Question 1: Calculate the value of y when a=3 and x=4 in the equation y=3ˣ.

Answer: 81

Question 2: Plot the graph of y=2ˣ for x from -2 to 2 on the grid.

Answer: Students plot on grid

Question 3: Evaluate y=5ˣ when x=0 and x=1.

Answer: 1 and 5

Question 4: Construct a graph of y=0.5ˣ and describe how it differs from y=2ˣ.

Answer: Graph of y=0.5ˣ decreases as x increases, opposite to y=2ˣ which increases.

Question 1: If y=4ˣ, for what value of x is y=16?

Answer: x=1/2

Question 2: Explain why the graph of y=3ˣ is always above y=2ˣ for positive x values.

Answer: Because 3ˣ grows faster than 2ˣ, so y=3ˣ values are larger for the same x > 0.

Question 3: Solve for x: 2ˣ = 8.

Answer: x=3

Question 4: Describe the behavior of y=0.1ˣ as x increases from -2 to 2.

Answer: It decreases rapidly toward zero as x increases.

Question 1: A bacteria culture doubles every hour. If the initial population is 100, write an expression for the population after x hours and plot it for x=0 to 5.

Answer: Population = 100×2ˣ

Question 2: The value of a car depreciates exponentially. If a car is worth £20,000 now and loses 10% of its value each year, write an expression for its worth after x years.

Answer: Value = 20000×(0.9)ˣ

Question 1: Given the graph of y=3ˣ, estimate the x-value when y=20. Show your working on the grid.

Answer: x≈2.73

Question 2: Derive the inverse function of y=2ˣ and explain its significance in graph plotting.

Answer: Inverse: x=log₂ y; it allows plotting y as a function of x in reverse.

Question 1: A student incorrectly states that y=5ˣ always increases as x increases. Identify and correct the mistake.

Answer: Incorrect because if 0< a<1, y=aˣ decreases; here, y=5ˣ increases, so the statement is true for a=5.

Question 2: Plot the graph of y=−2ˣ and describe its key features.

Answer: Graph is decreasing, always negative for x>0, positive for x<0, crosses y-axis at −1.