Example 5 from Algebra and Trigonometry, 6.4 Graphs of Logarithmic Functions

Graphing a Vertical Shift of the Parent Function y = logb(x)

Sketch a graph of $f (x) = \log_{3} (x) - 2$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Since the function is $f (x) = \log_{3} (x) - 2,$ we will notice $d = - 2.$ Thus $d < 0.$

This means we will shift the function $f (x) = \log_{3} (x)$ down 2 units.

The vertical asymptote is $x = 0.$

Consider the three key points from the parent function, $(\frac{1}{3}, −1),$ $(1, 0),$ and $(3, 1) .$
The new coordinates are found by subtracting 2 from the y coordinates.

Label the points $(\frac{1}{3}, −3),$ $(1, −2),$ and $(3, −1) .$
The domain is $(0, \infty),$ the range is $(- \infty, \infty),$ and the vertical asymptote is $x = 0.$

Figure 9

The domain is $(0, \infty),$ the range is $(- \infty, \infty),$ and the vertical asymptote is $x = 0.$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Sketch a graph of $f (x) = \log_{2} (x) + 2$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

How did it go?

Solution

Your turn

Solution

Your turn