MATH 109

Exam 3 Review

Jeopardy

Quads Log On Potent Potions Pieces of Eight Potpourri

100 100 100 100 100

200 200 200 200 200

300 300 300 300 300

400 400 400 400 400

500 500 500 500 500

Quads 100

• Find the focal point and vertical intercept for the quadratic function graphed below:

• Answer: focal pt (3,2.5) ; vertical int (0,6.5)

Quads 200

• Find an equation for the quadratic function graphed below in (i) vertex form (ii) factored form and (iii) standard form

• Answer: 424

1 2 xxf

264

1 xxxf

34

1 2 xxxf

Quads 300

• A relief package is dropped from an airplane and falls to the ground along a parabolic path starting at the vertex of the parabola. The package was released when the plane was directly above the marker at a height of 5 kilometers, and the package hits the ground 4.43 kilometers from the marker. If h is the height of the package when it is a horizontal distance d from the marker, find an equation for h in terms of d.

• Answer: 52548.0 2 dh

Quads 400• A 5.5 foot tall woman is shooting a free throw. The path

of the basketball is parabolic in shape (see below). The ball reaches a maximum height of 11.5 feet when the ball is 10 feet horizontally from the player. If the ball hits the front of the rim, which is 10 feet high, how far is the shooter from the front of the rim?

• Answer: 15 feet

Quads 500

• A skateboard company’s profits P(x) are a quadratic function of the price x they charge. If they sold skateboards for free, they would lose $37,500. If they charge either $5 or $35, they will break even (make no profit). What price should they charge per skateboard in order to make the most profit, and how much profit will they make?

• Answer: $20 per skateboard ; Max profit: $48214.29

Log On 100• Use rules of logarithms to explain how you can tell which graph is

• Answer: Graph 1 is the thin line graph. Graph 2 is the thick line graph. Graph 3 is the dotted graph. Expand each function with properties of logs to see vertical shifts up and down.

xy 5log 1 xy 5log 2 5 55log )3( xy

Log On 200

• Consider the functions and Fill in the table below:

13ln xxf 13ln xxg

f(x) g(x)

Domain

Range

Asymptotes

Intercepts

31x 0x

s'# real All s'# real All

31x 0x

0,0 0,3

1e

Log On 300• The graphs of two exponential functions of the form and two logarithmic functions of form are shown below.

Which pairs of functions are logarithmic?Which pairs of functions are exponential? Which exponential function has the largest b-value?Which logarithmic function has the smallest b-value?

xby xy blog

)(),( III)(),( IVIII

)(I)(III

Log On 400

• A recycling program is to be introduced to a local suburb. The cost of introducing such a program in thousands of dollars to p% of residents can be modeled by the logarithmic function: . Sketch a graph of C(p) in an appropriate viewing window. What is the vertical asymptote and what does it tell you?

• Answer: p = 100 ; cost become

infinitely large as the % of

residents on the program

approaches 100%.

300100log150 ppC

Log On 500

• A recycling program is to be introduced to a local suburb. The cost of introducing such a program in thousands of dollars to p% of residents can be modeled by the logarithmic function:

• The budget for the recycling program is $200 thousand dollars. What percentage of residents can be put on the recycling program?

• Answer: About 95% of residents

300100log150 ppC

Potent Potions 100

• Billy’s uncle has 17 coins in his pockets and the coins are either quarters or dollars. There is a total of $8.00 in his uncle’s pockets, and if Billy can find the correct number of each type of coin, he gets to keep the $8.00. Help Billy get rich.

• Answer: 12 quarters and 5 dollars

Potent Potions 200

• A new teacher has placed an order for textbooks. There are two types of textbooks, algebra and trigonometry. The algebra books are $18.50 and the trig books are $15. If the teacher orders 273 textbooks and spends $4536, how many books of each type did she buy?

• Answer: 126 algebra books ; 147 trig books

Potent Potions 300

• Willow needs a 43% sage powder for her spell. At the Magic Shop, she is able to buy a 32% sage powder and a 74% sage powder. If she needs 200 grams of her 43% sage powder, how many grams of the two available powders should she buy to make the powder for her spell?

• Answer: 147.62 grams of the 32% sage powder; 52.38 grams of the 74%

Potent Potions 400

• Allison has a 32% silver solution and a 73% silver solution. If she is going to mix the two together to get 60 grams of a 46% silver solution, how much of the 32% silver solution does she need?

• Answer: 39.51 grams of the 32% silver solution

Potent Potions 500

• Jimmy loves to go to the local burger joint where he can mix sodas to make his favorite drink which is 63% cola. His brother has already mixed a 6 ounce drink that is 37% cola. How many ounces of pure cola should Jimmy add to get his favorite drink?

• Answer: 4.22 ounces of pure cola

Pieces of Eight 100

• Find a formula for the piecewise function graphed below:

• Answer:

40if28

04if4

xx

xxxf

Pieces of Eight 200• ComGas, the Cambridge power company, charges its

customers $6.56 a month, plus an additional $0.87 per therm up to 28 therms. Any therms over 28 are charged at $0.58 per therm. (A therm is a unit of gas supplied). Express the monthly cost as a function of the number of therms used.

• Answer:

28if2858.036.2456.6

280if87.056.6

xx

xxxC

Pieces of Eight 300• A t-shirt printing company charges a set-up fee of $10 for

each order, plus the cost per shirt as shown in the table below:

• Express the cost of an order, C, as a piecewise function, n, the # of shirts ordered.

# shirts cost per shirt0-10 $10

11-20 $921-30 $8

over 30 $7

31if710

3021if810

2011if910

100if1010

nn

nn

nn

nn

nC

Pieces of Eight 400• A t-shirt printing company charges a set-up fee of $10 for

each order, plus the cost per shirt as shown in the table below:

• Sketch a graph of the cost of ordering up to 40 t-shirts.

# shirts cost per shirt0-10 $10

11-20 $921-30 $8

over 30 $7

Pieces of Eight 500

• The graph below shows a graduated tax system:

• Describe the tax system in words. [HINT: find a piecewise function for the graph]

• Answer: First $45000 of income is taxed at 5%. The next $50000 of income is taxed at 15%, and any income over $95000 is taxed at 30%.

Potpourri 100

• Consider the system of linear equations:

• Find the value of A so that the system has no solutions.

• Answer: A = 6

23

1142

xAy

yx

Potpourri 200

• Consider a power function of the form: Suppose p is EVEN and (-1,1) is a point on the power function. What can you say about k?

• Answer: k >0

pkxy

Potpourri 300

• A radioactive substance has a very long half-life. If you have 100 grams of the substance now, one year from now there will still be 99 grams left. Construct three exponential models by completing the table below.

xCay rxCey hx

Cy

2

1

xy 99.0100 xey 01.0100 69

2

1100

x

y

Potpourri 400

• Which function approaches zero faster as x goes to positive infinity?

x

y

2

110 10

1

xy

Potpourri 500

• Which of the following functions transforms the graph of left 1 unit and reflected over the x-axis? xy log

1

1log

xy

1

1log

xy 1log xy