Question

Graph the function. \(y=\log _3(x+2)\)

Short answer

The graph of \(y=\log_3(x+2)\) is a logarithmic curve crossing the point (-1,0) with a vertical asymptote at \(x = -2\). It increases as \(x\) increases but at a decreasing rate.

Step-by-step solution

Understand the Basic Logarithm Function

Firstly, recognize what the graph of \(y=\log_3(x)\) looks like: It passes through the point (1, 0), approaches but never crosses the y-axis given that it's a vertical asymptote, and it increases as \(x\) does but at a decreasing rate.

Adjust for the Horizontal Shift

All the \(x\)'s in the original function have been replaced with \((x+2)\)'s, meaning our graph will shift 2 units left. Therefore, instead of passing through (1,0), it will now pass through (-1, 0) and the vertical asymptote becomes \(x = -2.\)

Graph the Function

Plot the asymptote on the graph as the line \(x = -2\). Mark the point (-1, 0) and draw a curve following the same pattern as the base function \(y=\log_3(x)\), making sure it approaches but doesn't cross the asymptote.

Key concepts

Graphing Transformations

When working with logarithmic functions, understanding graphing transformations is key to mastering their visualization. A basic logarithmic function such as \( y = \log_3(x) \) transforms through various operations which affect its orientation and position on a coordinate plane. Transformations can compress, stretch, or reflect the graph. In general, within a function \( y = \log_b(x) \), adjusting the base \( b \) leads to steepness changes, whereas the inclusion of constants and modifications to \( x \) result in movements such as shifts or rotations. The core concept here is that every transformation maintains the function's inherent properties like its shape and vertical asymptotes, but changes its location or orientation in specific ways.

Horizontal Shifts

Horizontal shifts come into play when we modify the input of the logarithmic function. Essentially, any change in the form of \((x + h)\) within the function \( y = \log_b(x) \) results in a horizontal shift. If \( h > 0 \), the graph moves to the left by \( h \) units. Conversely, if \( h < 0 \), it shifts right. For the function \( y = \log_3(x + 2) \), this means we move the graph 2 units to the left. Consequently, the logarithm's intercept point on the x-axis shifts from (1, 0) to (-1, 0). This is critical: The function's shape remains unchanged, yet its position is adjusted horizontally, affecting which points the graph passes through.

Vertical Asymptotes

Vertical asymptotes are essential aspects of understanding logarithmic functions. They are imaginary lines the graph will approach but never cross. For logarithmic functions expressed as \( y = \log_b(x + h) \), the vertical asymptote is positioned at \( x = -h \). It represents a boundary where the logarithmic function does not exist. In our example, \( y = \log_3(x + 2) \), the vertical asymptote moves to \( x = -2 \) due to the horizontal shift. Asymptotes are crucial for recognizing limits of the function: as \( x \) approaches this line, the function approaches negative infinity. When graphing, draw a dotted line at this value to remind yourself of this invisible roadblock.