6. Exponential & Logarithmic Functions

Graphing Exponential Functions

6. Exponential & Logarithmic Functions

Graphing Exponential Functions: Videos & Practice Problems

Topic summary

Graphing exponential functions, such as $2^{x}$ , involves understanding key components like end behavior, intercepts, and asymptotes. The function increases as $x$ approaches infinity and approaches zero as $x$ approaches negative infinity, creating a horizontal asymptote at $0$ . Transformations, including shifts and reflections, can be applied to graph more complex functions, maintaining the domain of all real numbers and adjusting the range based on the asymptote's position.

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Graphs of Exponential Functions

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Graphs of Exponential Functions Video Summary

Graphing exponential functions, such as $ f(x) = 2^x $, is a straightforward process compared to polynomial and rational functions. To begin, we can generate ordered pairs by substituting values for $ x $. For instance, when $ x = 0 $, $ f(0) = 2^0 = 1 $, giving us the point (0, 1). Continuing with $ x = 1 $, we find $ f(1) = 2^1 = 2 $, leading to the point (1, 2). As $ x $ increases, the function values grow rapidly: $ f(2) = 4 $, $ f(3) = 8 $, and $ f(10) = 1024 $. This indicates that as $ x $ approaches infinity, $ f(x) $ also approaches infinity, demonstrating the end behavior of the graph.

For negative values of $ x $, we observe that $ f(-1) = 2^{-1} = \frac{1}{2} $ and as $ x $ decreases further, the function values approach zero but never actually reach it. This behavior indicates the presence of a horizontal asymptote at $ y = 0 $, which can be represented by a dashed line on the graph. The graph is continuous, meaning there are no breaks, and it passes the horizontal line test, confirming that it is a one-to-one function.

The overall shape of the graph resembles a stretched lowercase 'e', which can help in visualizing exponential growth. The domain of any exponential function is all real numbers, expressed as $ (-\infty, \infty) $. The range, however, depends on the asymptote; for $ f(x) = 2^x $, the range is $ (0, \infty) $, as the function values are always positive and never touch the asymptote.

When exploring other exponential functions of the form $ f(x) = b^x $, the base $ b $ significantly influences the graph's behavior. If $ b > 1 $, the graph will always increase, becoming steeper as $ b $ increases (e.g., from $ b = 2 $ to $ b = 5 $). Conversely, if $ 0 < b < 1 $, the graph will decrease, and it will become steeper as $ b $ approaches zero (e.g., from $ b = \frac{1}{2} $ to $ b = \frac{1}{4} $). Understanding these characteristics allows for a comprehensive grasp of exponential functions and their graphs.

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Graphing Exponential Functions Example 1

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Graphing Exponential Functions Example 1 Video Summary

To graph the function $ f(x) = \left(\frac{1}{2}\right)^x $, we start by evaluating the function at various values of $ x $. This process involves substituting different $ x $ values into the function to find corresponding $ y $ values, which we can then plot on a graph.

First, when $ x = 0 $, we calculate:

\[ f(0) = \left(\frac{1}{2}\right)^0 = 1 \]

This gives us our first point at $ (0, 1) $.

Next, for $ x = 1 $:

\[ f(1) = \left(\frac{1}{2}\right)^1 = \frac{1}{2} \]

We plot the second point at $ (1, \frac{1}{2}) $.

For $ x = 2 $:

\[ f(2) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

This gives us the point $ (2, \frac{1}{4}) $.

Continuing with $ x = 3 $:

\[ f(3) = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]

We plot this point at $ (3, \frac{1}{8}) $.

Now, we will evaluate negative values of $ x $. Starting with $ x = -1 $:

\[ f(-1) = \left(\frac{1}{2}\right)^{-1} = 2 \]

This gives us the point $ (-1, 2) $.

For $ x = -2 $:

\[ f(-2) = \left(\frac{1}{2}\right)^{-2} = 4 \]

We plot this point at $ (-2, 4) $.

Finally, for $ x = -3 $:

\[ f(-3) = \left(\frac{1}{2}\right)^{-3} = 8 \]

This gives us the point $ (-3, 8) $.

As we analyze the points plotted, we observe that as $ x $ increases, the $ y $ values decrease and approach zero, indicating that the graph approaches the x-axis but never touches it. This behavior suggests the presence of a horizontal asymptote at $ y = 0 $.

Additionally, we can recognize that the function $ f(x) = \left(\frac{1}{2}\right)^x $ can be rewritten as $ f(x) = 2^{-x} $. This transformation shows that the graph of $ \left(\frac{1}{2}\right)^x $ is a reflection of the graph of $ 2^x $ across the y-axis.

In terms of the domain and range of the function, the domain of all exponential functions is all real numbers, expressed as $ (-\infty, \infty) $. The range, determined by the asymptote, is from $ 0 $ to $ \infty $, denoted as $ (0, \infty) $, since the function never actually reaches zero.

In summary, the graph of $ f(x) = \left(\frac{1}{2}\right)^x $ exhibits exponential decay, approaches the x-axis as an asymptote, and has a domain of all real numbers and a range of positive values.

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Transformations of Exponential Graphs

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Transformations of Exponential Graphs Video Summary

Understanding how to graph exponential functions is essential, especially when dealing with more complex forms. The basic exponential function can be expressed as f(x) = b^x, where b is a positive constant. When faced with a more complicated function, such as g(x), we can utilize transformation rules rather than performing extensive calculations.

Transformations of exponential functions primarily involve reflections and shifts. A negative sign outside the function indicates a reflection over the x-axis, while a negative sign inside the function reflects over the y-axis. Additionally, the parameters h and k represent horizontal and vertical shifts, respectively. Specifically, h indicates how many units to shift horizontally, and k indicates how many units to shift vertically.

To graph a transformed function like g(x), the first step is to identify and graph the parent function, which in this case is f(x) = 2^x. Key points for this function include:

At x = -1, f(-1) = 1/2
At x = 0, f(0) = 1
At x = 1, f(1) = 2

These points can be plotted, and the graph will approach the horizontal asymptote at y = 0.

Next, we shift the horizontal asymptote according to the value of k. If k = -4, the new asymptote will be at y = -4. If there is no negative sign in the function, there is no reflection to consider. The next step involves determining the values of h and k from the transformed function. For example, if g(x) = 2^{(x - 1)} - 4, then h = 1 and k = -4.

To find the new points for g(x), shift the original points from the parent function: move each point 1 unit to the right and 4 units down. After plotting these new points, connect them smoothly, ensuring the graph approaches the new asymptote at y = -4.

Finally, the domain of any exponential function remains the same, extending from negative infinity to positive infinity. The range, however, is determined by the position of the graph relative to the asymptote. Since the graph is above the asymptote at y = -4, the range is [-4, ∞). If the graph were below the asymptote, the range would be (-∞, k].

By mastering these transformation techniques, you can confidently graph more complex exponential functions and understand their behaviors.

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Transformations of Exponential Graphs Example 1

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Transformations of Exponential Graphs Example 1 Video Summary

To graph the function $ g(x) = -\frac{1}{2^x} + 3 $ as a transformation of the parent function $ f(x) = \frac{1}{2^x} $, we start by understanding the transformations involved. The negative sign indicates a reflection over the x-axis, while the "+3" indicates a vertical shift upwards by 3 units.

First, we identify the parent function $ f(x) = \frac{1}{2^x} $. This function has a horizontal asymptote at $ y = 0 $. We can plot key points for this function: for $ x = -1 $, $ f(-1) = 2 $; for $ x = 0 $, $ f(0) = 1 $; and for $ x = 1 $, $ f(1) = \frac{1}{2} $. These points are (-1, 2), (0, 1), and (1, 0.5). Connecting these points gives us the graph of the parent function, which approaches the horizontal asymptote at $ y = 0 $.

Next, we apply the transformations to obtain $ g(x) $. The first step is to shift the horizontal asymptote from $ y = 0 $ to $ y = 3 $ due to the "+3" in the function. This means our new horizontal asymptote is now at $ y = 3 $.

Next, we reflect the points of the parent function over the x-axis because of the negative sign in front of the function. The reflected points will be: (-1, -2), (0, -1), and (1, -0.5). After reflecting, we then shift these points up by 3 units. The new coordinates become: (-1, 1), (0, 2), and (1, 2.5).

Now, we can sketch the graph of $ g(x) $ by connecting these transformed points with a smooth curve that approaches the new horizontal asymptote at $ y = 3 $. The graph will be below this asymptote, indicating that the function values are less than 3.

Finally, we determine the domain and range of the function. The domain of $ g(x) $ remains all real numbers, $ (-\infty, \infty) $. The range, however, is limited by the horizontal asymptote, which means the function values will approach but never reach 3. Therefore, the range is $ (-\infty, 3) $.

In summary, the graph of $ g(x) = -\frac{1}{2^x} + 3 $ is a reflection of the parent function $ f(x) = \frac{1}{2^x} $ that has been shifted up by 3 units, with a horizontal asymptote at $ y = 3 $, a domain of all real numbers, and a range of $ (-\infty, 3) $.

Problem

Graph the given function.
$g\left(x\right)=4^{-x}-1$

Problem

The graph for the function $f\left(x\right)=3^{x}$ is given below.

Match the given function, $g\left(x\right)$ , to its graph. $g\left(x\right)=-3^{x+2}+1$

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Here’s what students ask on this topic:

What is the domain and range of exponential functions?

The domain of exponential functions is all real numbers, meaning they can take any real value for x. This is because exponential functions are defined for all x values. The range, however, depends on the horizontal asymptote and the direction of the graph. For functions like $2^{x}$ , the range typically starts from zero (exclusive) to infinity, as the function approaches but never touches the asymptote at y=0. If transformations are applied, such as vertical shifts, the range will adjust accordingly, starting from the new asymptote set by the transformation value k to infinity.

How do transformations affect the graph of exponential functions?

Transformations can significantly alter the graph of exponential functions. Horizontal shifts, denoted by h, move the graph left or right, while vertical shifts, denoted by k, move it up or down. Reflections can occur over the x-axis or y-axis, depending on whether a negative sign is outside or inside the function. These transformations allow us to graph more complex functions without recalculating points. For example, in $2^{x - 1}$ - 4, the graph shifts right by 1 unit and down by 4 units. The domain remains all real numbers, but the range shifts according to the vertical transformation.

What is the significance of the horizontal asymptote in exponential functions?

The horizontal asymptote in exponential functions represents the limit behavior of the function as x approaches negative infinity. Typically, this asymptote is at y=0 for basic exponential functions like $2^{x}$ . The function approaches but never touches this line, indicating that as x becomes very negative, the function value gets closer to zero. This asymptote is crucial for understanding the range of the function, as it sets the lower boundary for y values. When transformations are applied, the asymptote shifts vertically, affecting the range of the transformed function.

How do you graph exponential functions using transformations?

To graph exponential functions using transformations, start by identifying the parent function, such as $2^{x}$ . Plot key points and the horizontal asymptote at y=0. Apply transformations: horizontal shifts (h units), vertical shifts (k units), and reflections (negative signs affecting x or y-axis). For example, in $2^{x - 1}$ - 4, shift the graph right by 1 unit and down by 4 units. Adjust the asymptote to y=-4. Connect the transformed points, ensuring the graph approaches the new asymptote. The domain remains all real numbers, while the range shifts according to the vertical transformation.

What is the end behavior of exponential functions?

The end behavior of exponential functions describes how the function behaves as x approaches positive or negative infinity. For functions like $2^{x}$ , as x goes to positive infinity, the function value increases rapidly, approaching infinity. Conversely, as x approaches negative infinity, the function value approaches zero but never touches it, due to the horizontal asymptote at y=0. This behavior is consistent across exponential functions, though transformations can shift the asymptote, affecting the range but not the overall end behavior.