Maxima and Minima
For each of these absolute value functions: is there an absolute maximum, minimum, or neither? If there is a minimum or maximum, specify the x value where it is located.
1. f(x) = |x| + 1
2. f(x) = |x – 1|
3. f(x) = 1 – |x|
4. f(x) = 1 – |2x|
5. f(x) = |x| + |x – 1|
Bonus question: graph the function in #5.
Intersections of Absolute Value Functions
Two absolute value functions are given. Find their intersection points. In every case, it helps to quickly sketch the graph to know what to expect.
1. f(x) = |x| and g(x) = 1 – |x|
2. f(x) = |x| and g(x) = |x – 3|
3. f(x) = |x| and g(x) = |2x| – 4
4. f(x) = |x| and g(x) = |x – 3|/2
5. f(x) = |x + 1| and g(x) = |x – 1|
|/4
5.
The Absolute Value Function
Sketch the graphs of the following absolute value functions.
1. y = f(x) = |x| + 1
2. y = g(x) = |x + 1|
3. y = h(x) = 1 – |x|
4. y = p(x) = – |x – 2|
5. y = q(x) = 3|x|
Solving Inequalities
Solve and graph each inequality on the number line.
1. |x – 2| < 1.
2. |x + 3| > 6
3. |2x| =< 8 (“=<” means “less than or equal to”)
4. 5 – |x – 2| >= 1 (“>=” means “greater than or equal to”)
5. 3|2x + 1| < 9
As Distances
The absolute value of x can be interpreted as its distance from zero. Similarly, |x – 5| is the distance of the number x from 5. Interpret the following absolute value expressions as distances.
1. |x – 10|
2. |x + 4|
3. |6x|
4. |2 – x|
5. |2 + x|