Problem 19 - 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 \(f\). \(y=f(x+4)-1\)

Step-by-Step Solution

Answer

Shift 4 units left and 1 unit down.

Step 1: Identify the Parent Function

The parent function is given as \( y = f(x) \). This represents the graph of the function before any transformations are applied.

Step 2: Horizontal Translation

The expression \( f(x+4) \) indicates a horizontal shift. Specifically, the graph of \( f(x) \) is shifted 4 units to the left because we replace \( x \) with \( x+4 \). Remember, \( x+c \) results in a shift to the left by \( c \) units.

Step 3: Vertical Translation

The \(-1\) outside of the function causes the graph to shift vertically. This means the graph is moved 1 unit down because we subtract 1 from the function value \( y \).

Key Concepts

Parent Function

A parent function is the simplest form of a set of functions that form a family. It serves as the base for variations of transformations, such as translations, stretches, and reflections.

• For many functions, the most basic form is often represented as a linear function or standard equation, for example, linear parent function: \( y = x \), or quadratic parent function: \( y = x^2 \).
• In our case, the parent function identified is \( y = f(x) \). While the exact form isn't given, this notation indicates the basic rule without transformations.

Understanding the parent function helps in predicting what the graph will look like before and after transformations are applied. Recognizing the parent function is the first step in analyzing the given equation and allows us to use it as a baseline for further modifications.

Horizontal Translation

Horizontal translation involves shifting the graph left or right. In algebraic terms, in the expression \( f(x+c) \) or \( f(x-c) \), adding or subtracting a constant shifts the graph.

• For the expression \( f(x+4) \), it indicates a leftward shift. The graph of the original function is moved 4 units to the left.
• Intuitively, this can seem reversed because of the positive sign. However, remember that changes inside the parentheses affect horizontal movement inversely.
• The horizontal shift does not alter the shape of the graph, only its position on the x-axis.

This concept helps visualize how functions can be mapped to different locations along the x-axis, which is crucial when obtaining different function solutions based on the parent form.

Vertical Translation

Vertical translation shifts a graph up or down. This is done by adding or subtracting a constant outside the function itself.

• In the expression, the \(-1\) outside the function \( y = f(x+4)-1 \) indicates a downward shift.
• Functions move up when a positive constant is added, and they move down when a negative constant is subtracted.
• This translation affects positions on the y-axis but maintains the graph's original shape and horizontal position.

Understanding vertical translation is essential in adjusting the output values of functions, which in turn alters the entire graph's visual height or depth relative to its original placement.