Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 27-32: If the point \(P\) is on the graph of a function \(f\), find the corresponding point on the graph of the given function. $$ P(3,-1) ; \quad y=2 f(x)+4 $$

Step-by-Step Solution

Answer

The corresponding point is \( (3, 2) \).

Step 1: Understand the transformation equation

We are given the point \( P(3, -1) \) on the graph of \( f(x) \) and need to find its corresponding point on the graph of \( y = 2f(x) + 4 \). This equation represents a vertical scaling and translation of the graph of \( f(x) \).

Step 2: Apply the vertical scaling factor

Since \( y = 2f(x) + 4 \), it involves a scaling by a factor of 2. The \( y \)-coordinate of the original function, \( f(x) = -1 \), is multiplied by 2. Thus, it becomes \( 2(-1) = -2 \).

Step 3: Apply the vertical translation

After scaling, the function is translated upwards by 4 units. Add 4 to the scaled \( y \)-value: \( -2 + 4 = 2 \).

Step 4: Determine the new coordinates

The new \( y \)-coordinate after scaling and translating is 2. The corresponding point on the graph of \( y = 2f(x) + 4 \) is \( (3, 2) \).

Key Concepts

Vertical Scaling

Vertical scaling is a transformation technique that changes the size of a graph along the y-axis. When we apply vertical scaling, we multiply the y-coordinates of all points on the graph by a scale factor. This scaling factor alters how much the graph stretches or compresses vertically.
A scale factor greater than 1 stretches the graph away from the x-axis, making it taller. For example, multiplying the y-coordinate by 2 means the function values become twice as large as their original values.

• If the scale factor is between 0 and 1, the graph compresses towards the x-axis.
• If negative, it includes a reflection over the x-axis, flipping the graph upside down as well.

In the given exercise, we applied a vertical scaling with a factor of 2. Each y-value of the function is doubled, changing the point from \( (3, -1) \) to \( (3, -2) \), thus stretching the graph upwards from its original position.

Vertical Translation

Vertical translation shifts a graph up or down along the y-axis. This transformation is achieved simply by adding or subtracting a constant to the y-values of the function. Unlike scaling, translation does not alter the shape or size of the graph, only its position.
In the equation \( y = 2f(x) + 4 \), the "+4" represents a vertical translation. Every point on the function graph is moved upwards by 4 units. Here’s how it works:

• Adding a positive constant shifts the graph up.
• Subtracting a positive constant, or adding a negative, shifts the graph down.

In the problem, after scaling point \( (3, -1) \) to \( (3, -2) \), the translation moves it to \( (3, 2) \), resulting in a vertically shifted graph.

Function Graph

A function graph visually represents the set of ordered pairs formed by elements of the domain (x-values) paired with elements of the range (y-values). It gives us a detailed view of how a function behaves over a set of inputs.
Understanding how a function graph changes under various transformations is crucial for analyzing functions. Graphs can illustrate:

• Key features like intercepts and turning points.
• The general shape and position of the function.
• Effects of scaling and translations intuitively.

The original exercise asked us to find the transformed point on the graph of \( y = 2f(x) + 4 \). By applying both vertical scaling and translation, the graph shifts and expands, altering how it appears without changing the x-coordinates of each point.

Coordinate Transformation

Coordinate transformation involves modifying the position or scale of a function graph by changing its coordinates. It encompasses both scaling and translation, providing a framework for understanding how each point on the graph adjusts.
This transformation follows a systematic approach:

• First, apply the scale factor to the y-coordinate, altering its magnitude.
• Then, translate by adding a constant, moving the position on the graph.

Both steps result in a new set of coordinates that define the transformed graph.
The given problem exemplifies a coordinate transformation, taking point \( (3, -1) \) from the graph of \( f(x) \) and mapping it to \( (3, 2) \) on \( y = 2f(x) + 4 \). Each step represents a logical procedure that simplifies understanding how these modifications affect the graph.