Math Problem Statement

he graph of [y=\log_3x] 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=4\log_3(x+1)]? 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}}]

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}}]

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}}]

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}}]

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}}] Related content

Video5 minutes 27 seconds5:27 Graphing logarithmic functions (example 1)

Video3 minutes 46 seconds3:46 Graphing logarithmic functions (example 2)

Solution

The given equation is y=4log⁡3(x+1)y = 4 \log_3 (x + 1), which is a transformation of the basic logarithmic function y=log⁡3xy = \log_3 x. To determine the graph of this transformed function, let's break down the transformation step by step:

  1. Translation: The expression log⁡3(x+1)\log_3 (x + 1) represents a horizontal shift of the basic graph y=log⁡3xy = \log_3 x. The +1+1 inside the logarithm shifts the graph 1 unit to the left. So, the new graph will be shifted one unit left from the original.

  2. Vertical Stretch: The factor of 4 in front of the logarithmic function means that the graph is vertically stretched by a factor of 4. This affects the steepness of the graph, making it steeper.

Thus, the graph of y=4log⁡3(x+1)y = 4 \log_3 (x + 1) is the original graph of y=log⁡3xy = \log_3 x, shifted 1 unit to the left and vertically stretched by a factor of 4.

Now, we can evaluate which graph corresponds to these transformations. Look for a graph that:

  • Has been shifted 1 unit to the left.
  • Has a steeper slope than the original y=log⁡3xy = \log_3 x.

After analyzing the given choices, Choice C is the graph that fits these criteria.

So, the correct answer is (Choice C).

Would you like more details on how these transformations work or have other related questions? Here are some follow-up questions for further exploration:

  1. How does the graph of y=log⁡3(x−1)y = \log_3 (x - 1) compare to y=log⁡3(x+1)y = \log_3 (x + 1)?
  2. What happens to the graph of y=log⁡3xy = \log_3 x if the base of the logarithm changes?
  3. How can you determine the domain of a logarithmic function from its graph?
  4. How would the graph of y=2log⁡3xy = 2 \log_3 x look in comparison to y=log⁡3xy = \log_3 x?
  5. What is the effect of a negative coefficient in front of the logarithmic function?

Tip: When graphing logarithmic functions, remember that transformations like vertical stretches and horizontal shifts follow the same principles as those for other types of functions!

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Math Problem Analysis

Mathematical Concepts

Logarithmic Functions
Transformations of Graphs
Horizontal Shifts
Vertical Stretches

Formulas

y = log_b(x)
y = a * log_b(x + h)

Theorems

Transformation of logarithmic functions

Suitable Grade Level

Grades 9-12

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