Saturday, April 04, 2020

Algebra 2 Problems of the Day (Jan 2020)

Daily Algebra 2 questions and answers.

More Algebra 2 problems.

January 2020, Part I

All Questions in Part II are worth 2 credits. Work must be shown. Partial credit is given.


31. Biologists are studying a new bacterium. They create a culture with 100 of the bacteria and anticipate that the number of bacteria will double every 30 hours. Write an equation for the number of bacteria, B, in terms of the number of hours, t, since the experiment began

Answer:
B(t) = 100(2)t/30

Start with 100 bacteria. It's doubling, so the base is 2. The time is measured in hours, but it doubles every 30 hours, so the exponent is t/30.





32. Graph y = x3 - 4x2 + 2x + 7 on the set of axes below.

Answer:
Put the equation in the calculator and check the table of values.
You'll see (-1, 0), (0, 7), (1, 6), (2, 3), (3, 4), (4, 15)
Also, you'll see (-2, 21), way below the screen, and (5, 42), which is way above.
Use this data to make the graph.
Also note that (2, 3) is not a minimum point, so when you sketch the curve, don't make it a turning point. Go through the point and curve after it.

Given the values, I used a scale of 1 on the x-axis and a scale of 2 on the y-axis. Label your scales.





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Friday, April 03, 2020

School Life #15

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(C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Odd is my choosing to switch back to black & white for no particular reason. It didn't make anything easier, which is why I went back to color before this.

These past episodes not based on any specific incident, but sometimes I hear the drama of middle school. Sometimes high school, too, but those students are usually aware that I'm still in the room. Although, sometimes, they just don't care that I'm in the room.

I also planned to have Daisy standing off the side, but there just wasn't any room for that. (x, why?) is mostly free format, but there are a few things that I've seen to lock in specific dimensions and a "look".




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Algebra 2 Problems of the Day (Jan 2020)

Daily Algebra 2 questions and answers.

More Algebra 2 problems.

January 2020, Part I

All Questions in Part II are worth 2 credits. Work must be shown. Partial credit is given.


29. Given the geometric series 300 + 360 + 432 + 518.4 + ..., write a geometric series formula, Sn, for the sum of the first n terms. Use the formula to find the sum of the first 10 terms, to the nearest tenth.

Answer:
Find the formula for Geometric Series in the back of the booklet:
Sn = (a1 - a1rn) / (1 - r), where r is the Common Ratio of the series.
360 / 300 = 1.2
432 / 360 = 1.2, etc.
r = 1.2

Sn = (300 - 300(1.2)10) / (1 - 1.2) = 7787.6...
S10 = 7787.6





30. Visible light can be represented by sinusoidal waves. Three visible light waves are shown in the graph below. The midline of each wave is labeled L.


Based on the graph, which light wave has the longest period? Justify your answer.

Answer:
Check the length between maximum and minimum.
A has a max at 60 and a min at 340. 340 - 60 = 280
B has a max at 180 and a min at 400. 400 - 180 = 220
C has a max at 380 and a min at 60. 380 - 60 = 320.
Light wave C has the longest period.





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Thursday, April 02, 2020

School Life #14

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(C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

On the bright side, everything is awkward at a certain awkward age.

What my old story lines were and where they were going? I was still deciding and working on plotting. Circumstances change, and, as usual, something that starts as an easy little side project starts to become more involved and complicated.

On the other hand, if I wanted to move forward with the new circumstances of the day, I thought I should wrap up old stuff before moving forward.

Like everything else, we'll see what tomorrow brings. Or doesn't.




Come back often for more funny math and geeky comics.




Algebra 2 Problems of the Day (Jan 2020)

Daily Algebra 2 questions and answers.

More Algebra 2 problems.

January 2020, Part I

All Questions in Part II are worth 2 credits. Work must be shown. Partial credit is given.


27. Describe the transformation applied to the graph of p(x) = 2x that form the new function q(x) = 2x - 3 + 4

Answer:
The graph would shift 3 units to the right and 4 units up.

More info:
The + 4 is a vertical move. Whatever calculations are done involving x, they have been completed by the time the 4 is added, so the shift is vertical.

Changing 2x to 2x - 3 means that the new function will have lower exponent values. It won't rise to the same level until the input is 3 higher. That is, 3 to the right, not the left.

You can graph the two functions in your calculator to see the difference. You can sketch them on the paper, if you want, but it isn't necessary. You just need to describe the transformation.





28. The parabola y = - 1/20(x - 3)2 + 6 has its focus at (3,1). Determine and state the equation of the directrix.
(The use of the grid below is optional.)

Answer:
The parabola has its vertex at (3, 6), which we know because it's written in vertex mode.
The vertex is the midpoint between of a line between focus and directrix.
6 - 1 = 5
6 + 5 = 11
The equation for the directrix is y = 11.





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Wednesday, April 01, 2020

No Fooling

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(C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

I've never done that thing were webcomics switch. So I just left the characters do it. Sort of.

In the early years, I really didn't know too many other comic writers. And the ones I knew, I couldn't really match their style. Plus, I would always forget until the end of March was practically here, and it would be too late.

Now, I'm not sure that comics still do this.




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Tuesday, March 31, 2020

Understood

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(C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Even numbers have problems. Maybe she was misunderstood?

Pretty explicit, actually.




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Algebra 2 Problems of the Day (Jan 2020)

Daily Algebra 2 questions and answers.

More Algebra 2 problems.

January 2020, Part I

All Questions in Part II are worth 2 credits. Work must be shown. Partial credit is given.


25. For n and p > 0, is the expression (p2n1/2)8SQRT(p5n4) equivalent to p18n6SQRT(p)? Justify your answer.

Answer:
Multiply the exponents in the parentheses of the first part of the first expression by 8. Multiply the exponents in the second part by 1/2. Then add the exponents of each matching variable.
(p16n4)(p5/2n2)
= p18 1/2n6
= p18n6SQRT(p)

Yes, they are equivalent.

Note: You must include the concluding statement to get full credit. However, the concluding statement, by itself, with no work, is not worth any points.





26. Show why x - 3 is a factor of m(x) = x3 - x2 - 5x - 3. Justify your answer.

Answer:
There are several methods.
You can show that x - 3 is a factor m(x) by calculating m(3), which will give you 0 if it is a factor.
m(3) = (3)3 - (3)2 - 5(3) - 3 = 27 - 9 - 15 - 3 = 0
Since 3 is a zero of the function, x - 3 is a factor.

You can also show this by factoring or dividing the function. You can use, for example, long division or the grid method.

Look at the image below.
Put x and -3 on the left and x3 in the first box.
Divide x into x3 and you get x2. Write that on top. Now multiply x2 by -3, which has a product of -3x3. Write that in the bottom of the first column.

We have -3x2 but the original expression has -x2, so we have to add 2x2 in the top box of the second column. Repeat the division and multiplication.

When you get to the last box, you have -3, which is what we need. There is no remainder.

(x - 3) is a factor of m(x) because it divides evenly with no remainder.





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Friday, March 27, 2020

Remote Learning III: Secant-Secant

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(C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

What do Math teachers have? Well, for one, T-shirts like that one!

Guesses to what my actual shirt says can be left in the comments.

So, if I reteach the same material year after year, I'm allowed to re-use the same puns, right? It's all new to them.

Since I said that I would explain on my blog, I guess I need to do that here.

As we see in Panel 2, above, we are Given Secant AB with point D on the circle, and AC with point E on the circle.
We want to prove that the products of these lengths are equal: (AB)(AD) = (AC)(AE)

If we draw chords CD and BE, we create triangles ABE and ACD, as shown in Panel 3.
Angles B and C both intercept the same arc, DE, and therefore they are congruent.
Angle A is congruent to itself because of the Reflexive Property.
Therefore, triangles ABE and ACD are similar.
If they are similar, then their corresponding sides are proportional.

So AB / AE = AC / AD
If we cross-multiply, we get: (AB)(AD) = (AC)(AE)

Or, in other words, "The whole line times the other part equals the other whole line times its outside part".

Actually, not very long, and could easily be included in an actual remote video, but not so much in a four- or six-panel comic page.




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Thursday, March 26, 2020

Algebra 2 Problems of the Day (Jan 2020)

Daily Algebra 2 questions and answers.

More Algebra 2 problems.

January 2020, Part I

All Questions in Part I are worth 2 credits. No work need be shown. No partial credit.


21. Which value, to the nearest tenth, is the smallest solution of f(x) = g(x) if f(x) = 3sin( 1/2 x) - 1 and g(x) = x3 - 2x + 1?

(1) -3.6
(2) -2.1
(3) -1.8
(4) 1.4

Answer: (2) -2.1
Put both equations in your graphing calculator. You should be able to see the results clearly enough to pick the correct choice. However, if you do not, you can use the Calculator functions. (2nd TRACE, option 5, on TI-83 and beyond)





22. Expressed in simplest a + bi form, (7 - 3i) + (x - 2i)2 - (4i + 2x2) is

(1) (3 - x2) - (4x + 7)i
(2) (3 + 3x2) - (4x + 7)i
(3) (3 - x2) - 7i
(4) (3 + 3x2) - 7i

Answer: (1) (3 - x2) - (4x + 7)i

(7 - 3i) + (x - 2i)2 - (4i + 2x2)
(7 - 3i) + (x2 - 4xi + 4i2) - (4i + 2x2)
x2 - 2x2 + 7 - 4 - 3i - 4i - 4xi
-x2 + 3 - 7i - 4xi
(3 - x2) - (4x + 7)i





23. Written in simplest form, the fraction (x3 - 9x)/(9 - x2), where x =/= 3, is equivalent to <

(1) -x
(2) x
(3) -x(x + 3) / (3 + x)
(4) x(x - 3) / (3 - x)

Answer: (1) -x
You should immediately realize that neither (3) nor (4) are in simplest term. In (3), the two binomials are the same, which results in 1. In (4), (x - 3)/(3 - x) reduce to -1.


(x3 - 9x)/(9 - x2)
(x)(x2 - 9)/(9 - x2)
(x)(-1)(9 - x2)/(9 - x2)
-x

Solve for y: (4) + y + (11) = 9
y + 15 = 9
y = -6





24. According to a study, 45% of Americans have type O blood. If a random number generator produces three-digit values from 000 to 999, which values would represent those having type O blood?

(1) between 000 and 045, inclusive
(2) between 000 and 444, inclusive
(3) between 000 and 449, inclusive
(4) between 000 and 450, inclusive

Answer: (3) between 000 and 449, inclusive
45% of 1,000 numbers is 450
Including 000 means that the 450th number would be 449.





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Tuesday, March 24, 2020

Remote Learning II: Two Chords

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(C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

Or you can fast-forward through the proof-y parts and just learn the rule and answer the questions.




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Monday, March 23, 2020

Algebra 2 Problems of the Day (Jan 2020)

Daily Algebra 2 questions and answers.

More Algebra 2 problems.

January 2020, Part I

All Questions in Part I are worth 2 credits. No work need be shown. No partial credit.


16. As θ increases from -π/2 to 0 radians, the value of cos θ will

(1) decrease from 1 to 0
(2) decrease from 0 to -1
(3) increase from -1 to 0
(4) increase from 0 to 1

Answer: (4) increase from 0 to 1
Cos(-π/2) = 0, Cos(0) = 1





17. Consider the following patterns:


I. 16, -12, 9, -6.75, …
II. 1, 4, 9, 16, …
III. 6, 18, 30, 42, …
IV. 1/2, 2/3, 3/4, 4/5, …

Which pattern is geometric?

(1) I
(2) II
(3) III
(4) IV

Answer: (1) I
The terms in pattern I have a common ratio: -12/16 = -3/4; 9/-12 = -3/4
Pattern II is quadratic. The terms are squares.
Pattern III is arithmetic. Each term is 12 more than the prior term.
Pattern IV is none of these. It has neither a common ratio or common difference.





18. Consider the system below.


x + y + z = 9
x - y - z = -1
x - y + z 21

Which value is not in the solution, (x,y,z), of the system?

(1) -8
(2) -6
(3) 11
(4) 4

Answer: (1) -8
Combine the first two equations:


x + y + z = 9
x - y - z = -1
2x = 8
x = 4

Combine the second and third equations:


x - y - z = -1
x - y + z 21
-2z = -22
z = 11

Solve for y: (4) + y + (11) = 9
y + 15 = 9
y = -6





19. Which statement regarding polynomials and their zeros is true?

(1) f(x) = (x2 - 1)(x + a) has zeros of 1 and -a, only
(2) f(x) = x3 - ax2 + 16x - 16a has zeroes of 4 and a, only.
(3) f(x) = (x2 + 25)(x + a) has zeros of +5 and -a.
(4) f(x) = x3 - ax2 - 9x + 9a has zeros at +3 and a.

Answer: (4) f(x) = x3 - ax2 - 9x + 9a has zeros at +3 and a.
Choice (1) has factors (x + 1)(x - 1) and a zero at -1.
In choice (2), f(4) = 64 - 16a + 64 - 16a = 128 - 32a, not zero.
In choice (3), x2 + 25 cannot be factored in (x + 5)(x - 5), so they are not zeros.
Choice (4) can be factored as follows:
f(x) = x3 - ax2 - 9x + 9a
f(x) = x2(x - a) - 9(x - a)
f(x) = (x2 - 9)(x - a)
f(x) = (x + 3)(x - 3)(x - a), x = +3 or a.





20. If a solution of 2(2x - 1) = 5x2 is expressed in simplest a + bi form, the value of b is

Answer: (2) SQRT(6)/5

2(2x - 1) = 5x2
4x - 2 = 5x2
5x2 - 4x + 2 = 0

x = ( -(-4) + SQRT( (-4)2 - 4(5)(2) ) / (2(5))

x = ( 4 + SQRT( 16 - 40 ) ) / (10)

x = ( 4 + SQRT( -24 ) ) / (10)

x = ( 4 + 2i SQRT( 6 ) / (10)

x = 2/5 + i SQRT( 6 ) / 5

The answer is choice (2) because they only wanted the value of b, not the entire term. Choice one includes i.





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