Problem 4 - Algebra 2

Question

What is the value of \(h\) in each translation? Describe each phase shift (use a phrase like 3 units to the left). $$ h(x)=f(x-3) $$

Step-by-Step Solution

Answer

The value of \(h\) is \(3\) and the phase shift is \(3\) units to the right.

Step 1: Identify the Value of h

In the given equation \(h(x)=f(x-3)\), we can compare this with our general function \(h(x) = f(x- h)\). It is clear that the value subtracted from \(x\) in the parenthesis is \(3\) so, \(h = 3\).

Step 2: Describe the Phase Shift

Since \(h > 0\), the function \(f(x)\) shifts \(h\) units towards the right. Therefore, our phase shift is \(3\) units to the right.

Key Concepts

Phase Shift

Phase shift, in the context of functions, refers to the horizontal movement of the graph of a function along the x-axis. When the inside of the function's argument is adjusted by adding or subtracting a value, it effectively shifts the graph of the function. Consider a basic function, like a sine or cosine wave, for visualization. When we write a function as \( h(x) = f(x - c) \), the graph of \( f(x) \) shifts by \( c \) units depending on the sign:

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

Phase shifts are particularly salient in trigonometric functions but can apply to all types of functions, be they linear, quadratic, or any other form. Understanding phase shifts allows for greater manipulation and application of function graphs in various mathematical and real-world contexts.

Horizontal Shift

A horizontal shift is a special kind of phase shift where the entire graph of a function moves left or right on the Cartesian plane. This shift occurs without any change in the function's shape or orientation. It's implemented by altering the function's input. If we have \( g(x) = f(x - h) \):

• A positive \( h \) means the graph shifts right.
• A negative \( h \) means the graph shifts left.

Horizontal shifts do not affect the function's vertical position or amplitude. They merely reposition the graph horizontally. For instance, in the example \( h(x) = f(x-3) \), the shift is 3 units right due to the subtraction of 3. This is crucial for graphing transformations and understanding the underlying geometry of function graphs.

Algebraic Function Transformation

Algebraic function transformation is a broader term encompassing various ways to alter a function's graph. These transformations include but are not limited to:

• Horizontal shifts, like moving the graph left or right.
• Vertical shifts, which move the graph up or down.
• Reflections, flipping it over a given axis.
• Stretches and compressions, altering the graph's width or height.

All these transformations are executed by adjusting the algebraic expression of the function. A shift such as \( h(x) = f(x - 3) \) is a classic example of a horizontal translation within this larger set of transformations. Algebraic transformations fundamentally alter how the function graphically presents itself, which can be used to model various real-world scenarios and mathematical concepts. Understanding these transformations helps in developing a deeper insight into function behavior and graph manipulation.