Problem 3 - Precalculus Mathematics for Calculus

Question

(a) To obtain the graph of \(g(x)=2^{x}-1,\) we start with the graph of \(f(x)=2^{x}\) and shift it \(______\) (upward/downward) 1 unit. (b) To obtain the graph of \(h(x)=2^{x-1}\), we start with the graph of \(f(x)=2^{x}\) and shift it to the \(_____\) (left/right) 1 unit.

Step-by-Step Solution

Answer

(a) downward; (b) right.

Step 1: Understanding the Shift

The function \(f(x) = 2^x\) is our starting point. We need to understand how the transformations affect this base function. For part (a), we examine a vertical shift, and for part (b), we look at a horizontal shift.

Step 2: Part (a): Vertical Shift Analysis

The function \(g(x) = 2^x - 1\) involves subtracting 1 from the original function \(f(x) = 2^x\). This subtraction represents a vertical shift. Therefore, \(g(x)\) is shifted 1 unit downward from \(f(x)\).

Step 3: Part (b): Horizontal Shift Analysis

The function \(h(x) = 2^{x-1}\) takes the original function \(f(x) = 2^x\) and modifies the exponent to \(x-1\). This indicates a horizontal shift. Specifically, the graph is shifted 1 unit to the right because the subtraction inside the exponent implies a shift in the opposite direction.

Key Concepts

Vertical Shift

When discussing function transformations, a vertical shift is one of the most intuitive shifts to understand. A vertical shift moves the graph of a function up or down without altering its shape, and it occurs when we add or subtract a constant from the function.
Take, for example, the function transformation from \(f(x) = 2^x\) to \(g(x) = 2^x - 1\). This transformation involves subtracting 1 from the entire function, which creates a vertical shift downwards.

• Subtracting 1 from \(2^x\) means every point on the graph of \(2^x\) is moved directly 1 unit downward.
• The subtraction does not affect the shape or the growth rate of the function; it merely changes its position on the graph.

To understand this better, imagine looking at a shadow that moves lower without changing the shape of the object casting it. That is how a vertical shift functions in graph transformations. When transforming a graph by shifting vertically, always keep in mind the direction:

• Adding a constant will shift the graph upward.
• Subtracting a constant will shift it downward.

Horizontal Shift

Unlike vertical shifts, horizontal shifts move the graph left or right. They change the position of the graph along the x-axis and occur when we change the variable inside the function’s exponent or argument.
In the exercise, the transformation from \(f(x) = 2^x\) to \(h(x) = 2^{x-1}\) involves modifying the exponent from \(x\) to \(x-1\).

• When you subtract 1 from the variable \(x\), it results in a horizontal shift to the right.
• This happens because inside changes affect the graph in the reverse direction of the operation. Subtracting inside causes a shift to the right.

Visually speaking, envision sliding a window shade to one side. The shade moves, but the structure outside doesn’t change in size or shape. They just "move" across the x-axis. Important reminders:

• Subtracting from \(x\) (i.e., \(x - k\)) results in a shift to the right by \(k\) units.
• Adding to \(x\) (i.e., \(x + k\)) shifts the graph to the left by \(k\) units.

Exponential Functions

Exponential functions are a fundamental type of mathematical function that describe many natural phenomena, such as population growth or radioactive decay. The basic form of an exponential function is \(f(x) = a^x\), where \(a\) is a positive constant and the exponent \(x\) is a variable.

• The base \(a\) must be positive and not equal to 1.
• When \(a > 1\), the function grows rapidly as \(x\) increases; this is called exponential growth.
• If \(0 < a < 1\), the function decreases as \(x\) increases; this is known as exponential decay.

In the exercise, we dealt with transformations of the exponential function \(f(x) = 2^x\). The number 2 indicates exponential growth since it is greater than 1. This function has characteristic properties, like a horizontal asymptote, typically the x-axis (y = 0), and goes through the point (0,1).
An exponential function's graph rises or falls dramatically, and transformations can shift it vertically or horizontally, as seen in our examples with \(g(x) = 2^x - 1\) and \(h(x) = 2^{x-1}\). Understanding these shifts allows us to apply exponential functions to various real-world contexts effectively. Remember:

• Vertical and horizontal transformations can make exponential functions model different scenarios, while the underlying principles of exponential growth or decay remain unchanged.