POWER FUNCTIONS AND EXPONENTIAL FUNCTIONS



POWER FUNCTIONS AND EXPONENTIAL FUNCTIONS

1. Graph [pic]using the window [-5,5] x [-10,10].

a. As x approaches infinity, y approaches_______.

b. As x approaches negative infinity, y approaches_______.

c. As x approaches zero from the left, y approaches_______.

d. As x approaches zero from the right, y approaches_______.

e. The equations of the asymptotes are ____________________.

2. Graph [pic]using the window [-5,5] x [-2,10].

a. As x approaches infinity, y approaches_______.

b. As x approaches negative infinity, y approaches_______.

c. As x approaches zero from the left, y approaches_______.

d. As x approaches zero from the right, y approaches_______.

e. The equations of the asymptotes are ____________________.

3. Graph [pic], [pic], and [pic]together using the window [0,2] x [0,2].

a. Why should we restrict the domain to [pic]?

b. In addition graph [pic]and [pic]. What is the relationship of all the graphs for

[pic]? For [pic]?

c. Which of the graphs above are concave up? Concave down? Would [pic]be concave up

or down? Would [pic]be concave up or down?

4. Graph [pic]and [pic] using the window [-25,25] x [-5000,5000].

a. Which of these functions seems to dominates the other?

b. Change the window to [0,50] x [0,25000]. Does your answer to

the previous question change?

c. Change the window to [0,100] x [0,250000]. Does your answer

to the previous question change?

5. Graph [pic] and [pic] using the window [-5000,5000] x [[pic],[pic]].

a. Which of these functions seems to dominates the other?

b. Change the window to [-10000,10000] x [[pic],[pic]]. Does your

answer to the previous questions change?

c. True or false. As long as the coefficients are greater than zero, the

function with the higher power always dominates.

6. Graph [pic] and [pic] using the window [-5,5] x [-10,10].

a. Which of these functions seems to dominate the other?

b. Change the window to [-1,10] x [-1,1000]. Does your answer change?

c. Change the window to [-1,15] x [-1,1000]. What happens?

d. Change the window to [-1,20] x [-1,10000]. Which function dominates?

7. Graph [pic] and [pic] using the window [1,7] x [0,0.4].

a. Are these functions increasing or decreasing?

b. Which function seems to approach the x-axis faster?

c. Compare the behaviors of the functions near [pic].

POLYNOMIAL & RATIONAL FUNCTIONS

1. Create a possible equation for the polynomial graphed below. Include the sign of the leading coefficient.

2. Create equations of rational functions with the following characteristics:

A. A horizontal asymptote of [pic] and a vertical asymptote of [pic].

B. No horizontal and no vertical asymptotes.

3. Match the function expressed in words with a graph and an equation. Find the horizontal asymptote for each,

A. Average cost of producing x items.

B. The oxygen content in a lake after dumping in fertilizer as a function of time. (The oxygen content decreases at

first, but then returns to its previous level.)

C. The amount of a drug in a body as a function of time. (Assume the drug was given by injection.)

D. The number of people purchasing a (trendy) new product as a function of time.

E. The number of people getting a particular disease during an epidemic as a function of time.

(i) [pic] (ii) [pic] (iii) [pic] (iv) [pic] (v) [pic]

a. b.

c. d.

e.

EXPONENTIAL FUNCTIONS

1. Determine which table illustrates an exponential function and which one illustrates a linear function. Find formulas for these two functions, then find a formula for the third function.

|[pic]|[pic] |

|-2 |-25.22 |

|0 |3.50 |

|2 |32.22 |

|4 |60.94 |

|6 |89.66 |

|[pic] |[pic] |

|0.5 |-1 |

|1 |0 |

|2 |1 |

|4 |2 |

|8 |3 |

|[pic]|[pic] |

|-3 |1.3310 |

|-1 |1.9167 |

|1 |2.7600 |

|3 |3.9744 |

|5 |5.7231 |

2. Determine which situation is linear and which is exponential. Find a formula for each.

A. A computer purchased for $3200 loses roughly 20% of its value each year.

B. A kitchen appliance purchased for $120 loses roughly $18 in value every two years.

3. Find a formula for each graph.

LOGARITHMS

1. Find (if possible):

A. [pic] B. [pic] C. [pic] D. [pic] E. [pic]

2. Sketch a graph of each function. Include the domain.

A. [pic] B. [pic]

3. You and a friend plan to purchase cars in September. The initial value of your car will be $34,000 and will depreciate 17% each year. The initial value of your friend’s car will be $16,500 and will

depreciate 12% each year. You agree to exchange cars when their values are equal.

A. How long do you need to wait? (to the nearest month) What is the value of your car?

B. What would your depreciation rate have to be in order for the values of the cars to match at the

end of 7 years? (assume your friend’s car still depreciates 12% each year)

TRIG FUNCTIONS

1. The following function describes the air temperature in Fairbanks, Alaska as a function of time.

[pic]

A. Without graphing this function, determine its period, amplitude, and average value.

B. Graph one period of this function (beginning with [pic]).

2. Find the exact value A. [pic] B. [pic]

3. Find exact values for each: A. [pic] B. [pic]

4. Simplify each: A. [pic] B. [pic]

5. Solve for the angle so that [pic]. In each case there are two solutions.

A. [pic] B. [pic] C. [pic] is undefined

6. Solve for the variable so that [pic]. Express your answers in radians.

A. [pic] B. [pic] C. [pic]

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