how to calculate this logarithmic function?

Question

Im having trouble in graphing this log function: $y=\log _{1/4}\left|x^2-5x+6\right|$

I found the intervals: $(-\infty, 2)$, $(2,3)$, $(3,\infty)$

Should I just give $x$ values and find $y$ to graph this or is there another way?

Answer 1

Use what you have computed:

On $(-\infty,2)\cup(3,\infty)$ the polynomial $x^2-5x+6$ is positive. So, the function is equal to $$\log_{1/4}(x^2-5x+6)$$

On $(2,3)$ the polynomial $x^2-5x+6$ is negative. So, the function is equal to $$\log_{1/4}((x-2)(3-x))$$

Now, apply to each of these the procedure to determine the main elements of the graph. For example these. Take the portions of those graphs lying on the relevant intervals and put the pieces together.