1. Section 2.6 Day 2: Special Functions
EQ: Can we interpret an absolute value function based on the vertex? Can we graph an absolute value function?

2. Absolute Value Function

3. Shifts of an absolute value graph
Shifts of an absolute value graph. General equation: y = a|x – h| + k Vertex: (h, k)
y = a|x – h| + k
If a is negative, it opens down.
If a is positive, it opens up.
If 𝑎>1 it’s stretched vertically. (narrower)
If 0<𝑎<1 it’s compressed vertically. (wider)
(+) means left
(−) means right
(+) means up
(−) means down

4. Example 1: Graph y = |x| + 1. Identify the domain and range.
Step 1: Find the vertex  (h, k); (0, 1)
Step 2: Plug in values smaller and greater than the vertex to find other points
Step 3: Graph the function. x y -2 3 -1 2 1
Domain: All real numbers
Range: 𝑦≥1

5. Example 2: Graph g(x) = |–2x| + 6. Identify the domain and range.
Step 1: Find the vertex  (h, k); (0, 6)
Step 2: Plug in values smaller and greater than the vertex to find other points
Step 3: Graph the function. x y -2 10 -1 8 6 1 2
Domain: All real numbers
Range: 𝑦≥6

6. You Try! Graph y = |-3x| + 3. Identify the domain and range.
Do on your own!

7. *Step 1: Graph the absolute value equation.
Graphing Absolute Value Inequalities (Section 2.8)
*Step 1: Graph the absolute value equation.
*Step 2: Determine whether the boundary line is dashed
(when the symbols < or > are used) or solid (when the symbols ≤ or ≥ are used).
*Step 3: Determine which region should be shaded.
(< and ≤, shade below the vertex, > and ≥, shade above the vertex.)

8. Example 9: Graph y ≥ |x| – 2. Solid line, above the graph x y -2 -1 1-1 1 2

9. You Try! Graph y < |x| + 3
Do on your own!