Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2011)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, January 2011

Part II: Each correct answer will receive 2 credits. Partial credit is available


32. The graph below represents the function y = f(x).

State the domain and the range of this function.

Answer:

The domian is the set of all x values, which run from -5 to 8, inclusive, of [-5,8].

The range is the set of all y values, which run from -3 to 2, inclusive, or [-3,2].

Note that you need square brackets because the endpoints are included.

Note that the y values are not dependent on the endpoints of the function. You want the maximum and the minumum values for y.

You could also express your answers using inequalities: -5 < x < 8 and -3 < y < 2. Again, make you use < and not <.

33.Express [see image] in simplest radical form.

Answer:

Look at the image below:

Put the entire fraction under the radical. Simplify the terms by dividing 108 by 6, subtracting 1 from 5, and subtracting 5 from 8. Next take the square root of the factors that are perfect squares. That leaves only 2y under the radical sign.

34. Assume that the ages of first-year college students are normally distributed with a mean of 19 years and standard deviation of 1 year.

To the nearest integer, find the percentage of first-year college students who are between the ages of 18 years and 20 years, inclusive.

To the nearest integer, find the percentage of first-year college students who are 20 years old or older.

Answer:

Students who are 18 years old are one standard deviation below the mean. Students who are 20 years old are one standard deviation above the mean. If you look at the diagram of the Normal Curve Standard Deviation, the percentage of students within one standard deviation from the mean is:

15.0 + 19.1 + 19.1 + 15.0 = 68.2 per cent.

68% are between 18 and 20.

The students who are 20 years old or older are more than 1 standard deviation away from the mean. According to the table, 9.2 + 4.4 + 1.7 + 0.5 + 0.1 = 15.9. So 16% of the first-year college students are 20 years old or older.

35. Starting with sin² A + cos² A = 1, derive the formula tan² A + 1 = sec² A.

Answer:

Use the identities you know.

In particular, tan = sin / cos and sec = 1 / cos.

sin² A + cos² A = 1

(sin² A)/(cos² A) + (cos² A)/(cos² A) = 1/(cos² A)

tan² A + 1 = sec² A

End of Part II.

More to come. Comments and questions welcome.

More Regents problems.

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