Problem 91 - A Graphical Approach to Precalculus with Limits

Question

Use transformations to explain how the graph of the given function can be obtained from the graphs of the square root function or the cube root function. $$y=\sqrt{9 x+27}$$

Step-by-Step Solution

Answer

Horizontally shift \(\sqrt{x}\) left by 3, then vertically stretch it by 3.

Step 1: Identify the Base Function

The given function is \(y=\sqrt{9x+27}\), and the base function is the square root function \(y=\sqrt{x}\). We'll transform this base function to obtain the given function.

Step 2: Simplify the Expression Inside the Square Root

Rewrite the expression inside the square root to reveal potential transformations. Notice, \(9x+27\) can be factored as \(9(x+3)\). So the function becomes \(y = \sqrt{9(x+3)}\).

Step 3: Apply Dilation Transformation

The factor \(9\) inside the square root can be taken out as \(3\) outside the root. Thus, \(y = \sqrt{9} \cdot \sqrt{x+3} = 3\sqrt{x+3}\). This dilation by 3 stretches the graph vertically.

Step 4: Apply Horizontal Translation

The expression \(\sqrt{x+3}\) means the graph is shifted horizontally. Specifically, \(\sqrt{x+3}\) translates the graph of \(\sqrt{x}\) left by 3 units.

Key Concepts

Understanding the Square Root Function

The base function involved in this exercise is the square root function, denoted as \(y=\sqrt{x}\). It is fundamental in mathematics, often appearing in various forms of equations and transformations. This function exhibits a non-linear relationship, where the output, \(y\), is the positive root of the input, \(x\).
A notable characteristic of the square root function is its domain: \(x\) must be non-negative, meaning \(x \geq 0\). This results in the graph starting at the origin and curving upwards to the right.

• The basic shape is a gentle curve, which flattens out as \(x\) increases.
• As a root function, it has an inherent symmetry about the vertical axis if mirrored with its negative counterpart.

Understanding the behavior of the square root function is crucial as it serves as a baseline for applying further transformations such as dilation and horizontal translation.

Dilation Transformation Explained

Dilation transformation helps in altering the size of a graph—either spreading it out or compressing it. For the given function \(y=\sqrt{9(x+3)}\), we've identified a dilation factor.
The expression \(\sqrt{9}\) simplifies to \(3\), meaning our transformation involves multiplying the base square root function by \(3\). Thus, the transformed function becomes \(y=3\sqrt{x+3}\).

• A positive dilation factor \(>1\) results in vertical stretching, making the graph taller or narrower.
• In contrast, a dilation factor between \(0\) and \(1\) would lead to compression.

In this specific example, the dilation factor of \(3\) stretches the graph vertically by a scale of \(3\). This means each point on the graph of \(y=\sqrt{x}\) moves three times higher on the \(y\)-axis, maintaining the general shape but extending the range of its values.

Understanding Horizontal Translation

Horizontal translation shifts the graph along the \(x\)-axis. The given function \(y=3\sqrt{x+3}\) involves a horizontal shift.
The expression \(x+3\) inside the square root indicates this translation. Specifically, the graph of the original square root function \(y=\sqrt{x}\) is moved horizontally.

• If inside expression is \(x + a\), the function shifts left by \(a\) units.
• Conversely, \(x - a\) would shift the graph right by \(a\) units.

In this case, the "+3" results in a translation of three units to the left. This means every point on \(y=\sqrt{x}\) now appears three units further left on the \(x\)-axis, maintaining the curvature and relative distances between points but altering the starting position of the graph.