Algebra 2 – exponential functions

Q: In 2010, River City had a population of 500. If the population has increased by 6% each year since then, what is the population this year, in 2014?

Answer: 631

Explanation:

Alternatively,

Calcululate 6% of 500 by doing .06 * 500 = 30.

Increase 500 by 30, and you find that the population is 530 in 2011.

Now, increase 530 by 6% (530*.06, added to 530, or shortcut is to do 530*1.06).

The population is 561.8 in 2012. Don’t round this number yet – not until you get to the end of the problem.

Doing this calculation 2 more times gives you 595.508 for 2013 and 631.23… for 2014.

Now, round to the nearest whole number: 631. It’s not the most efficient way, but you get the right answer.

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Precalculus – Probability

Q: In a group of 10 randomly chosen people, what is the probability that at least two people share the same birthday? (same month and day)

Answer: 12% 

Explanation: (For simplicity, this explanation doesn’t take into account people with leap-day birthdays, but it wouldn’t change the answer by much!)

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Precalculus – logarithms

Q: Solve the following equation involving logarithms:

Answer: x=4

Explanation:

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Algebra 1 – Find the equation of a line

Q: If the slope of the line through (-3,6) and (1,y) is 1/2, what is the value of y?

Answer:  8

Explanation: Slope is computed by the formula (y2 – y1) / (x2 – x1) … the change in y divided by the change in x.

So,

(y – 6) / (1 – -3) = 1 / 2

(y – 6) / 4 = 1 / 2

Multiply both sides by 4 and you have:

y – 6 = 2

y = 8

BUT, if you’re taking a test and having trouble remembering the method above, try this instead: Plot the point (-3,6) on a graph and move from that point with a slope of 1/2. Move up 1, right 2, up1, right 2… and repeat this pattern until your x-value is 1. Then, read the y-value at the same point. This is not my favorite way, but it works!

And one more thing!  If this were a multiple choice question, you could use my TI-84 Plus program ‘DISTANCE’ (which finds distance, midpoint, and slope for any two ordered pairs).  Substitute one of the answer choices for x, and see if the program reports that slope is 1/2.

 

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Algebra 2 – proportions

Q:  If x is inversely proportional to the square of y, and x = 4 when y = 1, find x when y = 2.

Answer:  1

Explanation:  We are told that x and y² are inversely proportional, so x • y² = k, where k is a constant.

Let’s find out what k is, using the given info:

4 • 1² = k

k = 4

Now,  x • y² = 4.

Plug in the new value for y, to find the new value for x:

x • y² = 4

x • 2² = 4

x • 4 = 4

x = 1

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Algebra 1, 2 – systems of equations

Q: Red pens cost twice as much as blue pens. Two blue pens and three red pens cost a total of $2.32. How much does a blue pen cost?

 Answer:  $0.29

 Explanation:  Let r be the cost of a red pen, and let b be the cost of a blue pen. 

Then, r = 2b and 2b + 3r = 2.32. Solve this system of equations. 

By substitution,

r + 3r = 2.32

4r = 2.32

r = 0.58

Plug r= 0.58 into the first equation (or second, but I like the first better):

0.58 = 2b

0.29 = b

 You can also use other algebraic methods such as elimination, Cramer’s Rule, etc.  Or use my TI-84 Plus program CRAMER once you rearrange the system to say: 2b – r = 0 and 2b + 3r = 2.32.

 

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Algebra 2 – linear functions

Q:  f(x) is a linear function with a y-intercept of 6. If f(-2) = 9, what is f(2)?

Answer:   3

Click ‘Like’ if you got it right!

Explanation:

f(x) = mx + b              (because it is linear)

m is the slope of the line passing through (0,6) and (-2,9).

m = (9-6) / (-2-0)

m = – 3/2

b is the y-intercept.  b = 6.

So, f(x) = – 3/2 x + 6

Now, find f(2):

f(2) = (- 3/2)(2) + 6 = 3

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Algebra 2 – Sequences

Q: What is the tenth term of the sequence: 4, 12, 36, 108, …?

Answer:  78732

Explanation:  It’s a geometric sequence because each term is multiplied by 3 to get the next term. The common ratio (r) is 3. You *could* just keep multiplying by 3 until you get to the tenth term. It’s not the most efficient way, but it works.

The ‘best’ way, if you know it, is to use the formula for the nth term of a geometric sequence:

2014-09-06_202336

Oh, and my TI-84 Plus program ‘NTHTERM’ will find this answer for you.

 

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Algebra 1, 2 – solving quadratic equations

Q: The square of a positive number is equal to 5 more than the product of 4 and the same number. What is the number?

Answer: 5

Explanation:

2014-09-04_231259

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Algebra 1, 2 – functions

Q: Which of the following represents a function?  (Choose all correct answers.)
 
a. {(4,2), (6,2), (-1,5), (3,-2), (-4,-7)}
b. {(4,2), (5,1), (-1,5), (5,-2), (4,-2)}
c. {(-1,-1), (1,1), (4,4), (3,3), (6,6)}
d. {(1,2), (3,2), (5,2), (7,2), (9,2)}
 
Answer: a, c, and d
Explanation:  A function must have only one y-value for each x-value.  Choice b has two points with the same x:  (4,2) and (4, -2) (and also two others).  All of the other choices have no repeated values for x, so they are functions.
 
Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.