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Tangent, Cotangent, Secant, Cosecant Graphs

Open the file TrigFunc.gsp found on the main page of the wiki, then answer the following questions as a comment:

1) What is the period of:

a) f(x) = tan x?
b) g(x) = csc x?
c) g(x) = sec x?
d) g(x) =cot x?

2) For what values of x is the given function undefined? Can you explain why this occurs (for each)? Why do these functions approach infinity / negative infinity at these asymptotes?

a) f(x) = tan x
b) g(x) = cot x
c) g(x) = csc x
d) g(x) = sec x

3) In what ways is the graph of f(x) = tan x related to g(x) = cot x? Explain

4) In what ways is the graph of f(x) = sin x related to g(x) = csc x? Explain

5) In what ways is the graph of f(x) = cos x related to g(x) = sec x? Explain

6) In what ways is the graph of g(x) = csc x related to g(x) = sec x? Explain

7) What is the range of csc x? Why?

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Sine, Cosine and Tangent Graphs

Answer these questions using terms such as domain, amplitude, period, x-intercepts and frequency.
1. How do the Sine and Cosine graphs change as "a" increases? As "a" decreases?
2. How do the Sine and Cosine graphs change as "b" increases? As "b" decreases?
3. What happens to the Sine and Cosine graphs when "a" goes from positive to negative?
4. How does the Tangent graph change as "a" increases? As "a" decreases?
5. How does the Tangent graph change as "b" increases? As "b" decreases?
6. What happens to the Tangent graph change when "a" goes from positive to negative?
7. Why does the Tangent graph drop off the bottom of the screen and reappear on the top?
8. What does the "h" variable do to the graphs (be specific)?
9. What does the "k" variable do to the graphs (be specific)?

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How Many Real Zeros Can a Polynomial Have?

1) Give examples of polynomials with real coefficients that have the following properties, or explain why it is impossible to find such a polynomial.


a) A polynomial of degree 3 that has no real zeros

b) A polynomial of degree 4 that has no real zeros

c) A polynomial of degree 3 that has three real zeros, only one of which is rational

d) A polynomial of degree 4 that has four real zeros, none of which is rational



2) What must be true about the degree of a polynomial with integer coefficients if it has no real zeros?

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Dividing Polynomials Warm-Up

P(x) = 3x^5 + 5x^4 – 4x^3 + 7x + 3

a) Find the quotient and remainder when P(x) is divided by x + 2

b) Find P(-2)

c) Find the quotient and remainder when P(x) is divided by x – 1

d) Find P(1)


f(x) = x^3 – 7x + 6

a) Find the quotient and remainder when f(x) is divided by x – 1

b) Find P(1)

c) Factor f(x) into its three linear factors


Answer the following questions as a group and post your answers as a comment (be sure to list all group members’ names at the top of the post).

1) How is the remainder when dividing a polynomial f(x) by x – c related to evaluating f(c)?

2) What is the significance of dividing a polynomial by x – c and having a remainder of zero?

3) For some polynomial P(x), if P(a) = 0 what else is true?


Answer the following questions without actually dividing:

4) What is the remainder when 6x1000 – 17x562 + 12x + 26 is divided by x + 1?

5) Is x – 1 a factor of x567 – 3x400 + x9 + 2?



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Inverse Functions

1) F = C*9/5 + 32 gives the conversion from degrees Celsius to Fahrenheit.

a) Find the inverse of the above function.

b) What does the inverse function do?



2) The function f(x) = .75x - 5 gives the cost of a pair of jeans after a 25% discount and a 5 dollar off coupon.

a) If the domain of the function is set to [0, 100], what is the range?


b) What is the equation for the inverse of f?


c) What does the inverse of f do?



3) The function g(t) = 2300 - 350t gives the distance from Los Angeles of an airplane flying to LAX from Philadelphia.

a) What is the domain and range of this function?


b) What is the equation for the inverse of g?


c) What is the domain and range of the inverse of g?


d) What does the inverse of g do?



4) Come up with or find a function (with a real context) of your own. Find the inverse of that function, and explain what the inverse function does.

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Polynomial Zeros

Can you find a degree 3 polynomial with the given number of x-intercepts? If so, give the function. If not, please explain briefly why you think you cannot.

a) 3 x-intercepts

b) 2 x-intercepts

c) 0 x-intercepts

d) 4 x-intercepts

Can you find a degree 4 polynomial with the given number of x-intercepts? If so, give the function. If not, please explain briefly why you think you cannot.

a) 3 x-intercepts

b) 2 x-intercepts

c) 0 x-intercepts

d) 4 x-intercepts

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Even and Odd Functions

1) What is an even function (you can look it up if you don't know)?





2) What is an odd function?





3) What must be true about the integer n if the function f(x) = x^n is:

a) an even function?





b) an odd function?




c) Why do you think the names "even" and "odd" were chosen for these function properties?

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Khan Homework 10/18

-Distance Formula

-Midpoint Formula

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NY Times Article

Read the following article:

http://learning.blogs.nytimes.com/2012/09/26/n-ways-to-apply-algebra-with-the-new-york-times/


Pick one of the listed ways to use math in life, and answer the questions.

For example, #1:

For a given house, explore questions like “How much would you end up paying over the entire term of the mortgage, and how does this compare to the value of the house?” and “How big is the difference in monthly payments between a 15-year mortgage and a 30-year mortgage?”

Do you think it is more cost effective to rent an apartment or buy a house? Read this article and play around with this Times Interactive to explore the answer.


Put together a brief report with your group that explains the question you investigated and your answer, including any calculations you needed to get that answer. Attach any data in an appendix, with sources cited informally (i.e. NY Times Website)


Individually, read the comments that subscribers posted at the bottom of the article. Pick one that resonates with you, and comment on this blog post why you agree or disagree with that person's comment. Please back up any statement you make with evidence that supports it (you'll notice many NY Times readers failed to do so).



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Khan Academy 2

The Complex Plane

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Khan Academy

Please complete (when it says "ready to go on") the following modules in Khan Academy:

Imaginary Unit Powers

Adding and Subtracting Complex Numbers

Multiplying Complex Numbers

Dividing Complex Numbers


Hints:

To add coach, click your name at the top right once you log in to visit your profile, then click coaches at bottom left of profile page.
You can search for a module by typing it into the search box in the top left of the practice screen.

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Solving Quadratics


When is it best to use factoring to solve a quadratic? When is it best to use completing the square? When is it best to use quadratic formula?

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ERA Problem


A pitcher's Earned Run Average (ERA) is the mean average number of runs he gives up per nine innings pitched, which is a full game of baseball. For example, if he's pitched six innings and given up four runs, that's an average of two runs per three innings. Extending that to nine innings, he'd give up six runs, so his ERA would be 6.00 at that point.

As the season goes on and he pitches more games, his ERA will constantly change in value, but it always represents the average number of runs he would give up over nine innings of pitching. If he's given up 11 runs after pitching a total of 18 innings, his ERA would be 5.50.

I recently attended a Red Sox game, and one pitcher had an ERA of 13.50 when he came into the game. After he got one batter out, his ERA dropped to 13.00. Note that since there are three outs in an inning, one out represents a third of an inning pitched.

How many innings had that pitcher pitched before he came into the game I saw?

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