Graph of y=log_2(-x)+3: Transformations Explained

Math Problem Statement

The graph of [y=\log_2x] is shown below.

[\small{2}] [\small{4}] [\small{6}] [\small{8}] [\small{\llap{-}4}] [\small{\llap{-}6}] [\small{\llap{-}8}] [\small{2}] [\small{4}] [\small{6}] [\small{8}] [\small{\llap{-}4}] [\small{\llap{-}6}] [\small{\llap{-}8}] [y] [x] Which of the following is the graph of [y=\log_2(-x)+3]? Choose 1 answer: Choose 1 answer: (Choice A) [\small{2}] [\small{4}] [\small{6}] [\small{8}] [\small{\llap{-}4}] [\small{\llap{-}6}] [\small{\llap{-}8}] [\small{2}] [\small{4}] [\small{6}] [\small{8}] [\small{\llap{-}4}] [\small{\llap{-}6}] [\small{\llap{-}8}] [y] [x] [\normalsize{\text{A}}]

[\small{2}] [\small{4}] [\small{6}] [\small{8}] [\small{\llap{-}4}] [\small{\llap{-}6}] [\small{\llap{-}8}] [\small{2}] [\small{4}] [\small{6}] [\small{8}] [\small{\llap{-}4}] [\small{\llap{-}6}] [\small{\llap{-}8}] [y] [x] [\normalsize{\text{A}}] (Choice B) [\small{2}] [\small{4}] [\small{6}] [\small{8}] [\small{\llap{-}4}] [\small{\llap{-}6}] [\small{\llap{-}8}] [\small{2}] [\small{4}] [\small{6}] [\small{8}] [\small{\llap{-}4}] [\small{\llap{-}6}] [\small{\llap{-}8}] [y] [x] [\normalsize{\text{B}}]

[\small{2}] [\small{4}] [\small{6}] [\small{8}] [\small{\llap{-}4}] [\small{\llap{-}6}] [\small{\llap{-}8}] [\small{2}] [\small{4}] [\small{6}] [\small{8}] [\small{\llap{-}4}] [\small{\llap{-}6}] [\small{\llap{-}8}] [y] [x] [\normalsize{\text{B}}] (Choice C) [\small{2}] [\small{4}] [\small{6}] [\small{8}] [\small{\llap{-}4}] [\small{\llap{-}6}] [\small{\llap{-}8}] [\small{2}] [\small{4}] [\small{6}] [\small{8}] [\small{\llap{-}4}] [\small{\llap{-}6}] [\small{\llap{-}8}] [y] [x] [\normalsize{\text{C}}]

[\small{2}] [\small{4}] [\small{6}] [\small{8}] [\small{\llap{-}4}] [\small{\llap{-}6}] [\small{\llap{-}8}] [\small{2}] [\small{4}] [\small{6}] [\small{8}] [\small{\llap{-}4}] [\small{\llap{-}6}] [\small{\llap{-}8}] [y] [x] [\normalsize{\text{C}}] (Choice D) [\small{2}] [\small{4}] [\small{6}] [\small{8}] [\small{\llap{-}4}] [\small{\llap{-}6}] [\small{\llap{-}8}] [\small{2}] [\small{4}] [\small{6}] [\small{8}] [\small{\llap{-}4}] [\small{\llap{-}6}] [\small{\llap{-}8}] [y] [x] [\normalsize{\text{D}}]

Solution

The equation given is $y = \log_2(-x) + 3$ . To understand the transformation, let's break it down:

$y = \log_2(x)$ : This is the standard logarithmic function with base 2. The graph of this function passes through the point (1,0), has a vertical asymptote at $x = 0$ , and increases as $x$ increases.
Transformation for $y = \log_2(-x)$ :
- The negative sign in front of $x$ reflects the graph across the y-axis.
- So, the graph of $y = \log_2(-x)$ will be a mirror image of the graph of $y = \log_2(x)$ reflected over the y-axis.
Transformation for $y = \log_2(-x) + 3$ :
- The $+3$ shifts the entire graph upwards by 3 units.

To summarize:

The graph of $y = \log_2(-x) + 3$ will be a reflection of $y = \log_2(x)$ across the y-axis and shifted upward by 3 units.

Now, we can use these transformations to find the correct graph.

Conclusion:

The graph of $y = \log_2(-x) + 3$ $y = lo g_{2} (- x) + 3$ will show:
- A reflection of the original graph across the y-axis.
- A vertical shift of 3 units upwards.

Given this, the graph that reflects these transformations will be Choice B.

Would you like a further explanation or clarification?

Here are a few related questions:

How does a reflection across the x-axis affect the graph of a logarithmic function?
What happens to the domain of the logarithmic function $y = \log_2(-x) + 3$ ?
How do you shift the graph of $y = \log_2(x)$ to the left or right?
What is the effect of adding a constant outside the logarithmic function?
How would you graph $y = \log_2(x)$ on a graphing calculator?

Tip: Always remember that the domain of $y = \log_b(x)$ is $x > 0$ , but for transformations, the domain can be adjusted accordingly.

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Math Problem Analysis

Mathematical Concepts

Logarithmic Functions
Graph Transformations
Reflections
Vertical Shifts

Formulas

y = log_2(x)
y = log_2(-x)
y = log_2(-x) + 3

Theorems

Reflection of graphs
Vertical shift of graphs

Suitable Grade Level

Grades 9-12

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