Question

a) Sketch the graph of \(y=\log _{3} x,\) and then apply, in order, each of the following transformations.Stretch vertically by a factor of 2 about the \(x\) -axis.Translate 3 units to the left. b) Write the equation of the final transformed image.

Short answer

The final transformed equation is \(y = 2 \log_{3} (x + 3)\).

Step-by-step solution

Title - Sketch the graph of the original function

The given function is \(y = \log_{3} x\). This is a logarithmic function with base 3. The graph will pass through the point (1,0) since \(\log_{3} 1 = 0\) and will grow slowly for positive x. It will approach negative infinity as x approaches 0 from the right.

Title - Apply a vertical stretch by a factor of 2

To stretch the graph vertically by a factor of 2 about the x-axis, multiply the function by 2. The new function becomes \(y = 2 \log_{3} x\). This will make the graph rise twice as fast for any given x-value.

Title - Translate the graph 3 units to the left

Translating the graph 3 units to the left involves replacing x with \(x + 3\). Adjust the equation to obtain the new function: \(y = 2 \log_{3} (x + 3)\). This shifts the entire graph to the left by 3 units.

Key concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They have the form \( y = \log_{b} x \) where \(b\) is the base of the logarithm. In these functions, \( y \) represents the exponent to which the base must be raised to produce \( x \).

Let’s break it down further:

• The notation \( y = \log_{b} x \) literally translates to 'y is the power you need to raise b to get x.'
  If \( b = 3 \), and \( x = 9 \), then \( y = \log_{3} 9 \) would be 2 because 3² = 9.
• Every logarithmic function has a vertical asymptote at x=0, meaning the function approaches but never touches this line.
• As \( x \) increases, the \( y \)-values grow slowly. In our example, the function grows slower than it would if the base were larger, like 10.

Logarithmic functions are incredibly useful, especially in scientific contexts, because they can transform multiplicative relationships into additive ones, making them simpler to analyze and understand.