Question 1: Calculate the value of y when a=2 and x=3 in the equation y = 2ˣ.

Answer: 8

Question 2: Plot the graph of y = 3ˣ for x values from -2 to 2. Use the grid to sketch the curve.

Answer: see sketch

Question 3: If y = 5ˣ, what is the value of y when x = -1? Show your working.

Answer: 0.2

Question 4: Explain why the graph of y = (1/2)ˣ decreases as x increases.

Answer: Because (1/2)ˣ is a fraction less than 1 raised to higher powers, resulting in smaller y-values as x increases.

Question 5: Construct a graph of y = 4ˣ on the grid. Identify the y-intercept and a point when x=1.

Answer: y-intercept = 1; y=4 when x=1

Question 6: A common misconception is that y = aˣ always increases as x increases. Identify a false statement related to this misconception.

Answer: For a < 1, y = aˣ decreases as x increases.

Question 7: Calculate y when a=0.5 and x=4.

Answer: 0.0625

Question 8: Plot the graph of y = 1.5ˣ for x from -2 to 2. Describe the shape and key features.

Answer: Exponential growth with y > 1 at x=0, increasing rapidly.

Question 9: A student claims that y = 2ˣ and y = -2ˣ are the same graph. Is this true? Explain why or why not.

Answer: No, because y=2ˣ is positive for all x, while y=-2ˣ is the reflection across the x-axis.

Question 10: Identify and fix the mistake in the following statement: 'Since y = 3ˣ, then y = -3ˣ must be decreasing.'

Answer: The mistake is assuming -3ˣ decreases; in fact, y=-3ˣ decreases as x increases, but y=3ˣ increases.

Question 11: When plotting y = aˣ, explain what happens when a=1 and when a<1, in terms of the graph's shape.

Answer: a=1 results in a horizontal line y=1; a<1 results in a decreasing curve.

Question 12: Challenge: Derive an expression for the y-value when x=5 in the equation y=0.8ˣ and evaluate it.

Answer: 0.8^5 = 0.32768