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

Step-by-Step Solution

Answer

The point \((2, 0)\) is on the graph of \(y = f(x) + 3\).

Step 1: Understanding the Given Information

We know that the point \((2, -3)\) is on the graph of \(y = f(x)\). This means when \(x = 2\), the function \(f(x)\) equals \(-3\).

Step 2: Analyzing the Transformation

The given transformation is \(y = f(x) + 3\). This transformation indicates a vertical shift of the graph upward by 3 units.

Step 3: Applying the Transformation to the Y-coordinate

Since the graph shifts vertically, we add 3 to the y-coordinate of the original point. Therefore, starting from \((2, -3)\), the new y-coordinate becomes \(-3 + 3 = 0\).

Step 4: Finding the New Point

With the x-coordinate remaining the same (\(x = 2\)) and the new y-coordinate calculated as 0, the transformed point on the graph of \(y = f(x) + 3\) is \((2, 0)\).

Key Concepts

Vertical Shift

A vertical shift refers to a type of transformation that impacts the position of a graph along the y-axis. It involves adding or subtracting a constant value to the function's output, thereby moving the graph either upwards or downwards. In this exercise, the transformation provided is of the form \( y = f(x) + 3 \), which clearly indicates a vertical shift.

To visualize a vertical shift, think of the graph as a picture on a sheet of paper. When we perform a vertical shift by adding 3, it's like sliding the picture up three units on the paper. The shape and orientation of the graph remain unchanged. Only its position along the y-axis is altered.

There's a simple rule:

• If the transformation is \( y = f(x) + c \) where \( c > 0 \), the shift is upward.
• If \( c < 0 \), the shift is downward.

This simple transformation doesn't affect the x-coordinates of points on the graph.

Graph of a Function

The graph of a function is a visual representation of all its ordered pairs \( (x, y) \). It shows how the function behaves over a range of values. For instance, the point \((2, -3)\) on the graph of \( y = f(x) \) tells us that when \( x = 2 \), the function outputs the value \( -3 \).

The only elements critical to forming the graph are the x-values and their corresponding y-values. In this case, transformations can alter these values and change the appearance of the graph. However, even after transformations like vertical shifts, the same set of x-values remains on the horizontal axis unless a horizontal transformation is applied.

Understanding how each point on a graph corresponds to its function is crucial for visualizing transformations. By sketching or analyzing graphs, we better grasp how the function behaves and the impact of different transformations.

Theorem 1.7

Theorem 1.7 generally assists in determining how function transformations modify the graph's points. Although the specifics of Theorem 1.7 aren't detailed in this problem, it describes properties of transformations such as translations, reflections, and scalings.

In our exercise, using Theorem 1.7 helps to formally analyze how the graph of \( y = f(x) \) changes under vertical transformations.

In a vertical shift like \( y = f(x) + 3 \), the theorem reinforces that the y-component of each point on the graph is modified while the x-component remains fixed, thereby shifting only vertically. This theorem helps ensure consistency and understanding across different types of transformations.

Coordinate Transformation

Coordinate transformation refers to the adjustment of points' coordinates on the graph due to transformations. Here, it involves evaluating how each point on the initial graph of \( y = f(x) \) shifts when applying the given transformation.

In our case, transformation \( y = f(x) + 3 \) results in modifying the y-coordinate by increasing it by 3 units. So, if a point on the initial function \( f(x) \) was \((2, -3)\), this transformation results in the new point \((2, 0)\) by updating the y-coordinate.

Coordination transformation is fundamental because it allows us to predict new points precisely on the transformed graph. It illustrates how each point's new location is determined through simple arithmetic adjustments, leading to a new and transformed graph positionarily.