Week 3: Absolute Values – Day 5

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.

Week 3: Absolute Values – Day 4

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.

Week 3: Absolute Values – Day 3

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|

Week 3: Absolute Values – Day 2

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

Week 3: Absolute Values – Day 1

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|

Week 2: Exponents – Day 5

1. If we know that  a100 = 250 , what is a?

2. What is  31/2 x 31/4 ?

3. Which is bigger,  21/2 or 21/3 ?

4. Which is bigger, 2-2 or 21/2 ?

5. Which is bigger, (1/2)-2 or (1/2)1/2 ?

Week 2: Exponents – Day 3

Negative Exponents, Parentheses, Signs and Simplification

“Simplifying” expressions with exponents usually means canceling out all common factors and rewriting them with only positive exponents.

1. Simplify 4 x 2-4.

2. Simplify -23 and 2-3. Are they the same number?

3. Simplify (-3)-2 and -3-2. Are they the same number?

4. Simplify 1 – 2-1

5. Simplify 1/2-3

Week 2: Exponents – Day 2

Give exponential answers to these questions – in other words, your answer can involve numbers  raised to an exponent.

1. On Monday, 10 bacteria live in a petri dish. Each bacterium divides into two bacteria every twenty-four hours. How many bacteria are in the dish the following Saturday?

2. What is the average rate of bacteria population growth in problem 1 between Monday and Tuesday? (The average growth rate, in bacteria per day, is the number of bacteria on Tuesday minus the number of bacteria on Monday divided by the number of days between Tuesday and Monday.)

3. What is the average rate of bacteria population growth in problem 1 between Friday and Saturday?

4. What is the average rate of bacterial population growth in problem 1 between Tuesday and Friday? (Think about what answer you expect to this problem. Do you expect the same answer as problem 2 or problem 3, or something different?)

5. How many bacteria are in the petri dish one month later (thirty days later)?

Bonus question: since 210 is approximately 1000, approximately how many bacteria are living in the dish after one month – a thousand? million? billion? trillion?

Week 2: Exponents – Day 1

1. If we know that  2100 x 2n = 2300   what is n?

2. If we know that  210 / 2n = 8, what is n?

3. If we know that  22n x 23 = 211 what is n?

4. If we know that 220 = 4n what is n?

5. If we know that 913 = 3n what is n?