Problem 69 - 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 \(g(x) = \sqrt{x+2}\) is a result from taking the original function \(f(x) = \sqrt{x}\) and shifting it 2 units to the left. The graph starts at (-2,0) and extends in a curve towards the positive x- and y-axis.

Step 1: Graph the function \(f(x) = \sqrt{x}\)

Start by graphing the basic function \(f(x) = \sqrt{x}\) where the principal square root of x is taken. This graph starts from the point (0,0) and extends towards positive x-axis and y-axis, creating an increasing curved line.

Step 2: Understand the transformation

The function \(g(x) = \sqrt{x+2}\) is a transformation of the function \(f(x) = \sqrt{x}\) where the graph shifts horizontally to the left by 2 units. This is due to the fact that adding a value inside the square root function results in a horizontal shift opposite to the direction of the value added.

Step 3: Graph the function \(g(x) = \sqrt{x+2}\)

To graph the function \(g(x) = \sqrt{x+2}\), simply take the graph of the function \(f(x) = \sqrt{x}\) and shift all points 2 units to the left. The graph starts from (-2,0) and extends towards the right, increasing along the y-axis in a similar curve to \(f(x) = \sqrt{x}\).

Key Concepts

Square Root Function

The square root function, often represented as \(f(x) = \sqrt{x}\), is a fundamental mathematical function that features prominently in algebra and precalculus. The principal square root function involves finding the non-negative square root of a number. It generates a half-curve that starts at the origin

• Begins at point (0,0).
• Curves gently upward, moving towards positive x and y directions.
• Always produces non-negative results, thus, it only exists for non-negative values of \(x\).

This function forms a lazy, gradual slope that represents each positive number's square root as you move along the x-axis. It’s a foundational graph in understanding more complex transformations, like shifts and stretches.

Horizontal Shift

A horizontal shift is a transformation that moves a graph left or right on the coordinate plane, without altering its shape. For functions, horizontal shifts occur when you add or subtract a number inside the function's argument. In the case of the function \(g(x) = \sqrt{x+2}\), here's how the shift works:

• By adding 2 inside the square root, you shift the entire graph to the left by 2 units.
• This shift is counterintuitive: adding a number pulls the graph in the negative x-direction.
• Starting point changes from (0,0) to (-2,0).

Understanding this basic principle of horizontal shifts helps in graphing and interpreting various transformed functions.

Function Graphing

Function graphing is a crucial skill in mathematics that aids in visualizing how functions behave and change over a set domain. To graph a given function like \(g(x) = \sqrt{x+2}\), follow these tips:

• Identify the base graph: For example, \(f(x) = \sqrt{x}\) is the base.
• Determine transformations: In our case, a horizontal shift left by 2 units.
• Apply the transformation: Shift the base graph accordingly.

The newly adjusted graph illustrates visually how transformations affect the original function, resulting in \(g(x)\). Graphing is often the step where many students solidify their understanding of both the properties of a function and the implications of transformations like shifts and scalings. Each point on the graph represents a detailed plot of the function’s behavior, hence why it's so valuable in understanding mathematical concepts.