Problem 3 - Algebra and Trigonometry

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 Shifts of Graphs

To solve these problems, we need to understand how certain changes in the equation affect the graph of a function. Vertical shifts involve adding or subtracting a constant within the function, while horizontal shifts involve adding or subtracting a constant to the variable.

Step 2: Analyze Vertical Shift for g(x)

The function given is \( g(x) = 2^x - 1 \). This can be seen as the function \( f(x) = 2^x \) minus 1. When you subtract a constant from a function, it shifts the entire graph downward by that constant. Hence, the graph of \( g(x) \) is the graph of \( f(x) \) shifted downward by 1 unit.

Step 3: Analyze Horizontal Shift for h(x)

The function given is \( h(x) = 2^{x-1} \). This is equivalent to shifting every \( x \)-value of the original function \( f(x) = 2^x \) to the right by 1 unit. Subtracting a constant from \( x \) inside the function's exponent results in a shift to the right by that constant.

Key Concepts

Vertical Shifts

Vertical shifts are an essential aspect of graph transformations that can be both easy to implement and understand. They occur when a constant is added or subtracted directly from the function's output (the function value).

• If you add a constant, the graph shifts upward by that constant number of units.
• If you subtract a constant, the graph shifts downward by that constant number of units.

For instance, in the function given by \( g(x) = 2^x - 1 \), we start with the base graph \( f(x) = 2^x \). By subtracting 1 from \( f(x) \), the entire graph of \( g(x) \) shifts downward by 1 unit.
This vertical shift affects each point on the graph equally, meaning every point on the graph is simply moved down one unit without altering the graph's general shape.

Horizontal Shifts

Horizontal shifts occur when transformations are applied to the variable inside the function itself. This type of shift affects the graph in a way that moves it horizontally along the x-axis.

• Subtracting a constant from the variable shifts the graph to the right.
• Adding a constant to the variable shifts the graph to the left.

In the example of \( h(x) = 2^{x-1} \), the function \( f(x) = 2^x \) is shifted to the right by 1 unit. This happens because the entire graph moves every point one unit along the x-axis.
While the direction of the shift might seem counterintuitive at first—subtracting moves right, adding moves left—remember that these shifts occur due to changes in the x-values themselves.

Exponential Functions

Exponential functions, characterized by their sharp curves and constant percent growth or decay, are written in the form \( f(x) = a^{x} \) where \( a \) is a constant base.

• They have a rapid increase when the base \( a \) is greater than 1.
• They decrease when the base is a fraction between 0 and 1.
• Their domain is all real numbers, but the range is typically positive numbers, reflecting their unrestricted growth or decline.

For example, in the function \( f(x) = 2^x \), the base \( 2 \) indicates exponential growth because it is greater than 1. This base remains constant for the functions \( g(x)=2^x-1 \) and \( h(x)=2^{x-1} \), but different shifts apply to modify their graphs.
Understanding exponential functions is crucial, as they model real-world phenomena like population growth, radioactive decay, and financial interest, providing a mathematical framework for understanding growth processes.