Problem 17 - Algebra and Trigonometry

Question

Graph both functions on one set of axes. $$ f(x)=4^{x} \quad \text { and } \quad g(x)=7^{x} $$

Step-by-Step Solution

Answer

Graph the exponential functions: \( f(x) = 4^x \) grows slower than \( g(x) = 7^x \).

Step 1: Understand the Functions

The functions given are exponential functions. The function \( f(x) = 4^x \) means the base is 4 and the variable is in the exponent. Similarly, \( g(x) = 7^x \) has a base of 7.

Step 2: Identify Key Points

Identify a few key points for each function. For example:- For \( f(x) = 4^x \), we have: \( f(0) = 1 \), \( f(1) = 4 \), \( f(2) = 16 \), \( f(-1) = 1/4 \).- For \( g(x) = 7^x \), we have: \( g(0) = 1 \), \( g(1) = 7 \), \( g(2) = 49 \), \( g(-1) = 1/7 \).

Step 3: Plot the Functions

Using the key points identified, plot both functions on the same graph. The x-axis should have values ranging from negative to positive, while the y-axis should be scaled to include the range of values produced by both functions.

Step 4: Analyze the Graph

Both functions will pass through the point (0,1) because anything raised to the power of zero is 1. Further, since 7 is greater than 4, \( g(x) = 7^x \) will grow faster than \( f(x) = 4^x \). The graph of \( g(x) \) should be steeper than that of \( f(x) \).

Key Concepts

Graphing Exponential Functions

Exponential functions are special because they show repeated multiplication. Understanding how they work is the first step to mastering them. When graphing, these functions create a curve that either rises or falls dramatically.

The functions we have, \(f(x) = 4^x\) and \(g(x) = 7^x\), are exponential because the variable \(x\) is in the exponent.
Consider how the graphs of \(y = b^x \) typically behave:

• For \(x = 0\), any base \(b\) results in a value of 1.This is why both functions pass through the point \((0, 1)\).
• If \(b > 1\), the function grows rapidly.The larger the base, the steeper the graph.This means \(g(x) = 7^x\) will be steeper than \(f(x) = 4^x\).

To graph these functions, choose a range of \(x\)-values and calculate corresponding \(y\)-values. Connect these points smoothly to form a curve.

Key Points Identification

Identifying key points can make graphing easier. These act as anchor points that help define the curve of the graph. For exponential functions, key points include: \(x = 0, 1, 2, -1, -2, \) and so on. Each has a specific value based on our functions.

• For \(f(x) = 4^x\):
  • \(f(0) = 1\)
  • \(f(1) = 4\)
  • \(f(2) = 16\)
  • \(f(-1) = 1/4\)
• For \(g(x) = 7^x\):
  • \(g(0) = 1\)
  • \(g(1) = 7\)
  • \(g(2) = 49\)
  • \(g(-1) = 1/7\)

These points give clarity on how the graphs will look.Compare these points across both functions to predict which grows faster or by how much.
Remember, the graphs also approach zero but never quite reach it as \(x\) becomes very negative.

Function Analysis

Let's dive into analyzing these exponential graphs. Seeing them side by side reveals important differences between \(f(x)\) and \(g(x)\).Examining the plot of both functions shows that:

• Both exponential functions intersect the y-axis at \((0,1)\).
• \(g(x) = 7^x\) grows faster than \(f(x) = 4^x\).This is because the base 7 is greater than base 4.Hence the steeper graph.

These observations confirm:

• A higher base in an exponential function equates to a steeper increase.
• Exponential functions are key in modeling rapid growth scenarios, like population or bank interest.

Understanding how their growth differs helps in solving real-world problems and choosing the right model for various data sets.