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Chapter 5: Problem 49

Explain how the graph of the given function can be obtained from the graph of y=\log _{2} x, and (b) graph the function. $$y=\log _{2}(-x)+1$$

Short Answer

Expert verified
Reflect \( y = \log_{2} x \) across the y-axis, then shift up by 1 unit.

Step by step solution

01

Recognize the Base Function

Start by identifying the base function, which is given as \( y = \log_{2} x \). This is the logarithmic function with base 2, which is defined only for positive values of \( x \).
02

Reflect Across the y-axis

The given function is \( y = \log_{2}(-x) + 1 \). Here, \( \log_{2}(-x) \) indicates a reflection of the base function \( y = \log_{2} x \) across the y-axis. This is because substituting \( -x \) in place of \( x \) inverts the graph horizontally.
03

Translate Vertically

The expression \( +1 \) in \( y = \log_{2}(-x) + 1 \) translates the graph of \( y = \log_{2}(-x) \) upwards by 1 unit along the y-axis.
04

Graph the Function

To graph \( y = \log_{2}(-x) + 1 \), start by sketching \( y = \log_{2}(-x) \), which is the reflection of \( y = \log_{2}x \) over the y-axis, resulting in the graph inhabiting the negative x-domain. Then, shift this entire graph up by 1 unit to account for the \(+1\) translation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reflection of Graph
When dealing with transformations of logarithmic functions, a reflection is one of the key changes that can occur. A reflection across the y-axis involves changing the input of the function from \( x \) to \( -x \). This change flips the graph horizontally.
  • For our function \( y = \log_2(-x) \), the reflection causes the graph originally on the right side of the y-axis to flip over to the left.
  • The domain changes to negative values, meaning instead of being defined for \( x > 0 \), it's now defined for \( x < 0 \).
This type of inversion is common in functions and isn鈥檛 limited to logarithms. Anytime you see \( -x \) in place of \( x \), expect the graph to perform this horizontal flip.
Vertical Translation
After reflecting a function, you might also need to perform a vertical translation. This is a vertical shift, moving the graph up or down along the y-axis.
  • In our function \( y = \log_2(-x) + 1 \), the '+1' indicates that after reflecting, the graph should be translated upwards by 1 unit.
  • This kind of translation adjusts the entire graph equally upwards, affecting every point on the graph.
Vertical translations are often adjustments to a function's base form, allowing it to model data more accurately or fit a desired position on the graph.
Base Function
In any transformation problem, recognizing the base function is crucial. Here, our base function is \( y = \log_2 x \). This is a standard logarithmic function where the input \( x \) must be positive.
  • The base of the logarithm, which is 2 in this case, affects the rate and shape of the logarithmic curve.
  • Understanding the base function allows you to predict how transformations like vertical shifts or reflections will affect the graph.
It serves as the starting point, or "blueprint," for any graphing transformations.
Graph Transformation
Graph transformations involve systematically altering a graph by shifts, reflections, stretches, or compressions. Each transformation corresponds to a change in the function鈥檚 equation.
  • The reflection across the y-axis, translating \( x \) to \( -x \), is a horizontal transformation.
  • Adding 1 to the function translates the graph vertically, raising it by 1 unit.
Understanding these principles is key to graphing transformations. By following this step-by-step approach, you gain clarity on how the graphical representation of a function changes with its mathematical equation.

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Most popular questions from this chapter

Radioactive cesium 137 was emitted in large amounts in the Chernobyl nuclear power station accident in Russia on April \(26,1986 .\) The amount of cesium 137 remaining after \(x\) years in an initial sample of 100 milligrams can be described by $$A(x)=100 e^{-0.02295 x}$$ (Source: Mason, C., Biology of Freshwater Pollution, John Wiley and Sons.) (a) Estimate how much is remaining after 50 years. Is the half-life of cesium 137 greater or less than 50 years? (b) Estimate the half-life of cesium 137 .

In \(2012,17 \%\) of the U.S. population was Hispanic, and this number is expected to be \(31 \%\) in \(2060 .\) (Source: U.S. Census Bureau.) (a) Approximate \(C\) and \(a\) so that \(P(x)=C a^{x-2012}\) models these data, where \(P\) is the percent of the population that is Hispanic and \(x\) is the year. Why is \(a>1 ?\) (b) Estimate \(P\) in 2030 . (c) Use \(P\) to estimate the year when \(25 \%\) of the population could be Hispanic.

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Assume that \(f(x)=a^{x},\) where \(a>1\) If \(a=e,\) what is an equation for \(y=f^{-1}(x) ?\) (You need not solve for \(y .\) )

The revenue in millions of dollars for the first 5 years of mobile advertising is given by \(A(x)=42(2)^{x},\) where \(x\) is years after the industry started. (Source: Business Insider.) (a) Determine analytically when revenue was about \(\$ 400\) million. (b) Solve part (a) graphically. (c) According to this model, when did the mobile advertising revenue reach \(\$ 1\) billion?
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