Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 4: Problem 16
Sketch the graph of \(f\) $$f(x)=8(4)^{-x}-2$$
Short Answer
Expert verified
Sketch a decreasing curve crossing at \((0, 6)\) with asymptote \(y = -2\).
Step by step solution
01
Identify the components of the function
The function given is \(f(x) = 8(4)^{-x} - 2\). It is an exponential function in the form \(a(b)^{-x} + c\). Here, \(a = 8\), \(b = 4\), and \(c = -2\). This implies a transformation of the exponential function \(4^{-x}\) with a vertical stretch by a factor of 8 and a downward translation by 2 units.
02
Analyze the base exponential function
The base function is \(4^{-x}\). This is a decreasing exponential function because the exponent \(-x\) reverses the typical growth of \(4^x\). As \(x\) increases, \(4^{-x}\) decreases.
03
Apply the vertical stretch
Multiply the base function \(4^{-x}\) by 8, as indicated by the function \(8(4)^{-x}\). This effectively stretches the graph vertically by a factor of 8 compared to \(4^{-x}\).
04
Apply the vertical translation
Translate the entire graph of \(8(4)^{-x}\) downwards by 2 units, resulting in the final function \(f(x) = 8(4)^{-x} - 2\). This affects the vertical position of the graph, moving the horizontal asymptote from 0 to -2.
05
Determine key points
To sketch the graph, identify key points. Start with \(x = 0\): \(f(0) = 8(4)^0 - 2 = 8(1) - 2 = 6\). For \(x = 1\): \(f(1) = 8(4)^{-1} - 2 = 8(\frac{1}{4}) - 2 = 2 - 2 = 0\). For \(x = 2\): a smaller value, calculating further can help identify the curve.
06
Sketch the graph
Plot the points found: \((0, 6)\) and \((1, 0)\). Use these to sketch the curve, noting it approaches, but never reaches, an asymptote at \(y = -2\). The curve steadily decreases because of the negative exponent.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vertical Stretch
A vertical stretch occurs when you multiply an entire function by a given factor. In the function \( f(x) = 8(4)^{-x} - 2 \), the number 8 serves as the factor that stretches the graph vertically. Imagine taking the base graph of \( 4^{-x} \) and pulling it upwards away from the x-axis so that it grows taller and steeper.
This transformation affects every point on the graph by multiplying the y-values by 8. This makes the graph rise or fall more sharply compared to its base form.
This transformation affects every point on the graph by multiplying the y-values by 8. This makes the graph rise or fall more sharply compared to its base form.
- For example, if a point on the base graph at \((x, y)\) would normally be \((x, 1)\), after a vertical stretch by 8, it would be \((x, 8)\).
- This transformation does not move the graph horizontally, just vertically.
Vertical Translation
Vertical translation involves shifting all the points on a graph up or down by a certain number of units. For the function \( f(x) = 8(4)^{-x} - 2 \), the term '-2' indicates a downward translation by 2 units.
This means that every point on the transformed function will be moved 2 units downward from where it would be if the '-2' were not there.
This means that every point on the transformed function will be moved 2 units downward from where it would be if the '-2' were not there.
- For example, a point that was at \((x, 6)\) on the vertically stretched graph becomes \((x, 4)\) after the vertical translation.
- This downward shift impacts the horizontal asymptote as well, moving it from \( y = 0 \) to \( y = -2 \).
Asymptotes
Asymptotes are lines that a graph approaches but never quite touches or intersects. They give a sense of direction to the curve of your graph. In the function \( f(x) = 8(4)^{-x} - 2 \), the asymptote is horizontal at \( y = -2 \).
Initially, the base function \( 4^{-x} \) has an asymptote at \( y = 0 \). Due to the vertical translation of -2, this line moves downwards, which helps us understand the behavior of the function as \( x \) approaches infinity or negative infinity. This manifests as:
Initially, the base function \( 4^{-x} \) has an asymptote at \( y = 0 \). Due to the vertical translation of -2, this line moves downwards, which helps us understand the behavior of the function as \( x \) approaches infinity or negative infinity. This manifests as:
- The graph will get closer and closer to this horizontal line but will never actually touch it.
- Identifying asymptotes is crucial in graphing techniques because they help define the boundary behavior of the graph.
Graphing Techniques
Graphing techniques are methods used to accurately depict the path of a function's graph. For \( f(x) = 8(4)^{-x} - 2 \), we consider vertical stretching, vertical translation, and asymptotic behavior.
When graphing this function:
When graphing this function:
- First, plot the base function \( 4^{-x} \), understanding it decreases as \( x \) increases.
- Apply the vertical stretch by multiplying y-values by 8 to capture the steepness.
- Translate downwards by 2 units to reflect the final function.
- Recognize and draw the asymptote at \( y = -2 \).
- Verify your graph by plotting key points, such as where the graph crosses the y-axis.
Key Points Identification
Identifying key points helps in forming a detailed graph, capturing essential behavior and intersections. For the given function \( f(x) = 8(4)^{-x} - 2 \), critical points guide the graphing process.
To identify these points:
To identify these points:
- Calculate \( f(0) = 6 \) to find where it crosses the y-axis at \((0, 6)\).
- Find \( f(1) = 0 \) giving another plot point at \((1, 0)\).
- Consider additional points for a complete picture, such as \( x = 2 \), which would help illustrate how fast the graph approaches the asymptote.
