Problem 57 - Algebra 2

Question

Graph each pair of functions on the same screen. Then compare the graphs, listing both similarities and differences in shape, asymptotes, domain, range, and \(y\) -intercepts. $$ y=3^{x} \quad y=3^{x+1} $$

Step-by-Step Solution

Answer

The functions are similar in shape, asymptotes, domain, and range, but differ in horizontal position and \( y \)-intercept.

Step 1: Understand the Functions

We are given two functions: \( y = 3^x \) and \( y = 3^{x+1} \). The first function is an exponential function with base 3, and the second is also an exponential function, but it's shifted horizontally.

Step 2: Graph the Functions

Plot the first function, \( y = 3^x \), which is an increasing exponential curve. Next, plot \( y = 3^{x+1} \), which is the same exponential curve shifted one unit to the left on the x-axis compared to \( y = 3^x \).

Step 3: Compare Shapes and Intercepts

Both graphs are exponential curves that rise steeply. They have similar shapes: increasing, smooth, and continuous curves. The \( y \)-intercept for \( y = 3^x \) is 1 (when \( x = 0 \)) while for \( y = 3^{x+1} \) it is 3 (since \( 3^{0+1} = 3 \)).

Step 4: Analyze Asymptotes

Both functions have a horizontal asymptote at \( y = 0 \) because as \( x \) approaches negative infinity, \( y = 3^x \) and \( y = 3^{x+1} \) approach zero.

Step 5: Determine Domain and Range

The domain for both functions is all real numbers, \( (-\infty, \infty) \). The range for both functions is \( (0, \infty) \), meaning the graphs lie above the x-axis.

Step 6: Summarize Similarities and Differences

**Similarities:** Both are exponential functions with a horizontal asymptote at \( y = 0 \), the domain is \( (-\infty, \infty) \), and the range is \( (0, \infty) \). **Differences:** \( y = 3^{x+1} \) is shifted one unit to the left and has a different \( y \)-intercept compared to \( y = 3^x \).

Key Concepts

Domain and Range

When graphing exponential functions like \( y = 3^x \) and \( y = 3^{x+1} \), understanding the domain and range is crucial. The **domain** refers to all possible input values \( x \) that you can plug into the function. In these cases, the domain is all real numbers \((-\infty, \infty)\), meaning you can input any real number into the exponent of 3.

The **range** refers to all possible output values. For both \( y = 3^x \) and \( y = 3^{x+1} \), the range is \((0, \infty)\).

• This is because, regardless of the exponent, the output is always positive.
• Exponential functions grow as \( x \) becomes larger and approach zero as \( x \) becomes smaller, but they never actually reach zero.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches but never touches. For the functions \( y = 3^x \) and \( y = 3^{x+1} \), the **horizontal asymptote** is at \( y = 0 \).

Here's why:

• As \( x \) approaches negative infinity, the values of \( 3^x \) and \( 3^{x+1} \) get closer and closer to zero.
• However, the values never quite reach zero, creating that imaginary line at \( y = 0 \).

Understanding asymptotes helps in sketching the general behavior of the graph as it extends left and right.

Function Transformation

The concept of function transformation involves shifting and modifying functions to create new graphs. For example, in \( y = 3^{x+1} \), the "\(+1\)" indicates a **horizontal shift**.

Here's how it works:

• The graph of \( y = 3^x \) is shifted one unit to the left, resulting in \( y = 3^{x+1} \).

These transformations do not change the core nature of the function like its exponential growth or asymptote but do change its position in the coordinate plane.

Function transformations are key to comparing graphs and understanding how changes in the equation affect the visual representation of a function.

Exponential Growth

Exponential functions like \( y = 3^x \) and \( y = 3^{x+1} \) are powerful demonstrations of **exponential growth**. This type of growth means the rate of increase becomes faster at larger values of \( x \).

Here’s what makes them unique:

• The base, which is 3 in these examples, indicates how steeply the graph will rise. A higher base results in faster growth.
• These functions will continue to grow increasingly fast without bound as \( x \) gets larger.

Understanding exponential growth is essential in various real-world applications—from calculating compound interest to modeling population growth.