Thursday, June 21, 2018

(x, why?) School Life #2

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(C)Copyright 2018, C. Burke.

Also, your fly is open.

The "funny" thing is that I wanted to use "Tell me something I don't know" for something totally different, as a setup for another punchline, later in the series.
I'll get there.




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Complete the Squares and Circles

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(C)Copyright 2018, C. Burke.

Why give an equation in standard form when you can make students work for it, right?

The scene will continue in the next couple of (x, why?) School Life, which are on the way! No, for real, this time.




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June 2018 Common Core Algebra I Regents (mult choice)

The following are some of the multiple questions from the recent January 2018 New York State Common Core Algebra I Regents exam.

June 2018 Algebra I, Part I

Each correct answer is worth up to 2 credits. No partial credit. Work need not be shown.


1. The solution to 4p + 2 < 2(p + 5) is

Answer: (4) p < 4
Distributive property and inverse operations.
4p + 2 < 2(p + 5)
4p + 2 < 2p + 10
2p + 2 < 10
2p < 8
p < 4
There is no multiplication or division by a negative, so there is no need to flip the inequality symbol.


2. If k(x) = 2x2 - 3*sqrt(x), the k(9) is

Answer: (4) 153
Substitution and Order of Operations.
2(9)2 - 3(9)^(.5) = 2(81) - 3(3) = 162 - 9 = 153


3. The expression 3(x2 + 2x - 3) - 4(4x2 - 7x + 5) is equivalent to

Answer: (2) -13x2 + 34x - 29
Distributive property (including distributing a minus sign) and Combining Like Terms.
3(x2 + 2x - 3) - 4(4x2 - 7x + 5)
3x2 + 6x - 9 - 16x2 + 28x - 20
-13x2 + 34x - 29
Once you got -13, you could have eliminated choices 3 and 4. If you didn't get either choice 1 or 2, go back and check your signs.


4. The zeros of the function p(x) = x2 - 2x - 24 are

Answer: (3) -4 and 6
Factoring to find zeros / roots / x-intercepts. What two factors of -24 have a sum of -2?
x2 - 2x - 24 = 0
(x - 6)(x + 4) = 0
x - 6 = 0 or x + 4 = 0
x = 6 or x = -4
I hope you didn't jump the gun after factoring and answer the question without solving for x, which flipped the signs.


5. The box plot below summarizes the data for the average monthly high temperatures in degrees Fahrenheit for Orlando, Florida.

The third quartile is

Answer: (2) 90
Box-and-whisker plots. Five-Number Summary.
The five-number summary for the plot shown are: Minimum is approximately 71. Q1 is 75. Median is approximately 83. Q3 is 90. Maximum is approximately 92.
"Approximately" because those numbers aren't labeled, but they can be inferred from the choices.
Not that the incorrect choices line up with another key portion of the plot.ut solving for x, which flipped the signs.


6. Joy wants to buy strawberries and raspberries to bring to a party. Strawberries cost $1.60 per pound and raspberries cost $1.75 per pound. If she only has $10 to spend on berries, which inequality represents the situation where she buys x pounds of strawberries and y pounds of raspberries?

Answer: (1) 1.60x + 1.75y < 10
Modeling inequalities
If x in the number of pounds of strawberries, which cost $1.60 per pound, then the first term is 1.60x. Eliminate choices 3 and 4.
She can't spend more than 10 dollars but and spend less than or exactly 10 dollars. So choice 1.


7. On the main floor of the Kodak Hall at the Eastman Theater, the number of seats per row increases at a constant rate. Steven counts 31 seats in row 3 and 37 seats in row 6. How many seats are there in row 20?

Answer: (1) 65
Sequences. Rate of Change.
If there are 6 more seats (37 - 31) when you go back 3 rows (6 - 3), then there is a rate of change of 2 seats per row.
If you go back another 14 rows (20 - 6), then there should be an additional 28 seats (14 * 2).
37 + 28 = 65 seats.
If x in the number of pounds of strawberries, which cost $1.60 per pound, then the first term is 1.60x. Eliminate choices 3 and 4.
She can't spend more than 10 dollars but and spend less than or exactly 10 dollars. So choice 1.


8. Which ordered pair below is not a solution to f(x) = x2 - 3x + 4?

Answer: (4) (-1, 6)
Graphing. Substitution. Evaluation.
Quickest way is to put the function into the graphing calculator and check the table of values. You will see that (-1, 8) is a solution, not (-1, 6).
If you change the settings, or use the Trace function, you will see that (1.5, 1.75) is a solution.


9. Students were asked to name their favorite sport from a list of basketball, soccer or tennis. The results are in the table below:
What percentage of the students chose soccer as their favorite sport?

Answer: (1) 39.6%
Statistics. Two-way frequency tables. Marginal frequencies. Percentages.
Find the number of student who prefer soccer. Find the total number of students. Divide the first by the second and multiply by 100%.
58 + 41 = 99 students like soccer
There are 42 + 84 + 58 + 41 + 20 + 5 = 250 total students
99 / 250 = 0.396 = 39.6%


10. The trinomial x2 - 14x + 49 can be expressed as

Answer: (1) (x - 7)2
Factoring. Perfect squares. Completing the squares
Even if you didn't recognize that this trinomial is a perfect square, you could have factored it quickly into (x - 7) and (x - 7), which is (x - 7)2.
Incorrect choices: Choice 2 would give + 14x as the middle term. Choice 3 has two conjugates, so there would be NO middle term. Choice 4 is just silly: -7 times 2 is not +49.


11. A function is definied as {(0,1), (2,3), (5,8), (7,2)}. Isaac is asked to create one more ordered pair for the function. Which ordered pair can be add(ed) to the set to keep it a function?

Answer: (4) (1, 3)
Functions. Relations.
You can't repeat the input (x) with a different output (y). Choices 1, 2, and 3 would cause the function to fail the Vertical-Line Test because they would duplicate x-values that already exist.


12. The quadratic equation x2 - 6x = 12 is rewritten in the form (x + p)2 = q, where q is a constant. What is the value of p?

Answer: (3) -3
Quadratic functions. Parabolas. Minimum value. Vertex.
Two notes: first, "q is a constant" means that it will be some number, but we really don't care what that number will be; second, take note of the fact that there is a plus sign (+) in the rewritten form, not the usual minus sing (-). You don't have to "flip the sign" when reading your answer.
To complete the square, take half of -6, and square it. Add that to both sides.
x2 - 6x = 12
x2 - 6x + 9 = 12 + 9
(x - 3)2 = 21
The constant q is 21, but that isn't important. The value of p is -3.
You can check by graphing that these two equations are equivalent.

Final note: most of the above was unnecessary. Once you found b/2, -6/2 = -3, you had the answer. The rest was just checking.


13. Which of the quadratic functions below has the smallest minimum value?

Answer: (2) [graph]
Quadratic functions. Parabolas. Minimum value. Vertex.
The table in Choice 4 has a minimum of -6, but the graph in Choice 2 has a minimum of -10, so choice 4 is eliminated.
If you graph h(x) and k(x), you will see that neither one has a minimum of less than -10.
You could also have found the axis of symmetry, and plug them into the function.
For choice 1, the axis of symmetry was x = -2 / 2 = -1, and h(-1) = (-1)2 + 2(-1) - 6 = -7
For choice 3, the axis of symmetry is (-5 + -2) / 2 = -3.5, and k(-3.5) = (-3.5 + 5)(-3.5 + 2) = (1.5)(-1.5) = -2.25


14. Which situation is not a linear function?

Answer: (4) A $12,000 car depreciates 15% per year.
Linear functions have a constant rate of change.
Choices 1, 2, and 3 have constant amounts per month, per mile and per hour.
Choice 4 decreases by 15% per year. This is exponential decay. After one year, the value will be smaller, so 15% of that value will be a smaller decrease.


15. The Utica Boilermaker is a 15-kilometer road race. Sara is signed up to run this race and has done the following trains runs:

I. 10 miles
II. 44,800 feet
III. 15,560 yards

Which run(s) are at least 15 kilometers.

Answer: (1) I, only
Unit conversion.
From the back of the test booklet: 1 mile = 5280 feet, 1 mile = 1760 yards, 1 kilometer = 0.62 miles.
15 kilometers * (0.62 miles / kilometer) = 9.3 miles
10 miles > 9.3 miles
44,800 feet / (5,280 feet / mile )= 8.48 miles
15,560 yards / (1,760 yards / mile) = 8.8 miles.
Only 10 miles is at least 15 kilometers.


16. If f(x) = x2 + 2, which interval describes the range of this function?

Answer: (3) [2, infinity)
Domain and range.
Range is the set of possible y-values. The vertex of this function is (0, 2). The range is all values of y greater than or equal to 2, y > 2, or [2, infinity).


17. The amount Mike gets paid weekly can be represented by the expression 2.50a + 290, where a is the number of cell phone accessories he sells that week. What is the constant term in this expression and what does it represent?

Answer: (3) 290, the amount he is guaranteed to be paid each week.
Linear functions.
The initial value (y-intercept, when graphing) is 290, the constant term. The rate of change is 2.50, which repeats for every accessory sold.


18. A cubic function is graphed on the set of axes below.
Which function could represent the graph?

Answer: (2) g(x) = (x + 3)(x + 1)(x - 1)
Zeroes of a function. Factored form.
The zeroes of the function are -3, -1 and 1. So the function should have the terms (x + 3)(x + 1)(x - 1).


19. Mrs. Allard asked her students to identify which of the polynomials below are in standard form and explain why.

I. 15x4 - 6x + 3x2 - 1
II. 12x3 + 8x - 4
III. 2x5 + 8x2 + 10x

Which student's repsonse is correct?

Answer: (3) Fred said II and III because the exponents are decreasing
The Standard form of a polynomial is the term with the highest exponent goes first, then the next highest exponent, and so on.
They are not ordered by coefficients.


20. Which graph does not represent a function that is always increasing over the entire interval -2 < x < 2?

Answer: (3) [graph]
The function in Choice 3 is decreasing when 0 < x < 2, so it doesn't increase over the entire interval specified in the question.
Choice 4 does not start decreasing until after x > 2.


21. At an ice cream shop, the profit, P(c), is modeled by the function P(c) = 0.87c, where c represents the number of ice cream cones sold. An appropriate domain for this function is

Answer: (2) an integer > 0
The domain should be an integer, not a rational number. Cones are sold as whole units. You wouldn't sell, for example, 3 1/2 cones.


22. How many real-number solutions does 4x2 + 2x + 5 have?

Answer: (3) zero > 0
Find the discriminant: b2 - 4ac = (2)2 - 4(4)(5) = 4 - 80 = -76.
There are no real solutions.
You could also graph this function. You will see that it never touches the x-axis, so it has no solutions. (The minimum occurs at (-.25, 4.75).)
Note that the answer "Infinitely many" is silly. A quadratic equation can only have 0, 1, or 2 solutions.
The only time is could be infinitely many is if both sides of the equation are quadratic expressions which are equivalent.


23. Students were asked to write a formula for the length of a rectangle by using the formula for its perimeter, p = 2L + 2W. Three of their responses are shown below.

Which response are correct?

Answer: (2) II and III, only> 0
To solve for L in terms of p and W, you need to use inverse operations to isolate L.
In this case, that means subtract 2w and then either divide by 2, or multiply by 1/2. So responses II and III are equivalent.
Response I isn't good because the 1/2 was only applied to the p term and not the W term.


24. If an = n(an-1 and a1 = 1, what is the value of a5?

Answer: (2) II and III, only> 0
a1 = 1,
a2 = 2(a1) = 2(1) = 2,
a3 = 3(a2) = 3(2) = 6,
a4 = 4(a3) = 4(6) = 24,
a5 = 5(a4) = 5(24) = 120.
Give yourself a pat on the back if you realized that this was the factorial function.

End of Part I

How did you do?

Questions, comments and corrections welcome.

Wednesday, June 20, 2018

Text From the Ex

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(C)Copyright 2018, C. Burke.

They were childhood friends, so it's probably something innocent, and possibly sad news.




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Tuesday, June 19, 2018

Straight Lines

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(C)Copyright 2018, C. Burke.

And, of course, it's the shortest distance between two pun-ts.

The Co-medians haven't taken to the stage in quite a while. About time they had a new gig.




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Monday, June 18, 2018

(x, why?) Mini: My Other Favorite Schools

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(C)Copyright 2018, C. Burke.

Particularly if is filled with fresh pasta.

I've always thought "scolapasta" was a great word from the first time I heard it, even though it sometimes sounded like "-basta" or even "-bassa". I had to look up the proper spelling.
Kind of a "duh!" moment -- of course, it would contain the word "pasta" since it's job is to contain the pasta!




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Sunday, June 17, 2018

Happy Fathers Day 2018!

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(C)Copyright 2018, C. Burke.

Someone's a little envious that they don't have kids yet.
Yes, this comic is a little late because despite being "my" day, it was still a busy one getting the house ready for company. And as much as I might have wanted to relax, that wasn't going to happen for the whole of the day -- and definitely not long enough for this to appear.
Which is probably a good thing, because this strip was originally so much more complicated in my head, and yet boiled down to the same thing.
I'm sure "Dad" ("Grandpa") will appear again soon.




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Saturday, June 16, 2018

Trigonometry Jones and the Running Gag, Part 2

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(C)Copyright 2018, C. Burke.

Since this is Part 2, it actually is a running gag now!

Part 1 appeared back in 2015, Comic 990.




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Friday, June 15, 2018

Opposite and Adjacent

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(C)Copyright 2018, C. Burke.

I have this conversation every term. Sometimes multiple times.




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Thursday, June 14, 2018

Axes of Symmetry

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(C)Copyright 2018, C. Burke.

This comic was a Gimli, er, I mean Gimme.




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Wednesday, June 13, 2018

Height Joke

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(C)Copyright 2018, C. Burke.

My colleague did this to me.
Hey -- that was one of the reasons I introduced this character. Probably the main one.
On a prior occasion I offered to help her get something from the shelf in her locker. She was standing on a chair. I was standing behind her. We were eye to eye.




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Tuesday, June 12, 2018

Art Room

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(C)Copyright 2018, C. Burke.

So this happened ...




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Monday, June 11, 2018

(x, why?) Mini: Skew

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(C)Copyright 2018, C. Burke.

Except for BBQ meats, where you just skewer it.

EDIT:

Skewin' It Right!!!!!

Ugh!! How did I miss that????



Here is the original image:






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Sunday, June 10, 2018

Pick-Up Lines

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(C)Copyright 2018, C. Burke.

If his lines had a point, it'd be an outlier.




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Saturday, June 09, 2018

(x, why?) School Life #1

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(C)Copyright 2018, C. Burke.

There's always drama behind the scenes at school. So naturally I waited until the school year was almost over to do this.

Whether these will be Blog Bonus comics or not, I haven't decided. I think it would be fun to tie them into the background of other strips. On the other hand, posting two strips on the comics-only site may be problematic because of the template that's use there. We'll see what develops

These comics will get their own numbers until I decide I want to name individual strips, which I don't know if I will. This is also (x, why?) #1323, but it times into (x, why?) #1321




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Friday, June 08, 2018

First Names

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(C)Copyright 2018, C. Burke.

Either it's a fat hamster or it's ironic.

Either it's a fat hamster or it's ironic.




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Thursday, June 07, 2018

Spread Sheets

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(C)Copyright 2018, C. Burke.

I needed to work on my delivery and wording and context ... and then I gave it to the kids.

What are the girls talking about? We'll find out really soon with (x, why?) School Life.




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Wednesday, June 06, 2018

Rocks Rock!

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(C)Copyright 2018, C. Burke.

And as a math teacher, I can appreciate a good complement.




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Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.

More Algebra 2 problems.

June 2017, Part I

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


10. A game spinner is divided into 6 equally sized regions, as shown in computations. the diagram below.
For Miles to win, the spinner must land on the number 6. After spinning the spinner 10 times, and losing all 10 times, Miles complained that the spinner is unfair. At home, his dad ran 100 simulations of spinning the spinner 10 times, assuming the probability of winning each spin is 1/6. The output of the simulation is shown in the diagram below.
Which explanation is appropriate for Miles and his dad to make?
1) The spinner was likely unfair, since the number 6 failed to occur in about 20% of the simulations.
2) The spinner was likely unfair, since the spinner should have landed on the number 6 by the sixth spin.
3) The spinner was likely not unfair, since the number 6 failed to occur in about 20% of the simulations.
4) The spinner was likely not unfair, since in the output the player wins once or twice in the majority of the simulations.

Answer: 3) The spinner was likely not unfair, since the number 6 failed to occur in about 20% of the simulations.
Since the number 6 didn't show up in 20% of the simulations, it is not unreasonable that it wouldn't occur in 10 spins of a spinner, even if it seems to be unlikely.





11. Which binomial is a factor of x4 - 4x2 - 4x + 8?
1) x - 2
2) x + 2
3) x - 4
4) x + 4

Answer: 1) x - 2
If you graph the function, it has two zeroes, at (approximately) x = 1.13 and 2. So (x - 2) is a factor.

Alternatively, you could calculate to see if there would be a remainder. Luckily, the first choice is the answer.





12. Given that sin2θ + cos2θ = 1 and sin θ = - SQRT(2)/5, what is a possible value of cos θ?


Answer: 2) -0.15x3 - 0.02x2 + 28x - 120
If sin θ = -SQRT(2)/5, sin2θ = (-sqrt(2)/5)2 = 2/25.
This means that cos2θ = 23/25, and cos θ = SQRT(23/25), which is +Sqrt(23)/5.





Comments and questions welcome.

More Algebra 2 problems.

Tuesday, June 05, 2018

A Question?

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(C)Copyright 2018, C. Burke.

Anyone? Anyone? Bueller? Bueller?

More than just a little semi-autobiographical.




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Monday, June 04, 2018

Imaginary

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(C)Copyright 2018, C. Burke.

I wonder if he could imagine asking a girl to the pizza parlor after school ....

I actually thought about having the new girl like Vaughn who likes Missy (who seemed indifferent) because that would create ...

A love triangle. What could be more mathematical than that?




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Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.

More Algebra 2 problems.

June 2017, Part I

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


7. The solution to the equation 4x2 + 98 = 0 is
1) + 7
2) + 7i
3) + 7*SQRT(2)/2
4) + 7i*SQRT(2)/2

Answer: 4) + 7i*SQRT(2)/2
This isn't the difference of squares. It has a sum, so when we move it to the other side of the equation, you have to take the square root of a negative number. That means an i in the answer.

4x2 + 98 = 0
4x2 = -98
x2 = -98/4
x2 = (-1)(2)(49)/(4)
x = +SQRT((-1)(2)(49)/(4))
x = + 7i*SQRT(2)/2





8. Which equation is represented by the graph shown below?


1) y = 1/2 cos 2x
2) y = cos x
3) y = 1/2 cos x
4) y = 2 cos 1/2x

Answer: 1) y = 1/2 cos 2x
At x = 0, the graph is at 0.5, or 1/2, so this eliminates choices 2 and 4, which would have amplitudes of 1 and 2, respectively. The cycle repeats at 1 pi, instead of 2 pi, so the answer is choice 1.





9. A manufacturing company has developed a cost model, C(x) = 0.15x3 + 0.01x2 + 2x + 120, where x is the number of items sold, in thousands. The sales price can be modeled by S(x) = 30 − 0.01x. Therefore, revenue is modeled by R(x) = x • S(x). The company's profit, P(x) = R(x) − C(x), could be modeled by
1) 0.15x3 + 0.02x2 - 28x + 120
2) -0.15x3 - 0.02x2 + 28x - 120
3) -0.15x3 + 0.01x2 - 2.01x - 120
4) -0.15x3 + 32x + 120

Answer: 2) -0.15x3 - 0.02x2 + 28x - 120
Substitute. Multiply. Combine like terms.
P(x) = R(x) − C(x)
P(x) = x • S(x) − C(x)
P(x) = x (30 − 0.01x) - (0.15x3 + 0.01x2 + 2x + 120)
P(x) = 30x − 0.01x2 - 0.15x3 - 0.01x2 - 2x - 120
P(x) = - 0.15x3 − 0.01x2 - 0.01x2 + 30x - 2x - 120
P(x) = - 0.15x3 − 0.02x2 + 28x - 120





Comments and questions welcome.

More Algebra 2 problems.

Friday, June 01, 2018

Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.

More Algebra 2 problems.

June 2017, Part I

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


4. The expression 6xi3(-4xi + 5) is equivalent to
1) 2x - 5i
2) -24x2 - 30xi
3) -24x2 + 30x - i
4) 26x - 24x2i - 5i

Answer: 2) -24x2 - 30xi
i2 = -1, i3 = -1i = -i, i4 = i2i2 =(-1)(-1) = 1

6xi3(-4xi + 5)
-24x2i4 + 30xi3
-24x2(1) + 30x(-i)
-24x2 - 30xi





5. If f(x) = 3|x| - 1 and g(x) = 0.03x3 - x + 1, an approximate solution for the equation f(x) = g(x) is
1) 1.96
2) 11.29
3) (-0.99, 1.96)
4) (11.29, 32.87)

Answer: 2) 11.29
Eliminate choices 3 and 4 because the solution is a single number, not an ordered pair.
There are up to three possible intersections. If you graph them, you will find them at approximately -0.99, 0.5, and 11.29.





6. Given the parent function p(x) =cos x, which phrase best describes the transformation used to obtain the graph of g(x) = cos(x + a) - b, if a and b are positive constants?
1) right a units, up b units
2) right a units, down b units
3) left a units, up b units
4) left a units, down b units

Answer: 4) left a units, down b units
Inside the parentheses, plus shifts to the left and minus shifts to the right. Outside of the parentheses, plus moves the graph up and minus moves the graph down.





Comments and questions welcome.

More Algebra 2 problems.

Second Difference

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(C)Copyright 2018, C. Burke.

He is a Ranger, so he'd know his domain and take it all in Stride.
But when it comes to lines, you know they never want to stop.




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