8) Exponential models

Question 1

Draw the graph for y = e^x ?

Answer 1

Question 2

Draw the graph for y = ln(x) ?

Answer 2

Question 3

What do the values of a in the equation y = a^x mean ?

Answer 3

• a > 1, then as x increases y increases; exponential growth
• 0 < a < 1, then as x increases y decreases, exponential decay

Question 4

What is the gradient of y = e^x ?

Answer 4

Gradient = e^x

Question 5

What is the gradient of y = e^kx ?

Answer 5

Gradient = ke^kx

Question 6

What is another form in which the equation y =a^x can be written ?

Answer 6

y = a^x can be written as y = e^kx

Question 7

Explain when transforming y = a^x into the form y = e^kx, the impact of the value of a ?

Answer 7

• a < 0, k is negative and therefore e^kx represents exponential decay
• a > 0, k is positive and therefore e^kx represents exponential growth

Question 8

State features of the function y = Ae^kt ?

Answer 8

• A is the initial value when t = 0 for an exponential model
• The rate of growth/ gradient is kAe^kt

Question 9

Explain why an exponential model may be not be a good accurate predictor of the population of bacteria in a dish after 12 hours ?

Answer 9

• Model assumes that the population will grow indefinitely, food and space however will become limiting factors
• Model is only an approximation, errors throughout the 12 hours will produce in an inaccurate prediction far off from the correct value

Question 10

What is the graph equation for y = kb^x ?

Answer 10

log y = log k + x log b

Question 11

What is the graph equation for y = ax^n ?

Answer 11

log y = log a + n log x