Chapter 3.4, Problem 45E

Solution

To match: The function with its graph.

  $Xscl=1;Yscl=1$

Window: $[-2,8]$

  $[-3,3]$

Given information: function $f(x)={\log }_{0.5}(x-2)$

Graph:

  

Calculation: As $x$ has a negative coefficient, this graph has been horizontally reflected.

As the base of the logarithm is $0.5M1$ , the graph has not been vertically reflected.

The only graph that matches this behaviour is graph (d).

To find the window dimensions, locate the intercepts of the graph.

The $x$ -intercept is found as:

  $\begin{array}{c}lo{g}_{0.5}(x-2)=0\\ x-2={0.5}^0\\ x-2=1\\ x=3\end{array}$

And therefore each tick in the $y$ -axis is equal to 1 unit, and so $Xscl=1$ and the horizontal dimensions are $[-2,8]$ .

The graph does not intercept the $y$ -axis, so we use another point. Note that at the right edge of the graph, $x=8$ , we have that.

  $\begin{array}{c}y={\log }_{0.5}6\\ =\frac{\log 6}{\log 0.5}\\ =2.585\end{array}$

Then each tick in the $y$ -axis is equal to 1 unit, and therefore $Yscl=1$ and the vertical dimensions are $[-3,3]$ .