Problem 12 - College Algebra with Corequisite Support

Question

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). \(y=f(x+3)\)

Step-by-Step Solution

Answer

The graph of the function shifts left by 3 units.

Step 1: Identify the Original Function

The original function is given as \( f(x) \). It represents the standard function before any transformation is applied. In this context, we don't have an explicit form. Instead, we focus on how transformations affect its graph.

Step 2: Analyze the Function Notation

The function \( y = f(x + 3) \) indicates a transformation applied to the original function \( f(x) \). The expression \( (x + 3) \) represents a horizontal shift of the function.

Step 3: Determine the Direction of Shift

The expression \( f(x+3) \) involves \((x + 3)\), which translates to shifting the graph to the left by 3 units. A function in the form \( f(x + c) \) moves to the left if \( c \) is positive.

Step 4: Conclude the Transformation

The transformation from \( f(x) \) to \( f(x + 3) \) results in the entire graph of \( f(x) \) being shifted to the left by 3 units on the x-axis.

Key Concepts

Horizontal Shift

In function transformations, a horizontal shift occurs when the input value of the function is altered before applying the function itself. Let’s look at the example function transformation: \( y = f(x+3) \). By focusing on \( f(x+3) \), we identify a horizontal shift. Adding a positive number inside the function with \( x+3 \) causes a leftward shift opposite to what might be intuitively expected.

• If we have \( f(x+c) \) where \( c > 0 \), the function shifts horizontally to the left by \( c \) units.
• Conversely, if \( c < 0 \), the function shifts to the right by \( |c| \) units.

Grasping these transformations is essential for graphing functions accurately. It helps in anticipating how function graphs will behave under different transformations. This is useful for both visualising complex functions and solving algebraic equations.

Graph of a Function

A graph of a function visually represents the set of all pairs \(( x, y )\) that satisfy the function’s equation. Exploring how transformations, like shifts, affect the graph, allows you to understand the functional behavior deeply. With the transformation \( f(x+3) \), you might ask, what happens to the graph? The graph of the function \( f(x) \) simply slides in the coordinate plane.

• The basic shape of the graph remains unchanged.
• There are no changes in size, shape, or orientation.
• Only the position changes, correlating with the transformation applied.

Understanding and predicting these alterations are crucial in mathematics, whether you're plotting points, constructing models, or analyzing data patterns.

Algebraic Transformations

Algebraic transformations involve altering the algebraic form of a function to translate, stretch, compress or reflect its graph. When discussing horizontal shifts, we're engaging with one of these algebraic transformations. In the case of \( y = f(x+3) \), the transformation is purely horizontal. Still, other types of transformations can also occur:

• Vertical shifts, involving terms like \( f(x) + k \) shifting up or \( f(x) - k \) shifting down.
• Scaling transforms where \( af(x) \) stretches or compresses the function.
• Reflections over the axes, such as \( f(-x) \) for a horizontal reflection.

Mastering these basic transformations prepares you to tackle more advanced manipulations, enhancing problem-solving skills and mathematical intuition for more complex equations in calculus and beyond.