Chapter 4 Test



Find the vertex and state the domain and range.

a) f(x)=x2-6x+1 ( b) f(x)= -2(x-5)(x+3) (

c) [pic](

1. An arena seats 2000 people and charges $10 per ticket. At this price all the tickets are sold. The team owner wants to increase the ticket prices. A survey indicated that for every one-dollar increase in price, the number of tickets sold will decrease by 100.

a) Find the ticket price that will yield the greatest revenue.(

b) Find the largest revenue.(

2. The demand function for a new product is p(x)=-5x+22, where x is the number of items sold in thousands and p(x) is the price in dollars. The cost function is C(x) = 3x+15.

a) Find the profit function.(

b) Find the value of x that maximizes profit.(

c) Find the break-even quantities.(

3. Find the number of different real roots for the following equations:

a) y=3(x-2)2-4 ( b) y=2x2-5x+4(

4. Find the value of “k” so the following has two different real roots.

a) f(x)=kx2-8x+2 ( b) f(x)=9x2+kx+4(

5. Identify all sets of numbers to which each number belongs.

(Natural-N, Whole W, Integers Z, Rational Q, Irrational Q, Real R, Complex C)

a) 0.85 ( b) [pic] ( c) 2+5i (

6. Solve each quadratic equation for x, x Є C.

a) x2-2x+3=0 ( b) 3x(x+4)= -14 (

7. Simplify the following:

a) (3i)2 ( b) [pic] (

9. Given 1-2i is a root of a quadratic equation. Find the other root and write a quadratic equation that has these roots.(

1. Factor fully

( a) [pic][pic] ( b) [pic] ( c) [pic]

2. Find the vertex by partial factoring. (No decimals!)

( a) [pic] ( b) [pic] ( c) [pic]

( d) [pic] ( 3. Graph [pic]from above 2b).

4. Convert to vertex form. (No decimals!)

( a) [pic] ( b) [pic]

( 5. Graph [pic] from above 4a).

( 6. State the type of roots for each equation.

a) [pic] b) [pic] c) [pic]

7. Solve where [pic].

( a) [pic] ( b) [pic]

8. Solve using absolute values.

( a) [pic] ( b) [pic]

9. Graph using transformation of the graph [pic].

( [pic]

( 10. Write as a complex number.

a) [pic] b) [pic]

1. Graph the following functions using transformations.

a) [pic] b) [pic]

c) [pic] d) [pic]

2. Given a function [pic], where the domain is [pic]

a) What is the domain and range of [pic]

b) What is the domain and range of [pic]

3. Given [pic], graph the function [pic]

4. Solve the following,

a) [pic] b) [pic]

5. Given the function [pic], find the range where, [pic].

1. Find the roots where, [pic]

a) [pic] b) [pic]

2. Find the value of k in the following quadratic functions that satisfies;

a) [pic] b) [pic]

has two different real roots has one real root

3a) Given [pic], find [pic] and graph both [pic], and [pic].

b) Given the [pic], graph the reciprocal.

4. The height of a javelin can be determined by the equation,

[pic], where [pic] is in meters off the ground

and [pic] is in seconds after it is released

a) How long will it take for the javelin to hit the ground?

b) What is the highest point the javelin reaches?

c) If the average speed of the javelin is [pic] what distance is achieved by the javelin?

5. A minimum of 800 yearbooks must be produced. The price for each is $17.50 if the minimum is to be produced. For each additional 10 books sold the cost is reduced by 10 cents per book. If you were the publisher how many books sold will give you the maximum revenue?

Solve for [pic] where [pic],

2. Determine the value of p so that, [pic], has one root.

3. A lifeguard marks off a rectangular swimming area at a beach with 200m of rope. What is the maximum area of water she can enclose? (Note: the side along the beach does not need to be roped off.)

. Simplify each rational expression. State any restrictions on the variables.

a) [pic] b) [pic]

c) [pic] d) [pic]

5. Write the following as a complex number in the form [pic].

a) [pic] b) [pic]

6. Solve for w and express in the form [pic]. [pic]

7. Given the function [pic], graph the reciprocal [pic].

8. Suppose the distance from Seoul to Detroit is 9 000km. 2 airplanes depart at the same time, one from each city heading towards the other. The average air speeds of the both planes are the same at 850km/h. If it took the plane from Detroit to Seoul an extra

1 hour and 15 minutes, find the speed of the air current. (Hint: draw a diagram with a chart.)

a) [pic] b) [pic] c) [pic]

. Determine the quadratic equation with the following information:

a) Vertex [pic] and passing through the point [pic]

b) Passing through the points [pic] and [pic] with optimum value [pic]

3. Given the following quadratic, determine the vertex by the method indicated.

Completing the square Partial factoring

a) [pic] b) [pic]

4. Convert [pic] to vertex form. (Any method.)

1. Given a quadratic of the form [pic], how can you determine if the optimum value is a maximum?

2. For any quadratic in standard form, what do you get from partial factoring? Why would you do this? Explain.

1. Massey’s student council has decided to invite a hypnotist to the school. A couple years ago they sold 1000 tickets for a price of $4. A survey indicates that for every ticket price increase of 50 cents, the number of tickets sold will decrease by 50. What ticket price will result in the greatest revenue for the school?

2. A lifeguard has 300 ft of rope and wants to rope off two equal rectangular beaches that border the water (as shown in the diagram). Find the dimensions of the total beach that maximizes the area.

1. The hypotenuse of a right angle triangle is 34 cm. One of the sides is 14 cm longer than the other side. Find the lengths of the other two sides.

1. Given the following functions,

a) [pic], find[pic] b) [pic], find [pic]

2. Given, [pic] find the following;

a) [pic] b) [pic]

c) [pic] d) [pic]

3. Given the function [pic], determine the following,

a) [pic]

b) [pic]

c) [pic]

d) [pic]

e) [pic] f) [pic] g) [pic]

4. Find the range and domain of the following relation AND state whether each is a function or not.

a) b) [pic]

5. Solve the following inequalities and graph the solution on a number line.

a) [pic] b) [pic] c) [pic]

6. Given [pic],

a) graph [pic] and its

inverse on the same grid

b) find the equation for the inverse c) range of [pic], [pic]

1. Given the following quadratic equations determine the vertex.

a) [pic] b) [pic] c)[pic]

2. Find the vertex of the quadratic equation given by, [pic]

3. Graph [pic], by stretching and shifting the original quadratic [pic]

4. Given, [pic] find the following;

a) [pic] b) [pic] c) [pic]

d) [pic] e) [pic]

5. Find the range and domain of the following relation AND state whether each is a function or not.

a) b) [pic]

7. Make a rough sketch of [pic] for [pic], to find the range and domain.

8. Solve the following inequalities and graph the solution on a number line.

a) [pic] b) [pic] c) [pic]

1. Expand and simplify.

a) [pic] b) [pic]

2. Factor fully.

a) [pic] b) [pic] c) [pic]

3. Evaluate each expression.

a) [pic] b) [pic] c) [pic]

4. Simplify.

a) [pic] b) [pic]

5. Solve.

a) [pic] b) [pic]

c) [pic] d) [pic] e) [pic]

6. Find the first 3 (three) terms of the following general terms;

a) [pic] b) [pic] , [pic]

7. Find the general term of the following sequences;

a) 4, 7, 10, 13, … b) [pic] [pic] [pic] 24, … c) [pic] [pic] [pic], [pic] …

8. The following terms are terms in an arithmetic sequence. Find the general term.

[pic]

[pic]

9. Determine the number of terms in the following geometric sequence.

[pic], [pic] [pic] … [pic] [pic] 093 442

1. There are 28 days in February 2002. Suppose I was given one penny on Feb. 01 then 2 pennies on Feb. 02, then 4 pennies on Feb. 03 and so on for the entire month (doubled each day). How much money will I have at the end of the month in total?

2. How long would it take for a colony of ants to triple in size if they grow at a rate of 0.98 % every week?

3. Suppose a coin estimated to be $ 23.58 has been increasing in value by 4.5% each year for the last little while. If this trend continues, how much would this coin be worth in 36 years?

4. Mr. White currently has a fair chunk of money stashed in his mutual funds (a type of savings account). It was calculated that he would have $109 200 in 17 years when he will retire. If the rate of return is 8.9%/a compounded monthly how much does he have in his mutual funds today?

5. Credit card companies charge simple interest on the full amount of all purchases, if the entire amount is not paid in full on the due date. If a purchase of $879.49 was made on Nov. 13 and the full amount was not paid on Dec. 05 (due date). How much interest is charged on Dec. 05, if the rate of interest is 18.9%?

6. An annual scholarship of $ 500 is to be given out every year for the next 50 years. How much must be invested today if an investment at 10.4 %/a compounded annually could found?

7. How much must Mrs. Romiens deposit each week to pay for her son’s first year’s education which is project to cost $ 13 000 in three years at 6.85%/a compounded weekly?

8. Being a new teacher Mr. Tran can only afford to put away $25 every week. He currently has $ 1200 in this account. Suppose the investment will grow at 10.2%/a compounded weekly how much will Mr. Tran have in 32 years when he finally retires?

9. Find the interest rate compounded semi-annually, that is equivalent to 7%/a compounded weekly?

1. Graph the function [pic], by first finding.

a) the vertex b) the y-intercept c) the zeros or (roots)

2. Sketch [pic], using transformations (shifts and stretches) from the original graph [pic].

3. Given [pic],

a) Find [pic]. b) Sketch the graph of both [pic], and [pic].

c) State the range and domain of the above 3a and 3b),

i) [pic] ii) [pic]

4. Given [pic], find the following.

a) [pic] b) [pic] c) [pic] 5. Given [pic], find.

a) [pic] b) [pic] c) [pic]

6a) Given the following graph [pic], graph the inverse [pic] and state the range and domain of both [pic], and [pic].

6b) Which of the above 6a) is/are function(s)? Circle one.

i) [pic] ii) [pic] iii) both iv) neither

7. Solve the following inequalities and graph a number line for each solution.

a) [pic] b) [pic]

8. Given [pic], graph this function for [pic]. State the range and domain for this graph.

1. Simplify the following: a) (xy-3)4 b) [pic]

c) [pic] d) [pic]

2. Define asymptote.

State all transformations and sketch:a) y=32x-4.

[pic]

A house was bought six years ago for $175 000. If real-estate values have been increasing at the rate of 4% per year, what is the value of the house now?

An element is decaying at the rate of 12% every 6 hours. If we have 100g now find:

a) how much remains after 12 hours?

b) how much was present 8 hours before now?

. For a biology experiment, there are 50 cells present. After 2 hours there are 1600 bacteria. How many bacteria would there be in 7 hours assuming a continuous growth rate.

) b)

2. Graph the function within the domain specified.

[pic] , [pic]

3. Solve in the domain stated.

a) [pic] , [pic] b) [pic] , [pic]

4. Given the equation [pic], where h the height of a ball in meter above the ground and t in seconds,

a) What is the maximum height of the ball.

b) Find 2 different times when the height of ball reaches [pic]

c) What is the height of the ball after [pic]? (Exact value!)

5. A car driving along a straight road at a constant speed runs over a nail that is on the ground. Suppose the nail is impaled in the tire at time [pic]sec (assume it does not puncture the tire). If the tire has a diameter of [pic] and rotates once every [pic];

a) Draw the graph of the height of the nail in cm in terms of time in sec. (3 rotations).

b) Write the equation of the above in terms of a cosine function.

c) Calculate the height of the nail after [pic].

a) [pic] b) [pic]

c) [pic] e) [pic]

2. Find the angle[pic] in the domain stated.

a) [pic] ,[pic] b) [pic]

c) [pic]

3. Solve for the missing variable(s).

a) b)

c) b)

5. A tree cast a shadow that is 3.5m in length. If the sun’s ray strikes the top of the tree at an angle of elevation of [pic] how tall is the tree?

6. From an air balloon looking in the same line of sight 2 boats are sighted. The boats are approximately 1km apart and the angle of depression from the balloon to one boat is [pic] and to the other boat is [pic]. Find the height of the air balloon. (1 decimal place)

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