Problem 6 - Precalculus

Question

Suppose (2,-3) is on the graph of \(y=f(x) .\) In Exercises \(1-18,\) use Theorem 1.7 to find a point on the graph of the given transformed function. $$ y=f(3 x) $$

Step-by-Step Solution

Answer

The new point on the graph is \( \left( \frac{2}{3}, -3 \right) \)."

Step 1: Identify the Transformation

The function given is of the form \( y = f(3x) \), which indicates a horizontal compression by a factor of 3. This means any \( x \) value on the original function \( y = f(x) \) will be divided by 3 to find the corresponding \( x \) value on \( y = f(3x) \).

Step 2: Apply Transformation to X-coordinate

Given that the point \((2, -3)\) is on the original function \( y = f(x) \), we need to adjust the \( x \) coordinate for the transformation. Divide the \( x \) coordinate by 3: \( x = \frac{2}{3} \).

Step 3: Preserve the Y-coordinate

The transformation \( y = f(3x) \) affects the \( x \) coordinates but does not change the \( y \) coordinate. Therefore, the \( y \) coordinate remains \( -3 \).

Step 4: State the New Point

Combine the transformed \( x \) coordinate and the unchanged \( y \) coordinate to find the new point on the graph of \( y = f(3x) \). The point is \( \left( \frac{2}{3}, -3 \right) \).

Key Concepts

Horizontal Compression

When dealing with functions, horizontal compression involves altering the input values of a function, resulting in the graph being squeezed horizontally. In the function \( y = f(3x) \), the compression factor is 3. This means that every \( x \) value on the original graph \( y = f(x) \) has to be divided by 3 to map onto the function \( y = f(3x) \). This essentially reduces the horizontal spread of the graph, making it appear tighter.

For example, if you have a point \((x, y)\) on the original function, the new point on the compressed graph would be \( \left( \frac{x}{3}, y \right) \). This adjustment only impacts the \( x \)-coordinate, keeping the \( y \)-coordinate unchanged. Understanding horizontal compression is essential for graphing transformations accurately.

Coordinate Transformation

Coordinate transformation is the process of changing the position of points on the graph through specific rules. With transformations, the location of points is altered either vertically or horizontally. In our case, the transformation involves a change in the \( x \)-coordinate of a point \((x, y)\) due to the function \( y = f(3x) \).

During the transformation process:

• The \( x \)-coordinate of the point is divided by 3 as a result of the horizontal compression.
• The \( y \)-coordinate remains the same since the transformation \( y = f(3x) \) does not affect it.

For example, if \((2, -3)\) is a point on \( y = f(x) \), it becomes \( \left( \frac{2}{3}, -3 \right) \) on \( y = f(3x) \), illustrating how only the \( x \)-coordinate is transformed.

Graphing Functions

Graphing functions involves creating a visual representation of the relationship between input \( x \) values and output \( y \) values defined by a function. When graphing transformations, understanding shifts, stretches, and compressions is crucial to accurately modify and visualize the function's graph.

Initially, graph the original function \( y = f(x) \). Upon applying transformations like horizontal compression, each original \( x \)-coordinate is transformed accordingly. For \( y = f(3x) \), this involves dividing \( x \) by 3, compressing points horizontally towards the y-axis.

• Start with the known points on the original graph.
• Apply the necessary transformations to these points.
• Plot the new points to visualize the transformed function.

By plotting these transformed points, you get the graph of the new function, giving insight into how the transformation affects the function's behavior.

Precalculus Problem-Solving

Precalculus provides tools to tackle complex mathematical problems, preparing you for calculus and beyond. Problem-solving in precalculus involves understanding and applying various mathematical concepts, including transformations, to analyze and interpret functions.

Solving exercises like the one presented requires a step-by-step approach:

• Identify the type of transformation applied to the function.
• Determine how the transformation affects individual coordinates.
• Apply these changes consistently across known points.
• Validate the new points by confirming they align with the transformation's expected outcome.

This methodological approach ensures accurate results and deepens your understanding of the function’s geometric transformations, crucial for advanced mathematics studies. By mastering such problem-solving skills, you're equipped to tackle similar tasks efficiently.