Problem 16 - 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. $$g(x)=2 \sqrt{x}$$

Step-by-Step Solution

Answer

The given function \(g(x) = 2 \sqrt{x}\) is derived from the basic function \(\sqrt{x}\) with a transformation of a vertical stretch by a factor of 2.

Step 1: Identification of the Basic Function

We first recognize the underlying basic function within \(g(x) = 2 \sqrt{x}\). The basic function here is \(\sqrt{x}\). The graph of \(\sqrt{x}\) begins at the origin (0,0) and grows slowly in positive x-direction.

Step 2: Identifying Transformation of the Basic Function

The number '2' before the square root in \(g(x)\) is not just any number, it's a multiplier to the basic function \(\sqrt{x}\). This means that every y-value in the basic graph will be multiplied by '2'. This is a vertical stretch of the basic function by a factor of 2. It will make the graph steeper.

Step 3: Sketch the Transformed Graph

Using the rules of transformations, sketch the graph of \(g(x) = 2 \sqrt{x}\) applying the identified transformation, which is a vertical stretch by factor of 2. The graph will be similar to the graph of \(\sqrt{x}\), but it will rise twice as fast.

Key Concepts

Basic Function Identification

Understanding the concept of a basic function is crucial when studying graph transformations. A basic function is a simple, fundamental function that serves as the building block for more complex functions through transformations like shifts, stretches, or reflections.When dealing with an expression such as \(g(x) = 2 \sqrt{x}\), identifying that the basic function is the square root function, \(\sqrt{x}\), is the first step. The graph of the basic square root function starts at the origin (0,0) and extends to the right, gradually increasing in the positive direction of the x-axis. It's known for its distinctive half-parabola shape, which graphically represents the set of all points \((x, y)\) where \(y = \sqrt{x}\).

• The graph of the square root function only exists for non-negative values of \(x\), as the square root of a negative number isn't defined in the set of real numbers.
• The square root graph is unique because it increases at a decreasing rate, which visually flattens out as \(x\) becomes larger.

A solid understanding of the shape and properties of basic functions like the square root allows students to easily grasp more complicated functions that are built from these foundations.

Vertical Stretch

A vertical stretch is one of the transformation techniques used to modify the graph of a function. To visualize this concept, imagine pulling the graph of a function upwards and downwards from its original position without altering the x-values.In the example \(g(x) = 2 \sqrt{x}\), the numeral '2' in front of the square root symbol indicates a vertical stretch. Mathematically, this means each y-value of the original square root function is multiplied by 2. The result of this transformation is a new graph, wherein each point is stretched away from the x-axis by a factor of 2.

• A vertical stretch doesn't affect the x-coordinates of the points on the graph but multiplies the y-coordinates by the stretching factor.
• The stretching factor greater than 1, such as 2 in this case, results in a graph that is steeper than the original basic function.

This transformation is a key concept in graphing as it allows for the manipulation of the function’s rate of change without shifting its position along the x-axis.

Square Root Function Graph

The graph of the square root function, \(\sqrt{x}\), is an essential concept in understanding the effect of transformations on functions. It's the graphical representation showing how the square root of x changes as x varies.When graphing the transformed function \(g(x) = 2 \sqrt{x}\), we apply a vertical stretch, as identified in the transformation step. The resultant graph will resemble the original square root graph but will rise more sharply. This is because every point \((x, y)\) on the basic graph becomes \((x, 2y)\) on the stretched graph.

• The vertical stretch does not modify the domain of the function; the square root function and its stretched counterpart are both defined for \(x \geq 0\).
• The graph’s starting point remains at the origin, but the 'curve' of the square root graph becomes less gradual as it ascends vertically at a faster pace due to the vertical stretch.

Recognizing the impact of this specific transformation on the square root graph is fundamental for students to predict and sketch graphs of transformed functions accurately.