Problem 11 - College Algebra

Question

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). $$ y=f(x+43) $$

Step-by-Step Solution

Answer

Shift the graph 43 units left.

Step 1: Identify the Original Function

The original function is given as \( f(x) \). This function represents the basic shape or curve that we will be transforming.

Step 2: Understand the Transformation Notation

The expression \( y=f(x+43) \) indicates a transformation of the original function \( f(x) \). The key part of this expression is \( x+43 \).

Step 3: Determine the Type of Transformation

The term \( x+43 \) corresponds to a horizontal shift. When \( x \) is replaced with \( x+c \), where \( c \) is a constant, it implies a horizontal shift.

Step 4: Identify the Shift Direction and Magnitude

Since the expression is \( x+43 \), it is a horizontal shift to the left by 43 units. This is because adding a positive number inside the function's argument results in moving the graph in the opposite direction.

Step 5: Describe the Transformation

The graph of the function \( y=f(x+43) \) is the graph of \( f(x) \) shifted 43 units to the left. No vertical shifts, stretches, or reflections occur here.

Key Concepts

Horizontal Shift

A horizontal shift occurs when you move a function's graph left or right on the coordinate plane. This transformation takes place inside the function's argument, typically affecting the variable \( x \). When you see an expression like \( f(x+c) \), it means the graph of the function is shifted horizontally.

• If \( c \) is positive, the function shifts to the left.
• If \( c \) is negative, the function shifts to the right.

For the given exercise, since we have \( f(x+43) \), the graph moves 43 units to the left. This might seem a little counterintuitive, but remember, adding a positive number within the function results in moving left.

Function Graph

A function graph visually represents the relationship of expected input-output pairs of a function. It's a way to see the behavior and general shape of a function. When you start with a basic function like \( f(x) \), the graph shows how \( y \) values change as \( x \) changes.
The graph can provide insights into the function's properties, such as symmetry, intercepts, and slopes. Understanding how transformations modify these graphs is crucial for analyzing more complex functions.

Graph Transformation

Graph transformation refers to altering the initial graph's position, shape, or size. This includes translations (shifts), reflections, stretches, and compressions.
In the given exercise, we focus on translations, which involves shifting the graph without changing its shape. Graph transformations help in improving our understanding of function behavior by illustrating how variables interact. Whether it's a simple shift or a more complex stretch, recognizing the transformation type can make graph analysis more intuitive and less daunting.

Algebraic Transformations

Algebraic transformations involve changing the expression of a function to achieve graph transformations. This might include adding constants, multiplying the function, or complex operations.
Every algebraic change corresponds to a specific graphical change. For instance, \( y = f(x+43) \) is an algebraic transformation denoting a horizontal shift. These changes showcase the interplay between algebra and geometry, helping us predict graph behavior by looking at the function's formula. Understanding algebraic transformations is pivotal for graph sketching and accurately identifying transformations in graphs.