Chapter 8: Problem 60
Graph the exponential function. $$y=5^{x}$$
Short Answer
Expert verified
The graph of the function \(y=5^{x}\) has an x-intercept at x=0, reacts rapidly with increasing x, and approaches but never reaches 0 as x decreases.
Step by step solution
01
Understanding Exponential Functions
In this step, gain an understanding of what an exponential function looks like. Our function is \(y=5^{x}\), making it an exponential function. Exponential functions have the form \(y=a^{x}\), where 'a' is a positive constant not equal to one. Here, 'a' is 5. The graph of an exponential function will always pass through (0, 1) because any non-zero number raised to the power of 0 is 1.
02
Plotting Key Points on the Graph
Calculate a few points to plot on the graph. For our function, when x=0, y=1 (because any number raised to 0 is 1). These points are then plotted on the graph.
03
Drawing the Graph
The graph of \(y=5^{x}\) rises rapidly as 'x' increases, but as 'x' decreases, 'y' gradually approaches 0 without reaching it. So the graph is always above the x-axis but never touches it (i.e., the x-axis is an asymptote for the graph). Drawing in all the key features will get you the complete graph of the function.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Function Properties
Understanding the properties of exponential functions is essential in various fields of mathematics and science. An exponential function is defined as a mathematical expression of the form \(y=a^{x}\), where the base \(a\) is a positive constant and \(x\) is the exponent. One distinct property of these functions is that their rate of change increases or decreases multiplicatively. For instance, in the function \(y=5^{x}\), the base is 5, and as \(x\) increases, the value of \(y\) will grow quickly because the values are being repeatedly multiplied by 5.
Other significant properties include the fact that exponential functions will always cross the y-axis at \(y=1\) when the exponent \(x\) is 0, indicating that the point \((0, 1)\) will always be on the graph. Also, the graph will never touch the x-axis, meaning the x-axis is an asymptote. This is because the output of the function approaches zero but never reaches it as \(x\) goes towards negative infinity. Furthermore, if the base \(a\) is greater than 1, the graph will be increasing, and if \(0 < a < 1\), it will be decreasing. These properties help define the distinctive 'J-shaped' curve for a graph of an exponential function with a base greater than 1.
Other significant properties include the fact that exponential functions will always cross the y-axis at \(y=1\) when the exponent \(x\) is 0, indicating that the point \((0, 1)\) will always be on the graph. Also, the graph will never touch the x-axis, meaning the x-axis is an asymptote. This is because the output of the function approaches zero but never reaches it as \(x\) goes towards negative infinity. Furthermore, if the base \(a\) is greater than 1, the graph will be increasing, and if \(0 < a < 1\), it will be decreasing. These properties help define the distinctive 'J-shaped' curve for a graph of an exponential function with a base greater than 1.
Plotting Points on a Graph
When graphing functions, it's essential to plot key points accurately to build a precise representation of the function on a coordinate plane. To plot points for the exponential function \(y=5^{x}\), you should start by calculating the function's value at various \(x\) values. A helpful strategy is to choose a range of positive and negative values for \(x\) and corresponding \(y\) outputs to understand the function's growth and decay.
For instance, you can start at \(x=0\), which gives us the point \((0, 1)\) as mentioned previously. Moving on, you can choose integral values for \(x\) to keep calculations simple, like \(-2, -1, 1,\) and \(2\). This will give you the respective points based on the function's equation, which you can then place on the graph. Remember to plot these points accurately on a set of axes. It's crucial to note that the distance between points on the graph will not be uniform due to the nature of exponential growth or decay - with distances quickly increasing or decreasing depending on the value of \(x\).
For instance, you can start at \(x=0\), which gives us the point \((0, 1)\) as mentioned previously. Moving on, you can choose integral values for \(x\) to keep calculations simple, like \(-2, -1, 1,\) and \(2\). This will give you the respective points based on the function's equation, which you can then place on the graph. Remember to plot these points accurately on a set of axes. It's crucial to note that the distance between points on the graph will not be uniform due to the nature of exponential growth or decay - with distances quickly increasing or decreasing depending on the value of \(x\).
Graphing Exponential Equations
The process of graphing exponential equations involves more than just plotting points; it requires an understanding of the equation's properties and its behavior on a coordinate plane. Once you've plotted the key points of the function \(y=5^{x}\), you can start to visualize its curve. As you'll notice, for positive \(x\) values, the function rises quickly, reflecting exponential growth. Conversely, for negative \(x\) values, the curve approaches the x-axis asymptotically from above, portraying exponential decay.
When you connect the points, ensure that the line is smooth and continuous, reflecting the fact that exponential functions are continuous for all real numbers. The line should never cross the x-axis and should extend infinitely in both the positive direction of the y-axis (as \(x\) becomes more positive) and horizontally towards the x-axis (as \(x\) becomes more negative). Adding this to the plotted points and remembering the asymptotic behavior along the x-axis will yield a complete and accurate graph of the exponential function. By mastering these graphing techniques, you can better understand the impact of other transformations like shifts, stretches, and reflections on the exponential graph.
When you connect the points, ensure that the line is smooth and continuous, reflecting the fact that exponential functions are continuous for all real numbers. The line should never cross the x-axis and should extend infinitely in both the positive direction of the y-axis (as \(x\) becomes more positive) and horizontally towards the x-axis (as \(x\) becomes more negative). Adding this to the plotted points and remembering the asymptotic behavior along the x-axis will yield a complete and accurate graph of the exponential function. By mastering these graphing techniques, you can better understand the impact of other transformations like shifts, stretches, and reflections on the exponential graph.
