Pre-Calculus I - 1st Semester Review - Pt. 2

Zeros

100

Determine the zeros of the function:  y = 2x² + 5x - 3

x = 1/2

x = -3

200

Determine the possibilities for zeros:

y = 2x⁴ - 3x³ + x² - x + 10

possibilities:  +/- 2, +/- 1, +/-5, +/-10, +/-1/2, +/-5/2

300

Determine the zeros:  y = 4x³ - 5x

x = 0, x = +/- sqrt 5/2

400

Approximate where f(x) is greater than or equal to g(x).

f(x) = x² - 5x + 6       g(x) = 2x - 3

(- infinity, 1.697]

[ 5.303 , infinity)

500

Graph the following function, include zeros, intercept(s), asymptotes.

y = [(x-3)(x+2)(x+5)]/[(x+1)(x-2)]

[Graph]

Inequalities

100

Solve:    -x² + 4 is greater than zero

( -2, 2)

200

Approximate where f(x) = g(x)

f(x) = 2x - 1           g(x) = x² - 8

x = 3.828

x = -1.828

300

Determine where the function is increasing, decreasing, or constant.

y = (1/2)x³ - 5x

Increasing:  (-infinity, -1.826) U (1.826, infinity)

Decreasing:  (-1.826, 1.826)

400

Where does f(x) exceed g(x)?

f(x) = 3x² - 5x - 9

g(x) = x - 1

(-infinity, -0.915) U (2.915, infinity)

500

Where does g(x) exceed f(x)?

f(x) = 2x - 3

g(x) = x³ - 4x² + 5

(-1.414, 1.414) U (4, infinity)

Exponents & Logarithms

100

Solve:  1/64 = 2^x-1

x = -5

200

Solve.  Give an exact answer.

5^2x-1 = 2^-3x+4

x = (ln 16 + ln 5) / (ln 25 + ln 8)

300

Solve:  log₂ x + log₂(x + 4) = 5

x = 4

400

If x is a real number and resides between 0 and 1, compare:

A.  1/x²      B.  1/x³

B is greater than A.

500

Solve.  Give an exact answer.

log₂24 = x

x = 3 + log₂3

Application Problems

100

The table shows the time t in seconds required for a car to attain a speed of s miles per hour from a standing start.

Speed, s = 30, 40, 50, 60, 70, 80, 90

Time, t = 3.4, 5.0, 7.0, 9.3, 12.0, 15.8, 20.0

Determine the seconds required to hit 65 mph.

t = 10.22 secs.

200

Use an augmented matrix to solve the systems of linear equations:

3x - 2y + z = 15

-x + y + 2z = -10

x - y - 4z = 14

(5, -1, 2)

300

How many years will it take to double an investment of $5000 with an interest rate of 6.5% compounded monthly?

10.7 years

400

Margaret ordered 200 flowers for the church for Mother's Day.  In the mix were carnations at $1.50, roses for $5.75, and daisies for $2.60 each.  The total rang up at $589.50.  She did get 20 fewer roses than daisies.

How many of each flower did she order?

Carnations = 80

Roses = 50

Daisies = 70

500

Max had $24,500 to invest in three different accounts.  The first paid 4% interest, the second paid 5.5% interest, and the third paid 6%.  At the end of the year, he had made $1300.  If the amount of money in the lowest paying account was four times the amount in the middle account, how much money was in each account?

$8000 in the 4% account

$2000 in the 5.5% account

$14,500 in the 6% account

Matrices

100

Solve:   2x + 3y - 4z = -5

            -x - 2y + z = 1

             -4x + y + 2z = -9

(3, -1, 2)

200

Solve for X.     2X - A = B

A = [ 1  2  -1                  B = [ 5  -6   1

         3  1  -2                          -2  3  -4

        -1  4  -5]                        -3  8  -9]

X = [ 3   -2   0

        1    2   -3

       -2    6   -7]

300

Multiply A times B.

A = [ 2  4  -3]       B = [ -6  0  2

                                     5   1  -4

                                    -2   -1  8]

AB = [14  7  -36]

400

Multiply C x D

C = [-3  0  2                D = [-2  6

        1   3  -2                       3  -4

        5  -1   4]                      2   1]

  CD = [10  -16  

             3    -8

            -5    38]

500

Find the equation of the parabola y = ax² + bx + c passing through the points (-3, 13), (0, -4), 

and (3, -1).

y = 10/9x² - 7/3x + 4