Problem 2 - 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 is (-1, -3).

Step 1: Understand the Transformations

The function given is \( y=f(x) \) and the transformed function is \( y=f(x+3) \). The transformation \( x+3 \) indicates a horizontal shift. A positive value added to \( x \) translates the graph to the left by that value.

Step 2: Apply the Horizontal Shift

Given the original point \((2,-3)\), we need to adjust the \(x\)-coordinate according to the transformation. The transformation \( f(x+3) \) means we subtract 3 from the initial \(x\)-coordinate. So, the new \(x\)-coordinate is calculated as: \(2 - 3 = -1\).

Step 3: Find the Point on the Transformed Graph

The \(y\)-coordinate remains unchanged because the transformation only affects the \(x\)-coordinate. Therefore, the point on the transformed graph is \((-1, -3)\).

Key Concepts

Horizontal Shift

When dealing with functions, a horizontal shift occurs when we adjust the input part of the function. For example, given a function transformation like \( y=f(x+3) \), our graph will move horizontally. But why exactly does this happen?

• The transformation \( f(x+3) \) might seem counterintuitive. Instead of the graph moving to the right, it shifts to the left. This is because we're altering the input by adding 3. To maintain the same output values, we need to decrease the \( x \)-values by 3.
• This means every point on the graph of \( y=f(x) \) sees its \( x \)-coordinate reduced by 3 for the transformed function. Therefore, a point originally at \( (2, -3) \) will change to \( (2-3, -3) = (-1, -3) \) for \( y=f(x+3) \).

It’s essential to recognize that horizontal shifts impact the \( x \)-value and are all about moving the graph side-to-side without changing its shape or \( y \)-values.

Graph Translations

Translations in graphing refer to slides of entire graphs either horizontally, vertically, or both. They are among the fundamental transformations applied in the coordinate plane.

• Horizontal translations shift the graph left or right, while vertical translations move it up or down.
• The function transformation \( y=f(x+3) \) is specifically a horizontal shift, translating the graph 3 units to the left.

These transformations don't alter any other feature of the graph. For example, the size, orientation, and shape remain the same. Recognizing graph translations is crucial for understanding more complex transformations and in solving algebraic problems involving adjusted graphs.

Coordinate Transformations

Coordinate transformations involve systematically changing each point’s coordinates using a rule or a function. In mathematical graphs, a transformation can be thought of as translating, rotating, scaling, or reflecting the graph.

• For the transformation \( y=f(x+3) \), we're focusing on a translation in the horizontal direction. Here, we subtract 3 from the \( x \)-coordinates of all points in the original graph.
• This adjustment results in each point, including our example \((2, -3)\), shifting to \((-1, -3)\) on the new graph.

Coordinate transformations are essential in context as they enable you to manipulate and interpret graphs in various practical tasks, such as modeling real-world phenomena. Recognizing how a transformation affects each point can provide insights into the larger behavior of functions.