Problem 17 - Precalculus : Building Concepts and Connections

Question

Sketch the graph of each function. $$f(x)=4^{x}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = 4^x\) is an upward-sloping curve that passes through the points (-1, 0.25), (0, 1), and (1, 4) and gets infinitely close to, but never touches, the x-axis as \(x\) becomes increasingly negative.

Step 1: Understand the nature of the function

First, we need to understand the key characteristics of exponential functions. An exponential function is a mathematical function of the following form: \(f(x) = a^x\), where 'a' is a positive constant different from 1. For our function, \(a = 4\). The graph of an exponential function will always pass through the point (0, 1), because any nonzero number raised to the power of 0 is 1.

Step 2: Determine a few key points

Next, we plug in a few values to get some points to plot on the graph. For \(x = -1\), \(f(x) = 4^{-1} = 0.25\). For \(x = 0\), we already know that \(f(x) = 1\), and for \(x = 1\), \(f(x) = 4^1 = 4\). Therefore, three points we can plot on the graph are (-1, 0.25), (0, 1), and (1, 4).

Step 3: Sketch the graph

Plot the points determined above on an x-y graph. Point when \(x = -1\) plots at 0.25 on the y-axis, when \(x = 0\) it plots at 1 on the y-axis and when \(x = 1\) it plots at 4 on the y-axis. In the end, connect the points to show increasing exponential growth.

Step 4: Recognize key functions characteristics

It's crucial to consider that the function has a horizontal asymptote at \(y = 0\). Thus, as \(x\) approaches negative infinity, \(y\) gets infinitely closer to but never reaches zero. Also, as \(x\) increases, \(y\) increases in an exponential pace. This growth is reflected in the way the graph steeply increases for positive x-values.