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1. The following diagram shows part of the graph of an exponential function f(x) = a–x, where x [pic] [pic].

[pic]

(a) What is the range of f ? _________________________

(b) Write down the coordinates of the point P. _________________________

(c) What happens to the values of f(x) as elements in its domain increase in value?

2. The figure below shows the graphs of the functions y = x2 and y = 2x for values of x between –2 and 5. The points of intersection of the two curves are labelled as B, C and D.[pic]

3. The figure below shows the graphs of the functions f (x) = 2x + 0.5 and g (x) = 4 − x2 for values of x between –3 and 3.

[pic]

(a) Write down the coordinates of the points A and B. ______________________

(b) Write down the set of values of x for which f (x)< g (x). ______________________

4. The following graph shows the temperature in degrees Celsius of Robert’s cup of coffee, t minutes after pouring it out. The equation of the cooling graph is f (t) =16 + 74 × 2.8−0.2t where f (t) is the temperature and t is the time in minutes after pouring the coffee out.

[pic]

(a) Find the initial temperature of the coffee. _________________

(b) Write down the equation of the horizontal asymptote. _________________

(c) Find the room temperature. _________________

(d) Find the temperature of the coffee after 10 minutes. _________________

If the coffee is not hot enough it is reheated in a microwave oven. The liquid increases in temperature according to the formula: T = A × 21.5t

where T is the final temperature of the liquid, A is the initial temperature of coffee in the microwave and t is the time in minutes after switching the microwave on.

Number 4 continued…

(e) Find the temperature of Robert’s coffee after being heated in the microwave for 30 seconds after it has reached the temperature in part (d).

_____________________

(f) Calculate the length of time it would take a similar cup of coffee, initially at 20[pic], to be heated in the microwave to reach 100[pic].

______________________

5. A function is represented by the equation f (x) = 3(2)x + 1.

The table of values of f (x), for – 3 [pic] x [pic] 2, is given below.

|x |–3 |–2 |–1 |0 |1 |2 |

|f (x) |1.375 |1.75 |a |4 |7 |b |

(a) Calculate the values for a and b. _________________

(2)

(b) On graph paper, draw the graph of f (x) , for – 3 [pic] x [pic] 2, taking 1 cm to represent 1 unit on both axes.

(4)

The domain of the function f (x) is the real numbers, [pic].

(c) Write down the range of f (x). _________________

(2)

(d) Using your graph, or otherwise, find the approximate value for x when f (x) = 10.

_________________

(2)

6. In an experiment researchers found that a specific culture of bacteria increases in number according to the formula

N = 150 × 2t,

where N is the number of bacteria present and t is the number of hours since the experiment began.

Use this formula to calculate

(a) the number of bacteria present at the start of the experiment; ___________________

(b) the number of bacteria present after 3 hours; _____________________

(c) the number of hours it would take for the number of bacteria to reach 19 200._______

7. The area, A m2, of a fast growing plant is measured at noon (12:00) each day. On 7 July the area was 100 m2. Every day the plant grew by 7.5%. The formula for A is given by

A = 100 (1.075)t

where t is the number of days after 7 July. (on 7 July, t = 0)

The graph of A = 100(1.075)t is shown below.

[pic]

7 July

(a) What does the graph represent when t is negative? ____________________

(b) Use the graph to find the value of t when A = 178. ______________________

(c) Calculate the area covered by the plant at noon on 28 July. _____________________

8. The value of a car decreases each year. This value can be calculated using the function

v = 32 000rt, t[pic],[pic]

where v is the value of the car in USD, t is the number of years after it was first bought and r is a constant.

(a) (i) Write down the value of the car when it was first bought. _____________

(ii) One year later the value of the car was 27 200 USD. Find the value of r. _______________

(b) Find how many years it will take for the value of the car to be less than 8000 USD. ______________

9. In an experiment it is found that a culture of bacteria triples in number every four hours. There are 200 bacteria at the start of the experiment.

|Hours |0 |4 |8 |12 |16 |

|No. of bacteria |200 |600 |a |5400 |16200 |

(a) Find the value of a. _______________

(b) Calculate how many bacteria there will be after one day. _____________

(c) Find how long it will take for there to be two million bacteria. ____________

10. The number of ants, N (in thousands), in a colony is given by [pic] where t is the time (in months) after the beginning of the colony [pic].

a) Calculate the initial number of ants at the start of the colony _________________

b) Calculate the number of ants present after 2 months ___________________

c) Find the time taken for the colony to reach 20,000 ants ____________________

d) Determine the equation of the horizontal asymptote of N(t) _________________

e) According to the function N(t), what is the largest number of ants the colony will never reach? _____________

11. The population affected by a virus grows at a rate of 20% per day. Initially there are 10 people affected.

a) Find the number of people affected after 1 day. _______________

b) Find the number of people affected after 1 week. Give your answer correct to the nearest whole number. _______________

c) Let the number of people affected after t days be given by [pic].

State the value of:

i) N _________________________ ii) a __________________________

d) Using the graphical display calculator sketch the graph of f(t) for [pic] showing clearly the value of the y- intercept.

e) Write down the range of f(t) for the given domain. Give your answer correct to the nearest whole number. __________________

f) Write down the equation of the horizontal asymptote of f(t). ___________________

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OJQJU[pic]^JmHnHu[pic]h@`±^J(jh@`±h@`±U[pic]^JmHnHtHWrite down the coordinates of the point A.

(b) Write down the coordinates of the points B and C.

(c) Find the x-coordinate of the point D.

(d) Write down, using interval notation, all values of x for which 2x ≤ x2.

Applying Exponential Equations to solve problems

f(x) = kaλx + c where λ is the rate of growth or decay and x is the time

f(x) = eλx + c where λ is the rate of growth or decay and x is the time

Appreciation: V = a(1 + r)t , a is initial value, r is the rate as a decimal, t is time

Depreciation: V = a(1 – r)t , a is initial value, r is the rate as a decimal, t is time

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