Precalculus- Review for Midterm

PreCalculus- Review for Midterm

On a separate sheet of paper, answer the following questions, showing all work:

Unit 1: Graphs & Functions

(1) State how many zeros P(x) = x5 + 2x4 – 6x3 – 8x2 + 9x – 7 can possibly have.

(2) If f(x) = 2x2 – x, find f(x + h).

(3) If f(x) = x2 – 2x and g(x) = 2x – 3, find [f ° g](x).

(4) Write an equation in standard form, Ax + By = C, of the line parallel to the graph of 2y – 6 = 0 and passing though (4, -1).

(5) Write an equation in standard form, Ax + By = C, of the line that passes through (3, 1) and is perpendicular to the line that passes through (-1, 1) and (1, 7).

(6) Given the relation {(2, 2), (3, -2), (2, -2), (-2, 3)}:

(a) State the domain and the range.

(b) State whether the relation is a function and explain why or why not.

(7) Given f(x) = (x – 1)2 + 2:

(a) Find the inverse, f -1(x).

(b) State whether the inverse is a function and explain why or why not.

(8) Determine algebraically whether the graph of 4x2 – 5y = 6 is symmetric with respect to the y-axis, the x-axis, the origin, or none of these.

(9) Determine algebraically whether the function f(x) = 7x2 – 2x is odd, even, or neither.

(10) Using interval notation, algebraically identify the domain of the following functions:

(a)

(b)

(11) Find the zero(s) of each function. If no zero exists, write none.

(a) f(x) = 3x + 5

(b)

(12) Graph the following on graph paper:

(a)

(b) 4x – 3y = 6

(c)

(d)

Unit 2: Rational Functions

(13) Given

(a) Determine algebraically the vertical asymptote(s)

(b) Determine algebraically the horizontal asymptote(s)

(14) Determine algebraically the oblique asymptote for

(15) Find the domain, in interval notation, and the x-intercepts for each rational function:

(a)

(b)

(c)

(16) Graph the following rational functions on graph paper, and state all relevant information (intercepts and asymptotes) about each function:

(a)

(b)

(c)

Unit 3: Limits & Continuity

(17) Graph the following piecewise functions on graph paper, and based on each of your graphs find the following using interval notation:

· Domain

· Range

· Intervals over which the function is increasing

· Intervals over which the function is decreasing

· Intervals over which the function is constant

(a) (b)

(18) Given the following piecewise function:

(a) Graph on graph paper

(b) Based on your graph find the following limits:

i. (ii) (iii)

(19) Algebraically determine the value each of the following limits or explain why it does not exist

(a) / (b) / (c)
(d) / (e) / (f)
(g) / (h) / (i)
(j) / (k) / (l)
(m) / (n) / (o)
(p) / (q) / (r)

Unit 4: Polynomial Functions

(20) Write the polynomial equation with the roots -4 and 2i.

(21) Write P(x) = 2x3 – 5x2 – 28x + 15 as a product of first-degree factors if -3 is a zero.

(22) Write P(x) = x3 + 6x2 – 9x – 54 as a product of first-degree factors if 3 is a zero.

(23) Write P(x) = x4 – 12x3 + 55x2 – 114x + 90 as a product of first-degree factors if 3 is a double zero.

(24) Given: (x5 + x3 + x) ¸ (x – 3):

(a) Use synthetic division to find the quotient and remainder.

(b) Is (x – 3) a factor of x5 + x3 + x? Explain why or why not.

(25) Given: (8x4 – 20x3 – 14x2 + 8x + 1) ¸ (x + 1):

(a) Use synthetic division to find the quotient and remainder.

(b) Is (x + 1) a factor of 8x4 – 20x3 – 14x2 + 8x + 1? Explain why or why not.

(26) Solve the following quadratic equations by completing the square:

(a) x2 + 10x + 35 = 0

(b) 4x2 + 6x – 3 = 0

(27) Complete the square to change the following quadratic functions to the form: and state the vertex for each parabola:

(a) f(x) = 2x2 – 8x + 4

(b) f(x) = -x2 + 3x – 4

Unit 5: Conic Sections

Write the standard form equation of each conic section using the given information:

(28) A parabola whose focus is at (2, 4) and equation of the directrix is x = 6.

(29) A hyperbola whose center is at (2, 4), length of vertical transverse axis is 24, and length of conjugate axis is 18.

(30) An ellipse whose center is at (-2, -4), length of vertical major axis is 24, and length of minor axis is 18.

(31) A hyperbola whose center is at the origin, length of horizontal conjugate axis is 8, and distance of foci from center

is 5.

(32) An ellipse whose center is at the origin, length of horizontal major axis is 16, and distance between foci is 12.

For 33-39:

(a) identify the conic section represented by each general form equation

(b) write the equation in standard form

(d) graph the equation

(33) x2 – 4y2 – 4x – 24y – 36 = 0 / (36) x2 + 8x + 8y = 0
(34) x2 + y2 + 12x + 10y + 45 = 0 / (37) 16x2 + 4y2+ 96x – 16y + 96 = 0
(35) 16x2 + 9y2 – 192x – 72y + 576 = 0 / (38) 25y2– 9x2 – 225 = 0
(39) y2 – 8x – 4y – 20 = 0

Unit 6: Exponential & Logarithmic Functions

(40) Solve each equation and round answers to four decimal places where necessary:

c. log2 x = 2

e. log3 (x + 5) = 2 log3 3

(41) In 2001, the moose population in a certain area is 12,000. The number of moose increases exponentially at a rate

of 9.1% per year. Predict the population in 2008.

(42) Aaron invests $1,500 at Exponential City’s Savings & Loan in an account bearing 8.5% compounded

continuously. Find the balance of his account after 18 months.

(43) How much should Emily invest now in a money market account if she wishes to have $50,000 in the account at

the end of 10 years if the account provides an interest rate of 7% that compounds interest quarterly?

(44) What was the interest rate on an account that took 18 years to double if interest was compounded continuously

and no deposits or withdrawals were made during the 18-year period? (Your answer should be a percentage

rounded to the nearest hundredth.)

(45) The number of garden snakes living Mathematical Meadow doubles every month. If there are 8 snakes present

initially:

(a) Express the number of snakes as a function of the time t.

(b) Use your answer from part (a) to find how many snakes are present after 1 year.

Unit 7: Applications of Functions

(46) The fixed costs per day for a doughnut shop are $250, and the variable costs are $1.35 per dozen doughnut produced. If x dozen doughnuts are produced daily, express the daily cost C(x) as a function of x.

(47) A car rental agency charges $0.42 per mile if the total mileage does not exceed 50. If the total mileage exceeds 50, the agency charges $0.42 for the first 50 miles plus $0.13 per mile for the additional mileage. There is an additional charge of $100.00 for the total mileage that exceeds 100 miles. Write the piecewise function f(x) that relates the total cost for renting a car driven x miles.

(48) A digital camera was purchased for $250 and is assumed to have a salvage value of $40 after 7 years. Its value has depreciated linearly during this period.

(a) Write a linear function V(t) that relates the value of the camera in dollars to time t in years.

(b) What would be the depreciated value of the camera after 2 years? 5 years?

(d) Sketch the graph and indicate the WINDOW used.

(49) Liz wants to buy a new flat screen television that will retain its value. She has a choice of buying the Luxury Brand television for $3,000 or the Expensive Brand television for $3,500. The Luxury Brand is assumed to have a depreciated value of $1,200 after three years, which represents a linear depreciation. The Expensive Brand depreciates in value exponentially at an average rate of 37.5% each year.

(a) Write a linear function L(t) that relates the value of the Luxury Brand television in dollars to time t in years.

(b) Write an exponential function E(t) that relates the value of the Expensive Brand television in dollars to time t in years.

(c) If Liz wants to be able to sell this flat screen television after 5 years, which model will have a higher value? Explain your answer.

(d) Sketch the graphs and indicate the WINDOW used.

(50) The total fencing used to make the five equal-sized rectangular pens is 460 feet.

(a) Write a perimeter (total fencing) equation.

(b) Write a function to express the total area A(x) in terms of x.

(d) Find the dimensions of x and r, to the nearest tenth, such that the diagram above would have a maximum area. What is the maximum area to the nearest tenth?

(e) Sketch the graph and indicate the WINDOW used.

(51) A box is to be made out of a piece of tin that measures 12 cm by 27 cm. Squares, x cm on a side, will be cut from each corner, and then the ends and sides will be folded up to create a box that will have no top.

(a) Write a function to express the volume of the box V(x) in terms

of x.

(b) What is the practical domain?

(c) Find the dimensions of the box, to the nearest tenth that would give the maximum volume. What is the maximum volume to the nearest tenth?

(d) Sketch the graph and indicate the WINDOW used.

(52) Polynomial Printers makes special-order T-shirts. They recently received two orders for a shirt designed for a math symposium. The first order was for 40 T-shirts at a cost of $295, and the second order was for 80 T-shirts at a cost of $565. Each order included a standard shipping and handling charge.

(a) Write a linear function P(x) that models the total cost of buying x T-shirts.

(b) What is the cost per T-shirt?

(53) The amount of tax that a corporation pays depends on its taxable income. The tax table shows the federal tax rate brackets for various levels of corporate income.

Federal Tax Brackets
Taxable Income ($) / Federal Tax Rate (%)
0 – 50,000 / 15
50,001 – 75,000 / 25
75,001 – 100,000 / 34
100,001 – 335,000 / 39

(a) What is the tax bracket for a corporation with a taxable income of $90,000?

(b) Graph the tax brackets for the different incomes.