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Magazine Chapter 6: Problem 38
Sketch the graph of \(f(x)=\left(\frac{1}{3}\right)^{x}\). Then refer to it and use earlier techniques to graph each finction. $$f(x)=\left(\frac{1}{3}\right)^{x}+4$$
Short Answer
Expert verified
Shift the graph of \( f(x) = (\frac{1}{3})^x \) up by 4 units.
Step by step solution
01
Understand the Base Function
The base function is \( f(x) = \left(\frac{1}{3}\right)^x \). This is an exponential decay function because the base \( \frac{1}{3} \) is between 0 and 1. As \( x \) increases, \( f(x) \) decreases towards zero.
02
Identify Key Points of Base Function
Select key points for the base function \( f(x) = \left(\frac{1}{3}\right)^x \). When \( x = 0 \), \( f(0) = 1 \); when \( x = 1 \), \( f(1) = \frac{1}{3} \); when \( x = 2 \), \( f(2) = \frac{1}{9} \); and when \( x = -1 \), \( f(-1) = 3 \). These help form the decreasing pattern of the graph.
03
Plot the Base Function
Plot the points from Step 2 and connect them to sketch the graph of \( f(x) = \left(\frac{1}{3}\right)^x \). The graph passes through points such as \( (0, 1) \), \( (1, \frac{1}{3}) \), and \( (-1, 3) \). It approaches the x-axis as \( x \) increases.
04
Understand the Vertical Shift
The function \( g(x) = \left(\frac{1}{3}\right)^x + 4 \) is the base function shifted vertically. Adding 4 means every value of \( f(x) \) is increased by 4 units. Thus, the horizontal asymptote is now \( y = 4 \) instead of \( y = 0 \).
05
Plot the Transformed Function
Adjust the graph of the base function by moving each point 4 units upward. For example, the point \( (0, 1) \) on the base function becomes \( (0, 5) \), and the horizontal asymptote is now \( y = 4 \). This sketch represents \( g(x) = \left(\frac{1}{3}\right)^x + 4 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Decay
Exponential decay is a type of function where the value decreases over time. The base function \(f(x) = \left(\frac{1}{3}\right)^{x}\) is a classic example of exponential decay. Since the base \(\frac{1}{3}\) is less than 1, the function's value decreases as \(x\) increases. This behavior is because multiplying by a fraction less than 1 makes the product smaller each time. Here are some key points to visualize exponential decay:
- When \(x = 0\), \(f(x) = 1\). This is because any number raised to the zero power is one.
- When \(x = 1\), \(f(x) = \frac{1}{3}\). Here, you see the beginning of the decay as the value starts to get smaller.
- When \(x = 2\), \(f(x) = \frac{1}{9}\). The value continues to decrease as the exponent increases.
- When \(x = -1\), \(f(x) = 3\). Negative exponents reverse the process, increasing the value since we're actually finding the reciprocal.
Vertical Shift
A vertical shift occurs when you add or subtract a constant from a function. In our example, \(g(x) = \left(\frac{1}{3}\right)^x + 4\), we add 4 to every value of the base function \(f(x) = \left(\frac{1}{3}\right)^{x}\). This simple addition translates the entire graph upward in the vertical direction by 4 units.Key points about vertical shifts:
- Every output value of the original function is increased by the same amount鈥攊n this case, 4. Thus \(f(x)\) is transformed to \(g(x) = f(x) + 4\).
- The horizontal asymptote shifts from \(y = 0\) to \(y = 4\). This means the baseline, where the graph stabilizes towards infinity, is now elevated by 4 units.
- Graphically, if the original function passed through \((0, 1)\), the shifted function would pass through \((0, 5)\).
Graphing Transformations
Graphing transformations help us analyze how functions change when modified. The function \(g(x) = \left(\frac{1}{3}\right)^x + 4\) showcases a combination of exponential decay and a vertical shift.Stages of graphing transformations:
- Start by plotting the base function \(f(x) = \left(\frac{1}{3}\right)^{x}\). Sketch points like \((0, 1)\), \((1, \frac{1}{3})\), and \((-1, 3)\) to form a visual guide.
- Envision the transformation by adjusting each point upwards by 4 units, aligning them to the transformation \(f(x) + 4\).
- Redraw the shifted points to obtain the graph of the transformed function \(g(x)\). For instance, point \((0, 1)\) moves to \((0, 5)\).
- Redefine the horizontal asymptote from \(y = 0\) to \(y = 4\), as the new baseline stabilization.
