Problem 19 - College Algebra

Question

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 left 4 units and down 1 unit.

Step 1: Identify the Parent Function

The parent function in this scenario is given as \( f(x) \). This is the original function before any transformations are applied. The original graph is defined as \( y = f(x) \).

Step 2: Determine Horizontal Shift

Examine the expression \( f(x+4) \) inside the function. The \( +4 \) indicates a horizontal shift. Recall that adding a positive number inside the function shifts the graph to the left by that number of units. Thus, \( f(x+4) \) shifts the graph of \( f \) left by 4 units.

Step 3: Identify Vertical Shift

Next, look at the minus 1 outside the function: \( -1 \). This shifts the graph vertically. Subtracting 1 from the function value lowers the entire graph by 1 unit. Thus, \( f(x+4)-1 \) represents a vertical shift downward by 1 unit.

Step 4: Combine Transformations

Combine the horizontal and vertical transformations. The new graph is the result of shifting the parent function \( f(x) \) left by 4 units and then down by 1 unit. The complete transformation is represented by the function \( y = f(x+4) - 1 \).

Key Concepts

Horizontal Shift

In function transformations, a horizontal shift involves moving the graph of a function left or right along the x-axis. This shift is determined by the value that is added or subtracted inside the function's argument.

For example, if we have a function expression like \( f(x+4) \), this indicates a horizontal shift. Here, you might think adding 4 would move the graph to the right, but it's the opposite. Adding a positive number inside the function actually shifts the graph to the left by that number of units.

Think of it this way:

• \( f(x+a) \): Shift the graph of \( f(x) \) to the left by \( a \) units.
• \( f(x-a) \): Shift the graph of \( f(x) \) to the right by \( a \) units.

This might seem counterintuitive at first, but imagine the x-values on the graph needing to reach previously occupied positions sooner (hence, a move to the left).

Vertical Shift

A vertical shift refers to moving the graph of a function up or down along the y-axis. This type of shift is indicated by numbers added or subtracted outside the function. In the expression \( f(x) - 1 \), the \(-1\) suggests a vertical shift.

To understand this:

• \( f(x) + b \): Shift the graph of \( f(x) \) upwards by \( b \) units.
• \( f(x) - b \): Shift the graph of \( f(x) \) downwards by \( b \) units.

For the function \( f(x+4) - 1 \), the vertical shift is achieved by subtracting 1, which lowers the entire graph by 1 unit. This kind of shift does not affect the x-values directly but changes the y-coordinate of each point on the graph.

Parent Function

The parent function is the simplest form of a function from which transformations are derived. It serves as the original or base function before any shifts, reflections, stretches, or compressions are applied.

In our example, the parent function is denoted as \( f(x) \). This function acts as the starting point for creating more complex or transformed functions like \( f(x+4) - 1 \).

Understanding the parent function is crucial because:

• It helps identify the effect of transformations clearly.
• Knowing the base form allows for easier manipulation and prediction of graph behavior after a transformation.

By recognizing the parent function \( f(x) \), you can systematically apply transformations, such as shifts, to find the new graph position or shape.