Problem 3 - 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. $$ y=f(x)-1 $$

Step-by-Step Solution

Answer

The point on the graph of the transformed function is (2, -4).

Step 1: Understanding Theorem 1.7

According to Theorem 1.7, if a function's graph is given by a transformation of the form \(y = f(x) + c\), it represents a vertical translation of the graph of \(y = f(x)\). In this context, if \(c\) is negative, the graph shifts downward. For the function \(y = f(x) - 1\), this implies a downward shift by 1 unit.

Step 2: Identifying the Original Point

We are given that the point \((2, -3)\) lies on the graph of the function \(y = f(x)\). This means when \(x = 2\), \(y = -3\).

Step 3: Applying the Transformation

With the transformation \(y = f(x) - 1\), every point on the original graph \((x, y)\) moves downward by 1 unit. Therefore, the new y-coordinate for a point \((x, y)\) will be \(y - 1\). For the given point \((2, -3)\), the new y-coordinate becomes \(-3 - 1 = -4\).

Step 4: Finding the Transformed Point

After applying the transformation to the original point \((2, -3)\), the new point on the graph of \(y = f(x) - 1\) is \((2, -4)\).

Key Concepts

Vertical Translation

A vertical translation in graph transformations involves shifting the entire graph of a function up or down along the y-axis. This type of transformation is integral to understanding how graphs can be moved without altering their basic shape.

For example, if you start with a function graph, like that of \( y = f(x) \), a vertical translation will bring about a change in the position of this graph in a straightforward manner.

### Understanding Vertical Movement

• If you add a positive constant \( c \) to a function, the entire graph moves up by \( c \) units.
• If you subtract a constant \( c \) from a function, the graph moves down by \( c \) units.

This transformation doesn't change the shape of the graph, just the positioning along the y-axis.

For instance, in our example with the function \( y = f(x) - 1 \), the graph shifts downwards by 1 unit. It's like lowering a picture on a wall without tilting or changing it.

Function Transformation

Function transformation involves altering the equation of a function to shift, stretch, compress, or reflect its graph. These transformations change how the graph appears, making them a powerful tool in mathematical visualization.

### Types of Function Transformations

• **Vertical translations:** Moves the graph up or down.
• **Horizontal translations:** Shifts the graph left or right.
• **Vertical scaling (stretching and compressing):** Changes the steepness of the graph.
• **Horizontal scaling:** Adjusts the width of the graph.
• **Reflections:** Flips the graph over a line, such as the x-axis or y-axis.

These transformations allow us to see the same function behaving differently across the coordinate system. This flexibility in altering a graph without distorting its core properties is essential for deeper mathematical insights.

Theorem 1.7

Theorem 1.7 is a pivotal concept in understanding graph transformations. It formally states the effect of adding or subtracting a constant from the value of a function. This theorem tells us that such modifications will lead to vertical translations.

### Applying Theorem 1.7 Consider the function \( y = f(x) + c \), where \( c \) is some constant:

• If \( c \) is positive, the graph shifts upwards by \( c \) units.
• If \( c \) is negative, the graph shifts downwards by the absolute value of \( c \).

This theorem is helpful because it provides a simple rule for modifying graphs. When applying Theorem 1.7, the transformation directly applies to the y-coordinates of points in the graph, translating them in the prescribed vertical direction. Such theorems make graph behavior predictable and manageable, simplifying the task of analyzing various functions.

Coordinate Shifts

Coordinate shifts refer to the changes in the coordinates of a point resulting from transformations applied to a function's graph. In simpler terms, these shifts explain how each point on the graph of a function is repositioned when the function undergoes specific transformations.

### Practical Example of Shifts Take the original point \((2, -3)\) from our problem statement:

• The transformation \( y = f(x) - 1 \) results in a coordinate shift for the y-value. Instead of \(-3\), it becomes \(-4\).

This shift indicates how transformations systematically adjust the graph's coordinates.

### Important Note:

• Coordinate shifts don't affect the intrinsic properties of the graph's shape.
• They are purely positional alterations that allow us to visualize the effect of a transformation in the coordinate plane.

Understanding coordinate shifts is crucial to graph transformation because they show the immediate, visible effects of applying transformations in algebraic form.