Question

Explain how the graph of \(\frac{1}{3}(y+2)=\log _{6}(x-4)\) can be generated by transforming the graph of \(y=\log _{6} x\).

Short answer

Shift right by 4, stretch vertically by 3, and shift down by 2.

Step-by-step solution

Understand the Base Graph

Start with the base graph of the function and understand its shape: \(y = \log_6{x}\). The graph of this function is a logarithmic curve that passes through the point (1, 0) and has a vertical asymptote at \(x=0\).

Horizontal Shift

To generate our target graph, apply a horizontal shift. The equation \(\log_6{(x-4)}\) represents a shift of 4 units to the right. This means every point \((x,y)\) on the base graph shifts to \((x+4, y)\).

Vertically Stretch and Translate

Next, solve the given equation for \(y\) to see the transformations more clearly: \(\frac{1}{3}(y+2) = \log_6{(x-4)}\). Multiply both sides by 3: \(y + 2 = 3\log_6{(x-4)}\). Now subtract 2: \(y = 3\log_6{(x-4)} - 2\). This indicates a vertical stretch by a factor of 3 followed by a vertical shift downward by 2 units.

Combine Transformations

Combine the shifts and stretches to sketch the final graph. Starting from \(y = \log_6{x}\), first shift the graph 4 units right to get \(y = \log_6{(x-4)}\). Then stretch the graph vertically by a factor of 3 to get \(y = 3\log_6{(x-4)}\). Finally, move the graph down by 2 units to achieve the desired function: \(y = 3\log_6{(x-4)} - 2\).

Key concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. If you think about the exponential function \(y = a^x\), the logarithmic function is \(y = \log_a{x}\). Here, 'a' is the base of the logarithm. For example, \(y = \log_6{x}\) means that 'a' is 6.
In a logarithmic function, like \(y = \log_6{x}\), the graph includes a vertical asymptote at \(x = 0\) and passes through the point \( (1, 0) \). The shape of the graph shows a gradual increase. 
It's important to remember that in the context of logarithms:

• The domain (all possible x-values) is \(0 < x < \infty\).
• The range (all possible y-values) is \(-\infty < y < \infty\).

Understanding these properties lays the foundation for handling more advanced transformations.

Graph Transformations

Graph transformations help us understand how the basic graph of a function changes when you modify the function's equation. The main types of transformations include shifts, stretches, and reflections.
For example, starting from the base graph of the logarithmic function \(y = \log_6{x}\), modifying x or y outside or inside the logarithm leads to different transformations.
When considering \(3\log_6{(x-4)} - 2 \), you see multiple transformations.

• The \( (x-4) \) causes a horizontal shift.
• The \( 3 \) in front of the logarithm represents a vertical stretch.
• The \(-2\) outside the logarithm indicates a vertical shift.

By breaking down and combining these transformations, you can accurately plot complex functions by transforming simpler ones.

Vertical and Horizontal Shifts

Vertical and horizontal shifts are common transformations that affect the position of a graph.
Horizontal shifts occur when you add or subtract a constant inside the function. For instance, swapping \(x \) with \( (x - 4) \) moves the graph to the right by 4 units.
This means if you have the base logarithmic graph \(y = \log_6{x}\), converting it to \(y = \log_6{(x-4)}\) results in a rightward shift.
Vertical shifts involve adding or subtracting a constant outside the function. For instance, with \(y = 3 \log_6{(x-4)} - 2\), the \(-2\) creates a downward shift by 2 units.
In essence, every point on the graph shifts straight down 2 units.
Combining these transforms accurately helps solve the given problem, leading to the final graph of the function.