# Week 3: Absolute Values – Answers

Week 3: Absolute Values – Day 5

1. f(x) = |x| + 1

Minimum occurs at x = 0. The minimum value of f(x) is f(0) = 1.

Note that in general, the minimum of an absolute value is zero. But the function f(x) is an absolute value plus a constant.

2. f(x) = |x – 1|

Minimum value is f(x) = 0. This occurs at x = 1.

3. f(x) = 1 – |x|

This has a maximum value of 1 at x = 0.

4. f(x) = 1 – |2x|

This has a maximum value of 1 at x = 0. (Graph it and see.)

5. f(x) = |x| + |x – 1|

This has a minimum value of 1. This occurs for x in the range 0 < x < 1.

Bonus question: graph the function in #5. See the graph at http://desmos.com

Week 3: Absolute Values – Day 4

Solve for intersection points.

1. f(x) = |x| and g(x) = 1 – |x|

f(x) is an upward “V” with slopes 1 and -1. g(x) is a downward “V” shifted up one unit. There will be two intersection points.

One point: x = 1 – x which corresponds to 2x = 1 or x = 1/2

Intersection point is (1/2, 1/2)

Second point: -x = 1 + x which corresponds to -2x = 1 or x = -1/2

Intersection point is (-1/2, 1/2)

(The technique used here is the piecewise definition of f(x) and g(x)).

Piecewise definition of f(x): f(x) = x if x is >= 0, -x if x < 0.

Piecewise definition of g(x): g(x) = 1 – x if x >= 0, 1 + x if x < 0.

2. f(x) = |x| and g(x) = |x – 3|

f(x) is the standard upward “V” with slope 1 and -1. g(x) is the upward “V” shifted 3 units to the right. The expectation is for only one intersection point, where the slope 1 part of f intersects the slope -1 part of g. (Sketch the graph to see this.)

Solving: x = -(x – 3) = -x + 3 which means 2x = 3, or x = 3/2.

Intersection point is (3/2, 3/2)

3. f(x) = |x|  and g(x) = |2x| – 4

f is the standard upward “V”. g(x) is an upward “V” with slopes 2 and -2, shifted down 4 units.

Expectation: two intersection points. Note: it is tempting to say there are 4 intersection points, but check the graphs and see that 4 points are impossible.

point one:  – x = -2x -4 or x = -4 therefore point is (-4, 4)

point two: x = 2x – 4 or -x = -4 which means x = 4 therefore point is (4, 4)

4. f(x) = |x| and g(x) = |x – 3|/2

This time, one branch of g(x) intersects two branches of f(x). The left branch of g(x) which has slope -1/2 will intersect both parts of f(x); the right branch of g(x) which has slope 1/2 will not intersect f(x). Sketch the graph and see.

first point (right branch of f): x = -(x – 3)/2 which means 2x = -x + 3 or 3x = 3 or x = 1

Intersection point is (1, 1)

second point (left branch of f): -x = -(x – 3)/2 which means -2x = -x + 3 which means -x = 3 or x = -3.

Intersection point is (-3, 3).

5. f(x) = |x + 1| and g(x) = |x – 1|

Two upward “V”s, one shifted left one unit, one shifted right one unit. Expect one intersection point: between the right branch of f and the left branch of g. No others. (Sketch graphs and see).

right branch of f = x + 1 = -(x – 1) = left branch of g

x + 1 = – x + 1 which means 2x = 0 or x = 0.

Intersection point is (0, 1).

Week 3: Absolute Values – Day 3

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| Week 3: Absolute Values – Day 2 Week 3: Absolute Values – Day 1

Interpret the following absolute value expressions as distances.

1. |x – 10|

The distance of x from 10.

2. |x + 4|

The distance of x from -4.

3. |6x|

|6x| = 6|x|

This is 6 times the distance of x from zero.

4. |2 – x|

|2 – x| = |x – 2|. This is the distance of x from 2.

(It’s also the distance of 2 from x, which is the same thing).

5. |2 + x|

The distance of x from -2.