Problem 91 - A Graphical Approach to College Algebra

Question

Concept Check 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

Shift left 3 units, then stretch vertically by factor of 3.

Step 1: Identify the Base Function

The given function is \(y=\sqrt{9x+27}\). This follows the form of the square root function \(y = \sqrt{x}\). We recognize the transformation involves scaling and translating this base function.

Step 2: Factor the Expression Inside the Square Root

Rewrite the expression under the square root. Begin by factoring the constant before \(x\).\[ y=\sqrt{9(x+3)} \] This helps in identifying how the transformations will affect the function graph.

Step 3: Apply Horizontal Transformation

The function \( y = \sqrt{9(x+3)} \) translates to \( y = \sqrt{9}\sqrt{x+3} \). The expression \( \sqrt{x+3} \) signifies a horizontal shift of the base graph \( y = \sqrt{x} \), 3 units to the left.

Step 4: Apply Vertical Stretch Transformation

The multiplier \( \sqrt{9} \) in the expression \( y = \sqrt{9}\sqrt{x+3} \) represents a vertical stretch by a factor of 3 of the graph of \( y = \sqrt{x+3} \).

Key Concepts

Square Root Function

The square root function is fundamental in algebra, playing a crucial role in various mathematical operations. It is expressed as \( y = \sqrt{x} \) and represents a relationship where each number is paired with its square root.

• The graph of this function takes the shape of a gentle curve, beginning at the origin (0,0) and extending to the right along the positive x-axis.
• This function only exists for non-negative x-values, as square roots of negative numbers fall into the realm of complex numbers in algebra.

Understanding this base function is important when dealing with more complex functions involving square roots, such as transformations that adjust the graph's position or shape on a coordinate plane.

Graph Transformations

Graph transformations involve modifying the graph of a base function to obtain a new function. For the function \(y=\sqrt{9x+27}\), transformations include both horizontal and vertical changes.First, let's explore horizontal transformations:

• The transformation \( y = \sqrt{9(x+3)}\) shows a horizontal shift. This results in moving the graph 3 units to the left, a crucial component for graph transformations.

Next, we consider vertical transformations:

• The presence of \(\sqrt{9}\) indicates a vertical stretch by a factor of 3. This makes the graph steeper compared to its base form, \( y = \sqrt{x} \).

These transformations offer a structured approach to adjust, stretch, or shift graphs, aiding in visualizing solutions and understanding the behavior of functions.

College Algebra

In college algebra, mastering transformations helps students tackle a wide range of problems. Understanding transformations like translation and stretching not only applies to the square root function but also to many function types. When students learn how to manipulate graphs:

• They gain the ability to predict the behavior of complex equations.
• They develop a deeper understanding of mathematical relationships and how variables interact.

By mastering these techniques, students strengthen their algebraic skills, preparing them for advanced topics in calculus and beyond. Graph transformations are therefore integral to a robust algebraic education, supporting logical reasoning and problem-solving capabilities.