Certainly! Let's examine the linear equation [tex]\( y = log_3 (x + 2) \)[/tex]. To determine which graph represents this equation, we should remember the properties of logarithmic functions and how they appear on a graph.
1. Basic Shape of Logarithmic Functions:
- Logarithmic functions generally have a shape where they increase slowly when [tex]\( x \)[/tex] is near 0 and then increase more rapidly as [tex]\( x \)[/tex] increases.
- The graph of [tex]\( y = \log_b(x) \)[/tex] (for [tex]\( b > 1 \)[/tex]) will pass through the point [tex]\( (1, 0) \)[/tex] because [tex]\( \log_b(1) = 0 \)[/tex].
- There will be a vertical asymptote at [tex]\( x = 0 \)[/tex] for the basic logarithmic function [tex]\( y = \log_b(x) \)[/tex], but since the function is [tex]\( \log_b(x + 2) \)[/tex], the vertical asymptote shifts left to [tex]\( x = -2 \)[/tex].
2. Transformations Specific to [tex]\( y = \log_3(x + 2) \)[/tex]:
- The shift by +2 inside the logarithm means the graph of [tex]\( y = \log_3(x) \)[/tex] is shifted left by 2 units.
- The vertical asymptote of the graph will be at [tex]\( x = -2 \)[/tex], not at [tex]\( x = 0 \)[/tex].
3. Characteristics to Check in Graphs:
- The graph should pass through the point such as when [tex]\( x = 1 \)[/tex], [tex]\( y = \log_3(1 + 2) = \log_3(3) = 1 \)[/tex].
- The graph will cross the x-axis at [tex]\( x = -1 \)[/tex] because [tex]\( y = \log_3(1) = 0 \)[/tex].
- The graph should have a vertical asymptote at [tex]\( x = -2 \)[/tex].
Let's summarize these points:
- Vertical asymptote at [tex]\( x = -2 \)[/tex].
- Passes through (1, 1) and (-1, 0).
- Shape should be logarithmic: slow increase near vertical asymptote and then steeper increase.
Given these properties, let's examine the provided options to identify the correct graph. We should choose the one that matches these characteristics of the logarithmic function.
However, the provided problem seems to have some issues with the representation of the graphs. Assuming the options labeled A, B, C, and D are the graphs, you'll need to choose the one that visually matches the characteristics described above.
If the actual graphs were visible to you and similar to what I've described, consider those key characteristics and match them to the graph that accurately depicts [tex]\( y = \log_3(x + 2) \)[/tex].
I hope this detailed explanation helps you identify the correct graph! If you have any graph illustrations or further clarifications, feel free to share for an even more precise match.
