Problem 19 - College Algebra

Question

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function \(f\). $$ y=f(x+4)-1 $$

Step-by-Step Solution

Answer

Shift the graph of \( f(x) \) 4 units left and 1 unit down.

Step 1: Understanding the Parent Function

The parent function here is given as \( y=f(x) \). This function is the original graph we will be transforming.

Step 2: Identify the Horizontal Shift

The term \( x+4 \) in the function \( f(x+4) \) indicates a horizontal shift of the graph. In this context, \( x+4 \) means that the graph of \( f(x) \) is shifted to the left by 4 units.

Step 3: Identify the Vertical Shift

The term \(-1\) outside the function notation as in \( f(x+4)-1 \) indicates a vertical shift. This specific term means that the graph of \( f(x+4) \) is shifted downward by 1 unit.

Step 4: Combine the Transformations

Combine both transformations to describe the complete shift: the graph of \( y=f(x+4)-1 \) is obtained by shifting the graph of \( y=f(x) \) 4 units to the left and 1 unit down.

Key Concepts

Horizontal Shift

When you're dealing with function transformations, a horizontal shift is a change that moves the graph left or right along the x-axis. Imagine holding the graph and sliding it sideways without lifting it from the plane. For the function given in the exercise, we see the term inside the function: \( f(x+4) \). This indicates a horizontal shift. But why to the left?

• If you see \(x+a\), the graph will shift to the left by \(a\) units.
• If it were \(x-a\), the graph would move to the right by \(a\) units.

In this case, \(x+4\) means we shift 4 units left. It might seem counterintuitive at first, but as you adjust, it'll become second nature. The key takeaway is: adding inside the function shifts horizontally to the left, and subtracting shifts to the right.Taking the time to thoroughly understand this concept will make it much easier to predict and apply horizontal shifts in any function.

Vertical Shift

A vertical shift moves the graph up or down along the y-axis, like raising or lowering a piece of paper without tilting it. This modification is signaled by terms added or subtracted outside of the function. In the function transformation from the exercise, the term \(-1\) modifies the graph vertically.

• When you see \(f(x)+b\), you'll shift the graph up by \(b\) units.
• If you encounter \(f(x)-b\), you push the graph down by \(b\) units.

So, for the function \(f(x+4)-1\), the \(-1\) means the graph is shifted down 1 unit. This type of transformation alters the position of the graph without changing its shape or direction. It directly affects the output values, moving every point on the graph an equal amount vertically. By understanding vertical shifts, you'll be well-equipped to intuitively adjust any function along the y-axis.

Parent Function

The parent function serves as the foundation or starting shape of a graph before any transformations are applied. In mathematical terms, it’s the most basic form of a function from which various transformations are made. The exercise begins with the parent function \(y=f(x)\). This is the "untransformed" version you begin with. Understanding the parent function is crucial:

• It's your baseline or "default" graph.
• All transformations, like shifts or stretches, happen relative to this function.

By identifying the parent function, you know exactly what original graph you're transforming. Whether it’s a straight line, a parabola, or something else, starting with the parent function helps you systematically apply any transformation — like horizontal or vertical shifts — and know exactly how the graph's position or shape will change. Recognizing the parent function encourages a deeper comprehension of function transformations in general.