs have the same shape, but the position of the latter graph is 2 units higher.

Step 1: Graph the Square Root Function

Draw or plot the function \(f(x)=\sqrt{x}\). This function starts at the origin (0,0) and increases as x increases, creating an upward curve.

Step 2: Understand the Transformation

Adding a constant to the function results in a vertical shift of the graph. Here, the constant is +2 which means our graph will move upwards by 2 units.

Step 3: Graph the Transformed Function

Shift every point of the original graph, \(f(x)=\sqrt{x}\), upwards by 2 units to graph \(g(x)=\sqrt{x}+2\). The graph still has the basic shape of the square root function but is now located 2 units higher on the y-axis.

Key Concepts

Understanding the Square Root Function

The square root function, represented as \(f(x) = \sqrt{x}\), is one of the fundamental building blocks in mathematics. This function is defined for non-negative values of \(x\) because the square root of a negative number is not a real number.
When you graph \(f(x) = \sqrt{x}\), you'll notice it starts at the point (0,0) and moves upward in a curve as \(x\) increases. This occurs because as \(x\) gets larger, its square root also becomes larger, but at a decreasing rate.
Key characteristics of this function include:

• Domain: \(x \geq 0\)
• Range: \(y \geq 0\)
• Shape: An increasing curve starting from the origin

The square root function is essential for understanding various transformations which we will explore further in this article.

The Concept of Vertical Shift

A vertical shift in a graph is a transformation that moves the graph up or down without changing its shape. It's a result of adding or subtracting a constant to the original function.
In our specific problem, the function transformation involves \(g(x) = \sqrt{x} + 2\). Here, the constant +2 signifies a vertical shift upwards by 2 units. Essentially, you're picking up the entire graph of \(f(x) = \sqrt{x}\) and moving it 2 units higher along the y-axis.
This type of transformation does not alter:

• the domain of the function, which remains \(x \geq 0\)
• the shape of the graph

However, the range of the function becomes \(y \geq 2\) because the entire curve is now positioned two units above the original. Understanding vertical shifts is fundamental in graphing functions as it helps in visualizing their spatial orientation.

Mastering Function Graphing

Graphing functions is a vital skill in mathematics that enables you to visualize and understand relationships between variables.
Using transformations, like the vertical shift explored above, simplifies this process by allowing us to start from a known function and apply changes systematically. This technique is particularly beneficial with complex functions built from simpler ones.
Here’s a quick guide to mastering function graphing:

• Start by graphing any known base function. For example, \(f(x) = \sqrt{x}\).
• Identify any transformations needed (like vertical shifts, stretches, or reflections).
• Apply these transformations step-by-step, first focusing on changes that impact one dimension like vertical shifts, then moving to others as needed.
• Check your final graph by looking at key points and ensuring they fit the transformed equations.

Function graphing may seem complex initially, but breaking it down into base functions and transformations makes it much more approachable and manageable.