Problem 25 - Algebra and Trigonometry

Question

23–26 ? Explain how the graph of g is obtained from the graph of f. (a) \(f(x)=\sqrt{x}, \quad g(x)=2 \sqrt{x}\) (b) \(f(x)=\sqrt{x}, \quad g(x)=\frac{1}{2} \sqrt{x-2}\)

Step-by-Step Solution

Answer

(a) Vertical stretch; (b) Horizontal shift and vertical compression.

Step 1: Understand the Basic Graph

The basic graph to consider here is that of function \( f(x) = \sqrt{x} \), which is the square root function. Its graph starts from the origin and increases slowly to the right, creating a smooth upward curve.

Step 2a: Transformation for Part (a)

For part (a), the transformation involves multiplying the square root function by 2 to get \( g(x) = 2\sqrt{x} \). This transformation results in a vertical stretch. Each point on the graph of \( f(x) = \sqrt{x} \) is moved upward by a factor of 2.

Step 3a: Sketch the Graph for Part (a)

Visualize the effect of the transformation. The points \( (1, 1) \) on \( f(x) \) will now be at \( (1, 2) \) on \( g(x) \), \( (4, 2) \) will transform to \( (4, 4) \), creating a taller graph that maintains the same general shape.

Step 2b: Identify Transformation for Part (b)

For part (b), we modify \( f(x) = \sqrt{x} \) to \( g(x) = \frac{1}{2}\sqrt{x-2} \). This involves two sequential transformations: a horizontal shift to the right by 2 units due to \( x-2 \), and a vertical compression by a factor of \( \frac{1}{2} \).

Step 3b: Horizontal Shift

Starting with the function being shifted to the right. The graph for \( \sqrt{x} \) will be moved 2 units to the right so that it begins at \( (2,0) \) instead of \( (0,0) \).

Step 4b: Vertical Compression

After the shift, apply the vertical compression. Multiply each y-value by \( \frac{1}{2} \). A point like \( (3, 1) \) on the shifted graph moves to \( (3, \frac{1}{2}) \).

Step 5b: Resultant Graph for Part (b)

The final graph of \( g(x) = \frac{1}{2}\sqrt{x-2} \) is a horizontally shifted and vertically compressed version of \( f(x) = \sqrt{x} \). It starts at \( (2, 0) \) and grows at half the rate of the original \( f(x) \).

Key Concepts

Square Root Function

The square root function is quite fundamental in mathematics, and it's represented as \( f(x) = \sqrt{x} \). This function plays an important role in graph transformations.

• Start: The graph of this function originates from the point (0,0).
• Shape and Direction: It increases gradually to the right, forming a smooth, upward-curving line.

The behavior of this function is unique because it only accepts inputs that are non-negative—the square root of a negative number isn't defined within the real number system.
This makes the domain of the graph all non-negative x-values. The range, similarly, includes all non-negative y-values.

Vertical Stretch

A vertical stretch is a transformation that changes the graph's height. Essentially, it stretches the graph vertically from its original position. In the case of our function \( g(x) = 2\sqrt{x} \), the vertical stretch occurs because each y-value in the \( f(x) = \sqrt{x} \) graph is multiplied by 2.

• Effect: The result is that each point on the graph is moved further away from the x-axis.
• Visualization: For instance, the point (1, 1) on the basic \( f(x) = \sqrt{x} \) graph becomes (1, 2) on \( g(x) \).

This effectively makes the graph appear taller while preserving its general shape. It's important to think of this stretch like pulling the graph upwards without altering its horizontal position.

Horizontal Shift

A horizontal shift moves the graph left or right within the coordinate plane. For the transformation seen in \( g(x) = \frac{1}{2}\sqrt{x-2} \), the modification \( x-2 \) causes the whole graph to shift 2 units to the right.

• Initial Move: The starting point of the graph shifts from (0,0) to (2,0).
• Effect: All other points on the graph move 2 units to the right as well.

It's helpful to visualize this shift in terms of where the graph starts and how it moves across the x-axis. Instead of beginning at zero horizontally, the graph has a new starting point and moves further out.

Vertical Compression

A vertical compression makes the graph appear shorter or squeezed vertically. In \( g(x) = \frac{1}{2}\sqrt{x-2} \), this occurs because every y-value is reduced by a factor of \( \frac{1}{2} \).

• Effect: Each point on the graph lowers in height, retaining the form of the curve but drawing it closer to the x-axis.
• Example: A point such as (5, 2) on the horizontally shifted graph becomes (5, 1) in the transformed graph.

Think of vertical compression as squeezing the graph from top to bottom. This affects how steeply the graph rises compared to the original, making it a crucial component in altering the graph’s appearance.