Graphing Absolute Value Functions

To draw the graph of $f(x)=\text{-}3|x-2|+3,$ we will make a table of values. To do so, we will assign random values to the $x\text{-}$variable and calculate the corresponding values of $y.$ Let's do it!

Now, we will plot and connect the obtained points. Do not forget that an absolute value function has a V-shaped graph.

The vertex of an absolute value function is the point at which its graph changes direction. Consider our graph, paying close attention to the coordinates of the vertex.

We see above that the vertex of the graph is the point $(2,3).$

Let's consider our graph again.

Note that the $x\text{-}$variable can take any value. Conversely, the $y\text{-}$variable takes values which are less than or equal to $3.$ With this in mind, we can write the domain and range of the function. $$\begin{array}{rl}\text{Domain:}& \text{all real numbers}\\ \text{Range:}& y\le 3\end{array}$$

The $x\text{-}$intercepts occur at the points at which the graph intersects the $x\text{-}$axis. Similarly, the $y\text{-}$intercept occurs at the point at which the graph intercepts the $y\text{-}$axis. Let's observe these points on the graph.

We see above that the $x\text{-}$intercepts are $1$ and $3,$ and the $y\text{-}$intercept is $\text{-}3.$