Problem 6 - Precalculus with Limits

Question

Match the rigid transformation of \(y=f(x)\) with the correct representation of the graph of \(h\), where \(c > 0\). (a) \(h(x) = f(x) + c\) (i) A horizontal shift of \(f\), \(c\) units to the right (b) \(h(x) = f(x) - c\) (ii) A vertical shift of \(f\), \(c\) units downward (c) \(h(x) = f(x + c)\) (iii) A horizontal shift of \(f\), \(c\) units to the left (d) \(h(x) = f(x - c)\) (iv) A vertical shift of \(f\), \(c\) units upward

Step-by-Step Solution

Answer

(a) is matched with (iv), (b) with (ii), (c) with (iii), and (d) with (i).

Step 1: Identify Vertical Shifts

A vertical shift is observed when a constant is added or subtracted directly to the function. So, based on this:\n(a) \(h(x) = f(x) + c\) corresponds to a vertical shift of \(f\), \(c\) units upward (option iv).\n(b) \(h(x) = f(x) - c\) corresponds to a vertical shift of \(f\), \(c\) units downward (option ii).

Step 2: Identify Horizontal Shifts

A horizontal shift is observed when a constant is added or subtracted to the variable (x) of the function. So, based on this:\n(c) \(h(x) = f(x + c)\) corresponds to a horizontal shift of \(f\), \(c\) units to the left (option iii).\n(d) \(h(x) = f(x - c)\) corresponds to a horizontal shift of \(f\), \(c\) units to the right (option i).

Key Concepts

Vertical Shift

A vertical shift occurs when you add or subtract a constant from the function itself. Imagine you have the function graph of \(y = f(x)\). If you then have \(h(x) = f(x) + c\), it means the graph of \(f(x)\) moves upward by \(c\) units. The entire function is lifted, much like raising a drawing up by a specific distance without changing its shape or direction.

Similarly, if you have \(h(x) = f(x) - c\), it implies that the graph of \(f(x)\) shifts downward by \(c\) units. This is like lowering the entire graph by that many units. It’s important to note that this movement is parallel to the y-axis and doesn’t involve any stretching or compression of the graph.

When working with vertical shifts:

• Addition of \(c\) moves the graph up.
• Subtraction of \(c\) moves the graph down.

This transformation is quite helpful in adjusting the baseline or starting point of a function, making it easier to fit certain conditions or parameters.

Horizontal Shift

A horizontal shift involves moving the entire graph of a function left or right along the x-axis. Unlike vertical shifts, this type of transformation modifies the input variable, \(x\), in the function. For instance, if you have the function \(y = f(x)\) and you transform it to \(h(x) = f(x - c)\), the graph shifts to the right by \(c\) units.

On the other hand, if you have \(h(x) = f(x + c)\), the graph will shift to the left by \(c\) units. This might seem counterintuitive initially—adding \(c\) moves it left—but remember, the change is happening inside the function's input, which alters how the values of \(x\) are perceived.

For horizontal shifts:

• \(h(x) = f(x - c)\) shifts the graph right by \(c\) units.
• \(h(x) = f(x + c)\) shifts the graph left by \(c\) units.

Horizontal shifts are pivotal in aligning functions accurately along the x-axis and are frequently used to model symmetrical behaviors around specific points.

Function Transformation

Function transformation is a broad concept that involves altering a function's graph using various techniques, such as translations (like vertical and horizontal shifts), reflections, stretches, and compressions. In the context of the current exercise, our focus is on rigid transformations, specifically shifts, which maintain the shape and orientation of the graph but change its position.

Understanding function transformations is essential for effectively analyzing and interpreting graph behaviors. Transformations allow us to:

• Adjust the function's starting point without altering its inherent properties.
• Graphically represent changes due to external conditions or modifications to the function's equations.
• Combine multiple graphs logically to approximate more complex phenomena.

Each transformation type plays a specific role, whether aligning a graph with an axis, shifting it between quadrants, or modeling real-world situations with precision. Mastery of function transformations can significantly enhance your mathematical toolkit, making it easier to solve various algebraic and calculus problems.