Problem 72 - College Algebra

Question

Begin by graphing the square root function, \(f(x)-\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)--\sqrt{x+1} $$

Step-by-Step Solution

Answer

The graph of the function \( h(x) = \sqrt{x + 1} \) is the graph of the square root function \( f(x) = \sqrt{x} \) shifted 1 unit to the left.

Step 1: Understand The Square Root Function

The square root function, \( f(x) = \sqrt{x} \), is only defined for \( x \geq 0 \), and it resembles half of a sideways parabola. The point (0,0) is the vertex of the parabola.

Step 2: Identify The Transformation

In the given function \( h(x) = \sqrt{x + 1} \), the transformation is \( x + 1 \). This is a horizontal shift. Since \( + 1 \) is inside the square root function, this implies a leftward shift of the graph by 1 unit.

Step 3: Plot The Transformed Function

Start by plotting the square root function \( f(x) = \sqrt{x} \), then shift every point in the graph 1 unit to the left to graph \( h(x) = \sqrt{x + 1} \). As a result, (0,0) moves to (-1,0), and this becomes the vertex of the transformed function.

Key Concepts

Understanding the Square Root Function

The square root function, denoted as \( f(x) = \sqrt{x} \), is fundamental in mathematics for various calculations and graphing techniques. This function is only defined for non-negative values of \( x \) (i.e., \( x \geq 0 \)), reflecting its domain limitation due to the properties of square roots.
Think of the graph of \( f(x) = \sqrt{x} \) as half of a parabola lying on its side, with its opening extending to the right. Its "vertex", which is the starting point of the graph, is located at the origin \((0, 0)\).

• Notice that as \( x \) increases, \( \sqrt{x} \) increases as well, but at a decreasing rate, indicating a curve that flattens as \( x \) grows.
• Understanding this basic shape and behavior of the square root function is critical as it forms the foundational graph before any transformations.

Insight into Horizontal Shifts

A horizontal shift in graphing involves moving the entire graph of a function left or right on the coordinate plane. When analyzing transformations within a function, such as \( h(x) = \sqrt{x + 1} \), the term \( x + 1 \) indicates a horizontal shift.
But how does this work?

• The expression \( x + 1 \) within the function means each x-value of the original function \( \sqrt{x} \) is effectively increased by 1. However, visually, this shift is counterintuitive, moving the graph to the left.
• Generally, \( x+c \) inside a function shifts the graph to the left by \( c \) units, while \( x-c \) shifts it to the right.

In this context, adding 1 within a square root, \( \sqrt{x+1} \), results in moving every point of the initial \( \sqrt{x} \) graph leftward by one unit, which is crucial to achieving the correct transformed graph.

Techniques for Function Graphing

Graphing functions often involves understanding and measuring transformations applied to basic functions like the square root function.
When graphing a transformed function such as \( h(x) = \sqrt{x + 1} \), it is essential to follow a step-by-step method:

• First, accurately sketch the basic graph you're starting with, in this case, \( f(x) = \sqrt{x} \).
• Next, apply the transformation, which here is a horizontal left shift by 1 unit.
• This means taking each point on the original graph and moving it leftward, ensuring consistent spacing and accuracy, especially for the graph's new vertex moving to \((-1,0)\).

Graphing transformations allow us to visually comprehend how algebraic changes affect function shapes, enhancing our understanding and application of mathematical functions.