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Magazine Chapter 5: Problem 46
Sketch a graph of \(y=f(x)\) $$ f(x)=4^{x} $$
Short Answer
Expert verified
Sketch a smooth curve passing through (0,1) and increasing rapidly, approaching the x-axis as an asymptote.
Step by step solution
01
Understand the Function
The given function is an exponential function, written as \( f(x) = 4^x \). This type of function involves a base raised to the power of \( x \), where the base here is 4. The function has a characteristic shape, usually showing rapid growth or decay.
02
Identify Key Characteristics
For the exponential function \( f(x) = 4^x \), note the following key characteristics:- The domain is all real numbers, \( x \in (-\infty, \infty) \).- The range is positive real numbers, \( y \in (0, \infty) \).- The y-intercept is at \( (0, 1) \), since \( 4^0 = 1 \).- The graph is an increasing function with no x-intercepts.
03
Plot Key Points
Choose some easy points to calculate their \( y \)-values and plot them:- At \( x = 0 \), \( f(0) = 4^0 = 1 \), so plot \( (0, 1) \).- At \( x = 1 \), \( f(1) = 4^1 = 4 \), so plot \( (1, 4) \).- At \( x = -1 \), \( f(-1) = 4^{-1} = 0.25 \), plot \( (-1, 0.25) \).- Continue this pattern for additional points like \( x = 2 \) and \( x = -2 \).
04
Draw the Graph
Using the points plotted in Step 3, draw a smooth curve starting from near zero on the left (asymptotically approaching the x-axis) and rising sharply as \( x \) increases. Make sure the curve passes through all the plotted points. The graph should steadily increase, reflecting the positive base of the exponential term.
05
Sketch the Asymptote
Since \( f(x) = 4^x \) never reaches zero, the x-axis (\( y = 0 \)) acts as a horizontal asymptote. Indicate this by noting the tendency of the graph to approach the x-axis but never touch it or cross it.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Techniques
When graphing exponential functions like \( f(x) = 4^x \), remember to follow a few key techniques to create an accurate representation. Start with plotting the easily calculated points that help form the framework of your graph.
- Begin by identifying the y-intercept, which for \( f(x) = 4^x \) is at \((0, 1)\), since any base raised to the power of zero equals one.
- Select additional x-values, such as -1, 1, and 2, to calculate corresponding y-values. These calculations will yield points like \((-1, 0.25)\), \((1, 4)\), and \((2, 16)\).
- Plot these points on your graph accurately. Ensure they form a smooth curve that appears to rise sharply as \(x\) increases.
- Remember the horizontal asymptote \( y = 0 \) as the curve approaches the x-axis but never crosses it.
Function Characteristics
Understanding the characteristics of exponential functions like \( f(x) = 4^x \) is key to mastering their behavior. These functions have specific traits that set them apart from other function types.
- The **domain** is all real numbers, \( x \in (-\infty, \infty) \), indicating that you can input any real number and receive a valid output.
- The **range** is \( y \in (0, \infty) \), meaning the output is always positive. This reflects the fact that powers of any positive number (like 4 in this case) are always positive.
- There is no x-intercept, as the graph never touches the x-axis. Since 4 raised to any real number doesn't equal zero, the function never hits y=0.
- The y-intercept is at \((0, 1)\), since \( 4^0 = 1 \). This point is crucial as it anchors the curve on the graph.
Exponential Growth
Exponential functions like \( f(x) = 4^x \) demonstrate exponential growth, a pattern that becomes quickly apparent when plotting these functions.
- The base of the function, 4 in this case, is greater than one, signaling that the function increases as x increases. This forms a distinctly steep curve upward.
- This type of growth is faster than linear growth. Even small changes in x lead to large changes in y. For example, from \( x=1 \) to \( x=2 \), there is a jump from y=4 to y=16.
- Exponential growth is often used to model real-world scenarios where quantities double rapidly, such as populations, financial investments, or the spread of a virus.
